The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. (See also Wikipedia.)

- 00-XX: General
- 00-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 00-02: Research exposition (monographs, survey articles)
- 00Axx: General and miscellaneous specific topics
- 00A05: General mathematics
- 00A06: Mathematics for nonmathematicians (engineering, social sciences, etc.)
- 00A07: Problem books
- 00A08: Recreational mathematics [See also 97A20]
- 00A09: Popularization of mathematics
- 00A15: Bibliographies
- 00A17: External book reviews
- 00A20: Dictionaries and other general reference works
- 00A22: Formularies
- 00A30: Philosophy of mathematics [See also 03A05]
- 00A35: Methodology of mathematics, didactics [See also 97Cxx, 97Dxx]
- 00A65: Mathematics and music
- 00A66: Mathematics and visual arts, visualization
- 00A67: Mathematics and architecture
- 00A69: General applied mathematics {For physics, see 00A79 and Sections 70 through 86}
- 00A71: Theory of mathematical modeling
- 00A72: General methods of simulation
- 00A73: Dimensional analysis
- 00A79: Physics (use more specific entries from Sections 70 through 86 when possible)
- 00A99: Miscellaneous topics

- 00Bxx: Conference proceedings and collections of papers
- 00B05: Collections of abstracts of lectures
- 00B10: Collections of articles of general interest
- 00B15: Collections of articles of miscellaneous specific content
- 00B20: Proceedings of conferences of general interest
- 00B25: Proceedings of conferences of miscellaneous specific interest
- 00B30: Festschriften
- 00B50: Volumes of selected translations
- 00B55: Miscellaneous volumes of translations
- 00B60: Collections of reprinted articles [See also 01A75]
- 00B99: None of the above, but in this section

- 01-XX: History and biography [See also the classification number –03 in the other sections]
- 01-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 01-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 01-02: Research exposition (monographs, survey articles)
- 01-06: Proceedings, conferences, collections, etc.
- 01-08: Computational methods
- 01Axx: History of mathematics and mathematicians
- 01A05: General histories, source books
- 01A07: Ethnomathematics, general
- 01A10: Paleolithic, Neolithic
- 01A12: Indigenous cultures of the Americas
- 01A13: Other indigenous cultures (non-European)
- 01A15: Indigenous European cultures (pre-Greek, etc.)
- 01A16: Egyptian
- 01A17: Babylonian
- 01A20: Greek, Roman
- 01A25: China
- 01A27: Japan
- 01A29: Southeast Asia
- 01A30: Islam (Medieval)
- 01A32: India
- 01A35: Medieval
- 01A40: 15th and 16th centuries, Renaissance
- 01A45: 17th century
- 01A50: 18th century
- 01A55: 19th century
- 01A60: 20th century
- 01A61: Twenty-first century
- 01A65: Contemporary
- 01A67: Future prospectives
- 01A70: Biographies, obituaries, personalia, bibliographies
- 01A72: Schools of mathematics
- 01A73: Universities
- 01A74: Other institutions and academies
- 01A75: Collected or selected works; reprintings or translations of classics [See also 00B60]
- 01A80: Sociology (and profession) of mathematics
- 01A85: Historiography
- 01A90: Bibliographic studies
- 01A99: Miscellaneous topics

- 03-XX: Mathematical logic and foundations
- 03-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 03-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 03-02: Research exposition (monographs, survey articles)
- 03-03: Historical (must also be assigned at least one classification number from Section 01)
- 03-04: Explicit machine computation and programs (not the theory of computation or programming)
- 03-06: Proceedings, conferences, collections, etc.
- 03Axx: Philosophical aspects of logic and foundations
- 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
- 03A10: Logic in the philosophy of science
- 03A99: None of the above, but in this section

- 03Bxx: General logic
- 03B05: Classical propositional logic
- 03B10: Classical first-order logic
- 03B15: Higher-order logic and type theory
- 03B20: Subsystems of classical logic (including intuitionistic logic)
- 03B22: Abstract deductive systems
- 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10]
- 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35]
- 03B35: Mechanization of proofs and logical operations [See also 68T15]
- 03B40: Combinatory logic and lambda-calculus [See also 68N18]
- 03B42: Logics of knowledge and belief (including belief change)
- 03B44: Temporal logic
- 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
- 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}
- 03B48: Probability and inductive logic [See also 60A05]
- 03B50: Many-valued logic
- 03B52: Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05]
- 03B53: Paraconsistent logics
- 03B55: Intermediate logics
- 03B60: Other nonclassical logic
- 03B62: Combined logics
- 03B65: Logic of natural languages [See also 68T50, 91F20]
- 03B70: Logic in computer science [See also 68-XX]
- 03B80: Other applications of logic
- 03B99: None of the above, but in this section

- 03Cxx: Model theory
- 03C05: Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05]
- 03C07: Basic properties of first-order languages and structures
- 03C10: Quantifier elimination, model completeness and related topics
- 03C13: Finite structures [See also 68Q15, 68Q19]
- 03C15: Denumerable structures
- 03C20: Ultraproducts and related constructions
- 03C25: Model-theoretic forcing
- 03C30: Other model constructions
- 03C35: Categoricity and completeness of theories
- 03C40: Interpolation, preservation, definability
- 03C45: Classification theory, stability and related concepts [See also 03C48]
- 03C48: Abstract elementary classes and related topics [See also 03C45]
- 03C50: Models with special properties (saturated, rigid, etc.)
- 03C52: Properties of classes of models
- 03C55: Set-theoretic model theory
- 03C57: Effective and recursion-theoretic model theory [See also 03D45]
- 03C60: Model-theoretic algebra [See also 08C10, 12Lxx, 13L05]
- 03C62: Models of arithmetic and set theory [See also 03Hxx]
- 03C64: Model theory of ordered structures; o-minimality
- 03C65: Models of other mathematical theories
- 03C68: Other classical first-order model theory
- 03C70: Logic on admissible sets
- 03C75: Other infinitary logic
- 03C80: Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48]
- 03C85: Second- and higher-order model theory
- 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)
- 03C95: Abstract model theory
- 03C98: Applications of model theory [See also 03C60]
- 03C99: None of the above, but in this section

- 03Dxx: Computability and recursion theory
- 03D03: Thue and Post systems, etc.
- 03D05: Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15]
- 03D10: Turing machines and related notions [See also 68Q05]
- 03D15: Complexity of computation (including implicit computational complexity) [See also 68Q15, 68Q17]
- 03D20: Recursive functions and relations, subrecursive hierarchies
- 03D25: Recursively (computably) enumerable sets and degrees
- 03D28: Other Turing degree structures
- 03D30: Other degrees and reducibilities
- 03D32: Algorithmic randomness and dimension [See also 68Q30]
- 03D35: Undecidability and degrees of sets of sentences
- 03D40: Word problems, etc. [See also 06B25, 08A50, 20F10, 68R15]
- 03D45: Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55]
- 03D50: Recursive equivalence types of sets and structures, isols
- 03D55: Hierarchies
- 03D60: Computability and recursion theory on ordinals, admissible sets, etc.
- 03D65: Higher-type and set recursion theory
- 03D70: Inductive definability
- 03D75: Abstract and axiomatic computability and recursion theory
- 03D78: Computation over the reals {For constructive aspects, see 03F60}
- 03D80: Applications of computability and recursion theory
- 03D99: None of the above, but in this section

- 03Exx: Set theory
- 03E02: Partition relations
- 03E04: Ordered sets and their cofinalities; pcf theory
- 03E05: Other combinatorial set theory
- 03E10: Ordinal and cardinal numbers
- 03E15: Descriptive set theory [See also 28A05, 54H05]
- 03E17: Cardinal characteristics of the continuum
- 03E20: Other classical set theory (including functions, relations, and set algebra)
- 03E25: Axiom of choice and related propositions
- 03E30: Axiomatics of classical set theory and its fragments
- 03E35: Consistency and independence results
- 03E40: Other aspects of forcing and Boolean-valued models
- 03E45: Inner models, including constructibility, ordinal definability, and core models
- 03E47: Other notions of set-theoretic definability
- 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]
- 03E55: Large cardinals
- 03E57: Generic absoluteness and forcing axioms [See also 03E50]
- 03E60: Determinacy principles
- 03E65: Other hypotheses and axioms
- 03E70: Nonclassical and second-order set theories
- 03E72: Fuzzy set theory
- 03E75: Applications of set theory
- 03E99: None of the above, but in this section

- 03Fxx: Proof theory and constructive mathematics
- 03F03: Proof theory, general
- 03F05: Cut-elimination and normal-form theorems
- 03F07: Structure of proofs
- 03F10: Functionals in proof theory
- 03F15: Recursive ordinals and ordinal notations
- 03F20: Complexity of proofs
- 03F25: Relative consistency and interpretations
- 03F30: First-order arithmetic and fragments
- 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
- 03F40: Gödel numberings and issues of incompleteness
- 03F45: Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25]
- 03F50: Metamathematics of constructive systems
- 03F52: Linear logic and other substructural logics [See also 03B47]
- 03F55: Intuitionistic mathematics
- 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]
- 03F65: Other constructive mathematics [See also 03D45]
- 03F99: None of the above, but in this section

- 03Gxx: Algebraic logic
- 03G05: Boolean algebras [See also 06Exx]
- 03G10: Lattices and related structures [See also 06Bxx]
- 03G12: Quantum logic [See also 06C15, 81P10]
- 03G15: Cylindric and polyadic algebras; relation algebras
- 03G20: Łukasiewicz and Post algebras [See also 06D25, 06D30]
- 03G25: Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35]
- 03G27: Abstract algebraic logic
- 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]
- 03G99: None of the above, but in this section

- 03Hxx: Nonstandard models [See also 03C62]

- 05-XX: Combinatorics {For finite fields, see 11Txx}
- 05-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 05-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 05-02: Research exposition (monographs, survey articles)
- 05-03: Historical (must also be assigned at least one classification number from Section 01)
- 05-04: Explicit machine computation and programs (not the theory of computation or programming)
- 05-06: Proceedings, conferences, collections, etc.
- 05Axx: Enumerative combinatorics {For enumeration in graph theory, see 05C30}
- 05A05: Permutations, words, matrices
- 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]
- 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]
- 05A16: Asymptotic enumeration
- 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]
- 05A18: Partitions of sets
- 05A19: Combinatorial identities, bijective combinatorics
- 05A20: Combinatorial inequalities
- 05A30: $q$-calculus and related topics [See also 33Dxx]
- 05A40: Umbral calculus
- 05A99: None of the above, but in this section

- 05Bxx: Designs and configurations {For applications of design theory, see 94C30}
- 05B05: Block designs [See also 51E05, 62K10]
- 05B07: Triple systems
- 05B10: Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13]
- 05B15: Orthogonal arrays, Latin squares, Room squares
- 05B20: Matrices (incidence, Hadamard, etc.)
- 05B25: Finite geometries [See also 51D20, 51Exx]
- 05B30: Other designs, configurations [See also 51E30]
- 05B35: Matroids, geometric lattices [See also 52B40, 90C27]
- 05B40: Packing and covering [See also 11H31, 52C15, 52C17]
- 05B45: Tessellation and tiling problems [See also 52C20, 52C22]
- 05B50: Polyominoes
- 05B99: None of the above, but in this section

- 05Cxx: Graph theory {For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15}
- 05C05: Trees
- 05C07: Vertex degrees [See also 05E30]
- 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
- 05C12: Distance in graphs
- 05C15: Coloring of graphs and hypergraphs
- 05C17: Perfect graphs
- 05C20: Directed graphs (digraphs), tournaments
- 05C21: Flows in graphs
- 05C22: Signed and weighted graphs
- 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]
- 05C30: Enumeration in graph theory
- 05C31: Graph polynomials
- 05C35: Extremal problems [See also 90C35]
- 05C38: Paths and cycles [See also 90B10]
- 05C40: Connectivity
- 05C42: Density (toughness, etc.)
- 05C45: Eulerian and Hamiltonian graphs
- 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
- 05C51: Graph designs and isomomorphic decomposition [See also 05B30]
- 05C55: Generalized Ramsey theory [See also 05D10]
- 05C57: Games on graphs [See also 91A43, 91A46]
- 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
- 05C62: Graph representations (geometric and intersection representations, etc.) {For graph drawing, see also 68R10}
- 05C63: Infinite graphs
- 05C65: Hypergraphs
- 05C69: Dominating sets, independent sets, cliques
- 05C70: Factorization, matching, partitioning, covering and packing
- 05C72: Fractional graph theory, fuzzy graph theory
- 05C75: Structural characterization of families of graphs
- 05C76: Graph operations (line graphs, products, etc.)
- 05C78: Graph labelling (graceful graphs, bandwidth, etc.)
- 05C80: Random graphs [See also 60B20]
- 05C81: Random walks on graphs
- 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30]
- 05C83: Graph minors
- 05C85: Graph algorithms [See also 68R10, 68W05]
- 05C90: Applications [See also 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15]
- 05C99: None of the above, but in this section

- 05Dxx: Extremal combinatorics
- 05D05: Extremal set theory
- 05D10: Ramsey theory [See also 05C55]
- 05D15: Transversal (matching) theory
- 05D40: Probabilistic methods
- 05D99: None of the above, but in this section

- 05Exx: Algebraic combinatorics
- 05E05: Symmetric functions and generalizations
- 05E10: Combinatorial aspects of representation theory[See also 20C30]
- 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]
- 05E18: Group actions on combinatorial structures
- 05E30: Association schemes, strongly regular graphs
- 05E40: Combinatorial aspects of commutative algebra
- 05E45: Combinatorial aspects of simplicial complexes
- 05E99: None of the above, but in this section

- 06-XX: Order, lattices, ordered algebraic structures [See also 18B35]
- 06-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 06-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 06-02: Research exposition (monographs, survey articles)
- 06-03: Historical (must also be assigned at least one classification number from Section 01)
- 06-04: Explicit machine computation and programs (not the theory of computation or programming)
- 06-06: Proceedings, conferences, collections, etc.
- 06Axx: Ordered sets
- 06A05: Total order
- 06A06: Partial order, general
- 06A07: Combinatorics of partially ordered sets
- 06A11: Algebraic aspects of posets
- 06A12: Semilattices [See also 20M10; for topological semilattices see 22A26]
- 06A15: Galois correspondences, closure operators
- 06A75: Generalizations of ordered sets
- 06A99: None of the above, but in this section

- 06Bxx: Lattices [See also 03G10]
- 06B05: Structure theory
- 06B10: Ideals, congruence relations
- 06B15: Representation theory
- 06B20: Varieties of lattices
- 06B23: Complete lattices, completions
- 06B25: Free lattices, projective lattices, word problems [See also 03D40, 08A50, 20F10]
- 06B30: Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12]
- 06B35: Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55]
- 06B75: Generalizations of lattices
- 06B99: None of the above, but in this section

- 06Cxx: Modular lattices, complemented lattices
- 06Dxx: Distributive lattices
- 06D05: Structure and representation theory
- 06D10: Complete distributivity
- 06D15: Pseudocomplemented lattices
- 06D20: Heyting algebras [See also 03G25]
- 06D22: Frames, locales {For topological questions see 54-XX}
- 06D25: Post algebras [See also 03G20]
- 06D30: De Morgan algebras, \L ukasiewicz algebras [See also 03G20]
- 06D35: MV-algebras
- 06D50: Lattices and duality
- 06D72: Fuzzy lattices (soft algebras) and related topics
- 06D75: Other generalizations of distributive lattices
- 06D99: None of the above, but in this section

- 06Exx: Boolean algebras (Boolean rings) [See also 03G05]
- 06E05: Structure theory
- 06E10: Chain conditions, complete algebras
- 06E15: Stone spaces (Boolean spaces) and related structures
- 06E20: Ring-theoretic properties [See also 16E50, 16G30]
- 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45]
- 06E30: Boolean functions [See also 94C10]
- 06E75: Generalizations of Boolean algebras
- 06E99: None of the above, but in this section

- 06Fxx: Ordered structures
- 06F05: Ordered semigroups and monoids [See also 20Mxx]
- 06F07: Quantales
- 06F10: Noether lattices
- 06F15: Ordered groups [See also 20F60]
- 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40]
- 06F25: Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80}
- 06F30: Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12]
- 06F35: BCK-algebras, BCI-algebras [See also 03G25]
- 06F99: None of the above, but in this section

- 08-XX: General algebraic systems
- 08-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 08-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 08-02: Research exposition (monographs, survey articles)
- 08-03: Historical (must also be assigned at least one classification number from Section 01)
- 08-04: Explicit machine computation and programs (not the theory of computation or programming)
- 08-06: Proceedings, conferences, collections, etc.
- 08Axx: Algebraic structures [See also 03C05]
- 08A02: Relational systems, laws of composition
- 08A05: Structure theory
- 08A30: Subalgebras, congruence relations
- 08A35: Automorphisms, endomorphisms
- 08A40: Operations, polynomials, primal algebras
- 08A45: Equational compactness
- 08A50: Word problems [See also 03D40, 06B25, 20F10, 68R15]
- 08A55: Partial algebras
- 08A60: Unary algebras
- 08A62: Finitary algebras
- 08A65: Infinitary algebras
- 08A68: Heterogeneous algebras
- 08A70: Applications of universal algebra in computer science
- 08A72: Fuzzy algebraic structures
- 08A99: None of the above, but in this section

- 08Bxx: Varieties [See also 03C05]
- 08B05: Equational logic, Mal′cev (Mal′tsev) conditions
- 08B10: Congruence modularity, congruence distributivity
- 08B15: Lattices of varieties
- 08B20: Free algebras
- 08B25: Products, amalgamated products, and other kinds of limits and colimits [See also 18A30]
- 08B26: Subdirect products and subdirect irreducibility
- 08B30: Injectives, projectives
- 08B99: None of the above, but in this section

- 08Cxx: Other classes of algebras

- 11-XX: Number theory
- 11-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 11-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 11-02: Research exposition (monographs, survey articles)
- 11-03: Historical (must also be assigned at least one classification number from Section 01)
- 11-04: Explicit machine computation and programs (not the theory of computation or programming)
- 11-06: Proceedings, conferences, collections, etc.
- 11Axx: Elementary number theory {For analogues in number fields, see 11R04}
- 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
- 11A07: Congruences; primitive roots; residue systems
- 11A15: Power residues, reciprocity
- 11A25: Arithmetic functions; related numbers; inversion formulas
- 11A41: Primes
- 11A51: Factorization; primality
- 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
- 11A63: Radix representation; digital problems {For metric results, see 11K16}
- 11A67: Other representations
- 11A99: None of the above, but in this section

- 11Bxx: Sequences and sets
- 11B05: Density, gaps, topology
- 11B13: Additive bases, including sumsets [See also 05B10]
- 11B25: Arithmetic progressions [See also 11N13]
- 11B30: Arithmetic combinatorics; higher degree uniformity
- 11B34: Representation functions
- 11B37: Recurrences {For applications to special functions, see 33-XX}
- 11B39: Fibonacci and Lucas numbers and polynomials and generalizations
- 11B50: Sequences (mod $m$)
- 11B57: Farey sequences; the sequences ${1^k, 2^k, \cdots}$
- 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]
- 11B68: Bernoulli and Euler numbers and polynomials
- 11B73: Bell and Stirling numbers
- 11B75: Other combinatorial number theory
- 11B83: Special sequences and polynomials
- 11B85: Automata sequences
- 11B99: None of the above, but in this section

- 11Cxx: Polynomials and matrices
- 11Dxx: Diophantine equations [See also 11Gxx, 14Gxx]
- 11D04: Linear equations
- 11D07: The Frobenius problem
- 11D09: Quadratic and bilinear equations
- 11D25: Cubic and quartic equations
- 11D41: Higher degree equations; Fermat's equation
- 11D45: Counting solutions of Diophantine equations
- 11D57: Multiplicative and norm form equations
- 11D59: Thue-Mahler equations
- 11D61: Exponential equations
- 11D68: Rational numbers as sums of fractions
- 11D72: Equations in many variables [See also 11P55]
- 11D75: Diophantine inequalities [See also 11J25]
- 11D79: Congruences in many variables
- 11D85: Representation problems [See also 11P55]
- 11D88: $p$-adic and power series fields
- 11D99: None of the above, but in this section

- 11Exx: Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63}
- 11E04: Quadratic forms over general fields
- 11E08: Quadratic forms over local rings and fields
- 11E10: Forms over real fields
- 11E12: Quadratic forms over global rings and fields
- 11E16: General binary quadratic forms
- 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
- 11E25: Sums of squares and representations by other particular quadratic forms
- 11E39: Bilinear and Hermitian forms
- 11E41: Class numbers of quadratic and Hermitian forms
- 11E45: Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
- 11E57: Classical groups [See also 14Lxx, 20Gxx]
- 11E70: $K$-theory of quadratic and Hermitian forms
- 11E72: Galois cohomology of linear algebraic groups [See also 20G10]
- 11E76: Forms of degree higher than two
- 11E81: Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24]
- 11E88: Quadratic spaces; Clifford algebras [See also 15A63, 15A66]
- 11E95: $p$-adic theory
- 11E99: None of the above, but in this section

- 11Fxx: Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45}
- 11F03: Modular and automorphic functions
- 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40]
- 11F11: Holomorphic modular forms of integral weight
- 11F12: Automorphic forms, one variable
- 11F20: Dedekind eta function, Dedekind sums
- 11F22: Relationship to Lie algebras and finite simple groups
- 11F23: Relations with algebraic geometry and topology
- 11F25: Hecke-Petersson operators, differential operators (one variable)
- 11F27: Theta series; Weil representation; theta correspondences
- 11F30: Fourier coefficients of automorphic forms
- 11F32: Modular correspondences, etc.
- 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]
- 11F37: Forms of half-integer weight; nonholomorphic modular forms
- 11F41: Automorphic forms on ${\rm GL}(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
- 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
- 11F50: Jacobi forms
- 11F52: Modular forms associated to Drinfel'd modules
- 11F55: Other groups and their modular and automorphic forms (several variables)
- 11F60: Hecke-Petersson operators, differential operators (several variables)
- 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
- 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
- 11F68: Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series
- 11F70: Representation-theoretic methods; automorphic representations over local and global fields
- 11F72: Spectral theory; Selberg trace formula
- 11F75: Cohomology of arithmetic groups
- 11F80: Galois representations
- 11F85: $p$-adic theory, local fields [See also 14G20, 22E50]
- 11F99: None of the above, but in this section

- 11Gxx: Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx]
- 11G05: Elliptic curves over global fields [See also 14H52]
- 11G07: Elliptic curves over local fields [See also 14G20, 14H52]
- 11G09: Drinfel'd modules; higher-dimensional motives, etc. [See also 14L05]
- 11G10: Abelian varieties of dimension $> 1$ [See also 14Kxx]
- 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22]
- 11G16: Elliptic and modular units [See also 11R27]
- 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
- 11G20: Curves over finite and local fields [See also 14H25]
- 11G25: Varieties over finite and local fields [See also 14G15, 14G20]
- 11G30: Curves of arbitrary genus or genus $\ne 1$ over global fields [See also 14H25]
- 11G32: Dessins d'enfants, Belyĭ theory
- 11G35: Varieties over global fields [See also 14G25]
- 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
- 11G42: Arithmetic mirror symmetry [See also 14J33]
- 11G45: Geometric class field theory [See also 11R37, 14C35, 19F05]
- 11G50: Heights [See also 14G40, 37P30]
- 11G55: Polylogarithms and relations with $K$-theory
- 11G99: None of the above, but in this section

- 11Hxx: Geometry of numbers {For applications in coding theory, see 94B75}
- 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07]
- 11H16: Nonconvex bodies
- 11H31: Lattice packing and covering [See also 05B40, 52C15, 52C17]
- 11H46: Products of linear forms
- 11H50: Minima of forms
- 11H55: Quadratic forms (reduction theory, extreme forms, etc.)
- 11H56: Automorphism groups of lattices
- 11H60: Mean value and transfer theorems
- 11H71: Relations with coding theory
- 11H99: None of the above, but in this section

- 11Jxx: Diophantine approximation, transcendental number theory [See also 11K60]
- 11J04: Homogeneous approximation to one number
- 11J06: Markov and Lagrange spectra and generalizations
- 11J13: Simultaneous homogeneous approximation, linear forms
- 11J17: Approximation by numbers from a fixed field
- 11J20: Inhomogeneous linear forms
- 11J25: Diophantine inequalities [See also 11D75]
- 11J54: Small fractional parts of polynomials and generalizations
- 11J61: Approximation in non-Archimedean valuations
- 11J68: Approximation to algebraic numbers
- 11J70: Continued fractions and generalizations [See also 11A55, 11K50]
- 11J71: Distribution modulo one [See also 11K06]
- 11J72: Irrationality; linear independence over a field
- 11J81: Transcendence (general theory)
- 11J82: Measures of irrationality and of transcendence
- 11J83: Metric theory
- 11J85: Algebraic independence; Gel'fond's method
- 11J86: Linear forms in logarithms; Baker's method
- 11J87: Schmidt Subspace Theorem and applications
- 11J89: Transcendence theory of elliptic and abelian functions
- 11J91: Transcendence theory of other special functions
- 11J93: Transcendence theory of Drinfel'd and $t$-modules
- 11J95: Results involving abelian varieties
- 11J97: Analogues of methods in Nevanlinna theory (work of Vojta et al.)
- 11J99: None of the above, but in this section

- 11Kxx: Probabilistic theory: distribution modulo $1$; metric theory of algorithms
- 11K06: General theory of distribution modulo $1$ [See also 11J71]
- 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63]
- 11K31: Special sequences
- 11K36: Well-distributed sequences and other variations
- 11K38: Irregularities of distribution, discrepancy [See also 11Nxx]
- 11K41: Continuous, $p$-adic and abstract analogues
- 11K45: Pseudo-random numbers; Monte Carlo methods
- 11K50: Metric theory of continued fractions [See also 11A55, 11J70]
- 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx]
- 11K60: Diophantine approximation [See also 11Jxx]
- 11K65: Arithmetic functions [See also 11Nxx]
- 11K70: Harmonic analysis and almost periodicity
- 11K99: None of the above, but in this section

- 11Lxx: Exponential sums and character sums {For finite fields, see 11Txx}
- 11L03: Trigonometric and exponential sums, general
- 11L05: Gauss and Kloosterman sums; generalizations
- 11L07: Estimates on exponential sums
- 11L10: Jacobsthal and Brewer sums; other complete character sums
- 11L15: Weyl sums
- 11L20: Sums over primes
- 11L26: Sums over arbitrary intervals
- 11L40: Estimates on character sums
- 11L99: None of the above, but in this section

- 11Mxx: Zeta and $L$-functions: analytic theory
- 11M06: $\zeta (s)$ and $L(s, \chi)$
- 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$
- 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
- 11M32: Multiple Dirichlet series and zeta functions and multizeta values
- 11M35: Hurwitz and Lerch zeta functions
- 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
- 11M38: Zeta and $L$-functions in characteristic $p$
- 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
- 11M45: Tauberian theorems [See also 40E05]
- 11M50: Relations with random matrices
- 11M55: Relations with noncommutative geometry
- 11M99: None of the above, but in this section

- 11Nxx: Multiplicative number theory
- 11N05: Distribution of primes
- 11N13: Primes in progressions [See also 11B25]
- 11N25: Distribution of integers with specified multiplicative constraints
- 11N30: Turán theory [See also 30Bxx]
- 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values
- 11N35: Sieves
- 11N36: Applications of sieve methods
- 11N37: Asymptotic results on arithmetic functions
- 11N45: Asymptotic results on counting functions for algebraic and topological structures
- 11N56: Rate of growth of arithmetic functions
- 11N60: Distribution functions associated with additive and positive multiplicative functions
- 11N64: Other results on the distribution of values or the characterization of arithmetic functions
- 11N69: Distribution of integers in special residue classes
- 11N75: Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx]
- 11N80: Generalized primes and integers
- 11N99: None of the above, but in this section

- 11Pxx: Additive number theory; partitions
- 11P05: Waring's problem and variants
- 11P21: Lattice points in specified regions
- 11P32: Goldbach-type theorems; other additive questions involving primes
- 11P55: Applications of the Hardy-Littlewood method [See also 11D85]
- 11P70: Inverse problems of additive number theory, including sumsets
- 11P81: Elementary theory of partitions [See also 05A17]
- 11P82: Analytic theory of partitions
- 11P83: Partitions; congruences and congruential restrictions
- 11P84: Partition identities; identities of Rogers-Ramanujan type
- 11P99: None of the above, but in this section

- 11Rxx: Algebraic number theory: global fields {For complex multiplication, see 11G15}
- 11R04: Algebraic numbers; rings of algebraic integers
- 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
- 11R09: Polynomials (irreducibility, etc.)
- 11R11: Quadratic extensions
- 11R16: Cubic and quartic extensions
- 11R18: Cyclotomic extensions
- 11R20: Other abelian and metabelian extensions
- 11R21: Other number fields
- 11R23: Iwasawa theory
- 11R27: Units and factorization
- 11R29: Class numbers, class groups, discriminants
- 11R32: Galois theory
- 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]
- 11R34: Galois cohomology [See also 12Gxx, 19A31]
- 11R37: Class field theory
- 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]
- 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
- 11R44: Distribution of prime ideals [See also 11N05]
- 11R45: Density theorems
- 11R47: Other analytic theory [See also 11Nxx]
- 11R52: Quaternion and other division algebras: arithmetic, zeta functions
- 11R54: Other algebras and orders, and their zeta and $L$-functions [See also 11S45, 16Hxx, 16Kxx]
- 11R56: Adèle rings and groups
- 11R58: Arithmetic theory of algebraic function fields [See also 14-XX]
- 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.)
- 11R65: Class groups and Picard groups of orders
- 11R70: $K$-theory of global fields [See also 19Fxx]
- 11R80: Totally real fields [See also 12J15]
- 11R99: None of the above, but in this section

- 11Sxx: Algebraic number theory: local and $p$-adic fields
- 11S05: Polynomials
- 11S15: Ramification and extension theory
- 11S20: Galois theory
- 11S23: Integral representations
- 11S25: Galois cohomology [See also 12Gxx, 16H05]
- 11S31: Class field theory; $p$-adic formal groups [See also 14L05]
- 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
- 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27]
- 11S45: Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16Hxx, 16Kxx]
- 11S70: $K$-theory of local fields [See also 19Fxx]
- 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)
- 11S82: Non-Archimedean dynamical systems [See mainly 37Pxx]
- 11S85: Other nonanalytic theory
- 11S90: Prehomogeneous vector spaces
- 11S99: None of the above, but in this section

- 11Txx: Finite fields and commutative rings (number-theoretic aspects)
- 11T06: Polynomials
- 11T22: Cyclotomy
- 11T23: Exponential sums
- 11T24: Other character sums and Gauss sums
- 11T30: Structure theory
- 11T55: Arithmetic theory of polynomial rings over finite fields
- 11T60: Finite upper half-planes
- 11T71: Algebraic coding theory; cryptography
- 11T99: None of the above, but in this section

- 11Uxx: Connections with logic
- 11Yxx: Computational number theory [See also 11-04]
- 11Y05: Factorization
- 11Y11: Primality
- 11Y16: Algorithms; complexity [See also 68Q25]
- 11Y35: Analytic computations
- 11Y40: Algebraic number theory computations
- 11Y50: Computer solution of Diophantine equations
- 11Y55: Calculation of integer sequences
- 11Y60: Evaluation of constants
- 11Y65: Continued fraction calculations
- 11Y70: Values of arithmetic functions; tables
- 11Y99: None of the above, but in this section

- 11Zxx: Miscellaneous applications of number theory
- 11Z05: Miscellaneous applications of number theory
- 11Z99: None of the above, but in this section

- 12-XX: Field theory and polynomials
- 12-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 12-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 12-02: Research exposition (monographs, survey articles)
- 12-03: Historical (must also be assigned at least one classification number from Section 01)
- 12-04: Explicit machine computation and programs (not the theory of computation or programming)
- 12-06: Proceedings, conferences, collections, etc.
- 12Dxx: Real and complex fields
- 12Exx: General field theory
- 12E05: Polynomials (irreducibility, etc.)
- 12E10: Special polynomials
- 12E12: Equations
- 12E15: Skew fields, division rings [See also 11R52, 11R54, 11S45, 16Kxx]
- 12E20: Finite fields (field-theoretic aspects)
- 12E25: Hilbertian fields; Hilbert's irreducibility theorem
- 12E30: Field arithmetic
- 12E99: None of the above, but in this section

- 12Fxx: Field extensions
- 12F05: Algebraic extensions
- 12F10: Separable extensions, Galois theory
- 12F12: Inverse Galois theory
- 12F15: Inseparable extensions
- 12F20: Transcendental extensions
- 12F99: None of the above, but in this section

- 12Gxx: Homological methods (field theory)
- 12Hxx: Differential and difference algebra
- 12Jxx: Topological fields
- 12J05: Normed fields
- 12J10: Valued fields
- 12J12: Formally $p$-adic fields
- 12J15: Ordered fields
- 12J17: Topological semifields
- 12J20: General valuation theory [See also 13A18]
- 12J25: Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10]
- 12J27: Krasner-Tate algebras [See mainly 32P05; see also 46S10, 47S10]
- 12J99: None of the above, but in this section

- 12Kxx: Generalizations of fields
- 12Lxx: Connections with logic
- 12Yxx: Computational aspects of field theory and polynomials
- 12Y05: Computational aspects of field theory and polynomials
- 12Y99: None of the above, but in this section

- 13-XX: Commutative algebra
- 13-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 13-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 13-02: Research exposition (monographs, survey articles)
- 13-03: Historical (must also be assigned at least one classification number from Section 01)
- 13-04: Explicit machine computation and programs (not the theory of computation or programming)
- 13-06: Proceedings, conferences, collections, etc.
- 13Axx: General commutative ring theory
- 13A02: Graded rings [See also 16W50]
- 13A05: Divisibility; factorizations [See also 13F15]
- 13A15: Ideals; multiplicative ideal theory
- 13A18: Valuations and their generalizations [See also 12J20]
- 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
- 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22]
- 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]
- 13A99: None of the above, but in this section

- 13Bxx: Ring extensions and related topics
- 13B02: Extension theory
- 13B05: Galois theory
- 13B10: Morphisms
- 13B21: Integral dependence; going up, going down
- 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)
- 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]
- 13B30: Rings of fractions and localization [See also 16S85]
- 13B35: Completion [See also 13J10]
- 13B40: Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25]
- 13B99: None of the above, but in this section

- 13Cxx: Theory of modules and ideals
- 13C05: Structure, classification theorems
- 13C10: Projective and free modules and ideals [See also 19A13]
- 13C11: Injective and flat modules and ideals
- 13C12: Torsion modules and ideals
- 13C13: Other special types
- 13C14: Cohen-Macaulay modules [See also 13H10]
- 13C15: Dimension theory, depth, related rings (catenary, etc.)
- 13C20: Class groups [See also 11R29]
- 13C40: Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12]
- 13C60: Module categories
- 13C99: None of the above, but in this section

- 13Dxx: Homological methods {For noncommutative rings, see 16Exx; for general categories, see 18Gxx}
- 13D02: Syzygies, resolutions, complexes
- 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
- 13D05: Homological dimension
- 13D07: Homological functors on modules (Tor, Ext, etc.)
- 13D09: Derived categories
- 13D10: Deformations and infinitesimal methods [See also 14B10, 14B12, 14D15, 32Gxx]
- 13D15: Grothendieck groups, $K$-theory [See also 14C35, 18F30, 19Axx, 19D50]
- 13D22: Homological conjectures (intersection theorems)
- 13D30: Torsion theory [See also 13C12, 18E40]
- 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
- 13D45: Local cohomology [See also 14B15]
- 13D99: None of the above, but in this section

- 13Exx: Chain conditions, finiteness conditions
- 13E05: Noetherian rings and modules
- 13E10: Artinian rings and modules, finite-dimensional algebras
- 13E15: Rings and modules of finite generation or presentation; number of generators
- 13E99: None of the above, but in this section

- 13Fxx: Arithmetic rings and other special rings
- 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations
- 13F07: Euclidean rings and generalizations
- 13F10: Principal ideal rings
- 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05]
- 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]
- 13F25: Formal power series rings [See also 13J05]
- 13F30: Valuation rings [See also 13A18]
- 13F35: Witt vectors and related rings
- 13F40: Excellent rings
- 13F45: Seminormal rings
- 13F50: Rings with straightening laws, Hodge algebras
- 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10]
- 13F60: Cluster algebras
- 13F99: None of the above, but in this section

- 13Gxx: Integral domains
- 13G05: Integral domains
- 13G99: None of the above, but in this section

- 13Hxx: Local rings and semilocal rings
- 13Jxx: Topological rings and modules [See also 16W60, 16W80]
- 13J05: Power series rings [See also 13F25]
- 13J07: Analytical algebras and rings [See also 32B05]
- 13J10: Complete rings, completion [See also 13B35]
- 13J15: Henselian rings [See also 13B40]
- 13J20: Global topological rings
- 13J25: Ordered rings [See also 06F25]
- 13J30: Real algebra [See also 12D15, 14Pxx]
- 13J99: None of the above, but in this section

- 13Lxx: Applications of logic to commutative algebra [See also 03Cxx, 03Hxx]
- 13Mxx: Finite commutative rings {For number-theoretic aspects, see 11Txx}
- 13M05: Structure
- 13M10: Polynomials
- 13M99: None of the above, but in this section

- 13Nxx: Differential algebra [See also 12H05, 14F10]
- 13Pxx: Computational aspects and applications [See also 14Qxx, 68W30]
- 13P05: Polynomials, factorization [See also 12Y05]
- 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
- 13P15: Solving polynomial systems; resultants
- 13P20: Computational homological algebra [See also 13Dxx]
- 13P25: Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
- 13P99: None of the above, but in this section

- 14-XX: Algebraic geometry
- 14-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 14-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 14-02: Research exposition (monographs, survey articles)
- 14-03: Historical (must also be assigned at least one classification number from Section 01)
- 14-04: Explicit machine computation and programs (not the theory of computation or programming)
- 14-06: Proceedings, conferences, collections, etc.
- 14Axx: Foundations
- 14Bxx: Local theory
- 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
- 14B07: Deformations of singularities [See also 14D15, 32S30]
- 14B10: Infinitesimal methods [See also 13D10]
- 14B12: Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10]
- 14B15: Local cohomology [See also 13D45, 32C36]
- 14B20: Formal neighborhoods
- 14B25: Local structure of morphisms: étale, flat, etc. [See also 13B40]
- 14B99: None of the above, but in this section

- 14Cxx: Cycles and subschemes
- 14C05: Parametrization (Chow and Hilbert schemes)
- 14C15: (Equivariant) Chow groups and rings; motives
- 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
- 14C20: Divisors, linear systems, invertible sheaves
- 14C21: Pencils, nets, webs [See also 53A60]
- 14C22: Picard groups
- 14C25: Algebraic cycles
- 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture
- 14C34: Torelli problem [See also 32G20]
- 14C35: Applications of methods of algebraic $K$-theory [See also 19Exx]
- 14C40: Riemann-Roch theorems [See also 19E20, 19L10]
- 14C99: None of the above, but in this section

- 14Dxx: Families, fibrations
- 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
- 14D06: Fibrations, degenerations
- 14D07: Variation of Hodge structures [See also 32G20]
- 14D10: Arithmetic ground fields (finite, local, global)
- 14D15: Formal methods; deformations [See also 13D10, 14B07, 32Gxx]
- 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
- 14D21: Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)[See also 32L25, 81Txx]
- 14D22: Fine and coarse moduli spaces
- 14D23: Stacks and moduli problems
- 14D24: Geometric Langlands program: algebro-geometric aspects [See also 22E57]
- 14D99: None of the above, but in this section

- 14Exx: Birational geometry
- 14E05: Rational and birational maps
- 14E07: Birational automorphisms, Cremona group and generalizations
- 14E08: Rationality questions [See also 14M20]
- 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
- 14E16: McKay correspondence
- 14E18: Arcs and motivic integration
- 14E20: Coverings [See also 14H30]
- 14E22: Ramification problems [See also 11S15]
- 14E25: Embeddings
- 14E30: Minimal model program (Mori theory, extremal rays)
- 14E99: None of the above, but in this section

- 14Fxx: (Co)homology theory [See also 13Dxx]
- 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
- 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38]
- 14F17: Vanishing theorems [See also 32L20]
- 14F18: Multiplier ideals
- 14F20: Étale and other Grothendieck topologies and (co)homologies
- 14F22: Brauer groups of schemes [See also 12G05, 16K50]
- 14F25: Classical real and complex (co)homology
- 14F30: $p$-adic cohomology, crystalline cohomology
- 14F35: Homotopy theory; fundamental groups [See also 14H30]
- 14F40: de Rham cohomology [See also 14C30, 32C35, 32L10]
- 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
- 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
- 14F45: Topological properties
- 14F99: None of the above, but in this section

- 14Gxx: Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx]
- 14G05: Rational points
- 14G10: Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture)
- 14G15: Finite ground fields
- 14G17: Positive characteristic ground fields
- 14G20: Local ground fields
- 14G22: Rigid analytic geometry
- 14G25: Global ground fields
- 14G27: Other nonalgebraically closed ground fields
- 14G32: Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
- 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
- 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
- 14G50: Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40]
- 14G99: None of the above, but in this section

- 14Hxx: Curves
- 14H05: Algebraic functions; function fields [See also 11R58]
- 14H10: Families, moduli (algebraic)
- 14H15: Families, moduli (analytic) [See also 30F10, 32G15]
- 14H20: Singularities, local rings [See also 13Hxx, 14B05]
- 14H25: Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
- 14H30: Coverings, fundamental group [See also 14E20, 14F35]
- 14H37: Automorphisms
- 14H40: Jacobians, Prym varieties [See also 32G20]
- 14H42: Theta functions; Schottky problem [See also 14K25, 32G20]
- 14H45: Special curves and curves of low genus
- 14H50: Plane and space curves
- 14H51: Special divisors (gonality, Brill-Noether theory)
- 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]
- 14H55: Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]
- 14H57: Dessins d'enfants theory {For arithmetic aspects, see 11G32}
- 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05]
- 14H70: Relationships with integrable systems
- 14H81: Relationships with physics
- 14H99: None of the above, but in this section

- 14Jxx: Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx}
- 14J10: Families, moduli, classification: algebraic theory
- 14J15: Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
- 14J17: Singularities [See also 14B05, 14E15]
- 14J20: Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
- 14J25: Special surfaces {For Hilbert modular surfaces, see 14G35}
- 14J26: Rational and ruled surfaces
- 14J27: Elliptic surfaces
- 14J28: $K3$ surfaces and Enriques surfaces
- 14J29: Surfaces of general type
- 14J30: $3$-folds [See also 32Q25]
- 14J32: Calabi-Yau manifolds
- 14J33: Mirror symmetry [See also 11G42, 53D37]
- 14J35: $4$-folds
- 14J40: $n$-folds ($n>4$)
- 14J45: Fano varieties
- 14J50: Automorphisms of surfaces and higher-dimensional varieties
- 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
- 14J70: Hypersurfaces
- 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
- 14J81: Relationships with physics
- 14J99: None of the above, but in this section

- 14Kxx: Abelian varieties and schemes
- 14K02: Isogeny
- 14K05: Algebraic theory
- 14K10: Algebraic moduli, classification [See also 11G15]
- 14K12: Subvarieties
- 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
- 14K20: Analytic theory; abelian integrals and differentials
- 14K22: Complex multiplication [See also 11G15]
- 14K25: Theta functions [See also 14H42]
- 14K30: Picard schemes, higher Jacobians [See also 14H40, 32G20]
- 14K99: None of the above, but in this section

- 14Lxx: Algebraic groups {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45}
- 14L05: Formal groups, $p$-divisible groups [See also 55N22]
- 14L10: Group varieties
- 14L15: Group schemes
- 14L17: Affine algebraic groups, hyperalgebra constructions [See also 17B45, 18D35]
- 14L24: Geometric invariant theory [See also 13A50]
- 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
- 14L35: Classical groups (geometric aspects) [See also 20Gxx, 51N30]
- 14L40: Other algebraic groups (geometric aspects)
- 14L99: None of the above, but in this section

- 14Mxx: Special varieties
- 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10]
- 14M06: Linkage [See also 13C40]
- 14M07: Low codimension problems
- 14M10: Complete intersections [See also 13C40]
- 14M12: Determinantal varieties [See also 13C40]
- 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]
- 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]
- 14M20: Rational and unirational varieties [See also 14E08]
- 14M22: Rationally connected varieties
- 14M25: Toric varieties, Newton polyhedra [See also 52B20]
- 14M27: Compactifications; symmetric and spherical varieties
- 14M30: Supervarieties [See also 32C11, 58A50]
- 14M99: None of the above, but in this section

- 14Nxx: Projective and enumerative geometry [See also 51-XX]
- 14N05: Projective techniques [See also 51N35]
- 14N10: Enumerative problems (combinatorial problems)
- 14N15: Classical problems, Schubert calculus
- 14N20: Configurations and arrangements of linear subspaces
- 14N25: Varieties of low degree
- 14N30: Adjunction problems
- 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
- 14N99: None of the above, but in this section

- 14Pxx: Real algebraic and real analytic geometry
- 14P05: Real algebraic sets [See also 12D15, 13J30]
- 14P10: Semialgebraic sets and related spaces
- 14P15: Real analytic and semianalytic sets [See also 32B20, 32C05]
- 14P20: Nash functions and manifolds [See also 32C07, 58A07]
- 14P25: Topology of real algebraic varieties
- 14P99: None of the above, but in this section

- 14Qxx: Computational aspects in algebraic geometry [See also 12Y05, 13Pxx, 68W30]
- 14Q05: Curves
- 14Q10: Surfaces, hypersurfaces
- 14Q15: Higher-dimensional varieties
- 14Q20: Effectivity, complexity
- 14Q99: None of the above, but in this section

- 14Rxx: Affine geometry
- 14R05: Classification of affine varieties
- 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
- 14R15: Jacobian problem [See also 13F20]
- 14R20: Group actions on affine varieties [See also 13A50, 14L30]
- 14R25: Affine fibrations [See also 14D06]
- 14R99: None of the above, but in this section

- 14Txx: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]

- 15-XX: Linear and multilinear algebra; matrix theory
- 15-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 15-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 15-02: Research exposition (monographs, survey articles)
- 15-03: Historical (must also be assigned at least one classification number from Section 01)
- 15-04: Explicit machine computation and programs (not the theory of computation or programming)
- 15-06: Proceedings, conferences, collections, etc.
- 15Axx: Basic linear algebra
- 15A03: Vector spaces, linear dependence, rank
- 15A04: Linear transformations, semilinear transformations
- 15A06: Linear equations
- 15A09: Matrix inversion, generalized inverses
- 15A12: Conditioning of matrices [See also 65F35]
- 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]
- 15A16: Matrix exponential and similar functions of matrices
- 15A18: Eigenvalues, singular values, and eigenvectors
- 15A21: Canonical forms, reductions, classification
- 15A22: Matrix pencils [See also 47A56]
- 15A23: Factorization of matrices
- 15A24: Matrix equations and identities
- 15A27: Commutativity
- 15A29: Inverse problems
- 15A30: Algebraic systems of matrices [See also 16S50, 20Gxx, 20Hxx]
- 15A39: Linear inequalities
- 15A42: Inequalities involving eigenvalues and eigenvectors
- 15A45: Miscellaneous inequalities involving matrices
- 15A54: Matrices over function rings in one or more variables
- 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
- 15A63: Quadratic and bilinear forms, inner products [See mainly 11Exx]
- 15A66: Clifford algebras, spinors
- 15A69: Multilinear algebra, tensor products
- 15A72: Vector and tensor algebra, theory of invariants [See also 13A50, 14L24]
- 15A75: Exterior algebra, Grassmann algebras
- 15A78: Other algebras built from modules
- 15A80: Max-plus and related algebras
- 15A83: Matrix completion problems
- 15A86: Linear preserver problems
- 15A99: Miscellaneous topics

- 15Bxx: Special matrices
- 15B05: Toeplitz, Cauchy, and related matrices
- 15B10: Orthogonal matrices
- 15B15: Fuzzy matrices
- 15B33: Matrices over special rings (quaternions, finite fields, etc.)
- 15B34: Boolean and Hadamard matrices
- 15B35: Sign pattern matrices
- 15B36: Matrices of integers [See also 11C20]
- 15B48: Positive matrices and their generalizations; cones of matrices
- 15B51: Stochastic matrices
- 15B52: Random matrices
- 15B57: Hermitian, skew-Hermitian, and related matrices
- 15B99: None of the above, but in this section

- 16-XX: Associative rings and algebras {For the commutative case, see 13-XX}
- 16-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 16-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 16-02: Research exposition (monographs, survey articles)
- 16-03: Historical (must also be assigned at least one classification number from Section 01)
- 16-04: Explicit machine computation and programs (not the theory of computation or programming)
- 16-06: Proceedings, conferences, collections, etc.
- 16Bxx: General and miscellaneous
- 16Dxx: Modules, bimodules and ideals
- 16D10: General module theory
- 16D20: Bimodules
- 16D25: Ideals
- 16D30: Infinite-dimensional simple rings (except as in 16Kxx)
- 16D40: Free, projective, and flat modules and ideals [See also 19A13]
- 16D50: Injective modules, self-injective rings [See also 16L60]
- 16D60: Simple and semisimple modules, primitive rings and ideals
- 16D70: Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation
- 16D80: Other classes of modules and ideals [See also 16G50]
- 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality
- 16D99: None of the above, but in this section

- 16Exx: Homological methods {For commutative rings, see 13Dxx; for general categories, see 18Gxx}
- 16E05: Syzygies, resolutions, complexes
- 16E10: Homological dimension
- 16E20: Grothendieck groups, $K$-theory, etc. [See also 18F30, 19Axx, 19D50]
- 16E30: Homological functors on modules (Tor, Ext, etc.)
- 16E35: Derived categories
- 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
- 16E45: Differential graded algebras and applications
- 16E50: von Neumann regular rings and generalizations
- 16E60: Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
- 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
- 16E99: None of the above, but in this section

- 16Gxx: Representation theory of rings and algebras
- 16G10: Representations of Artinian rings
- 16G20: Representations of quivers and partially ordered sets
- 16G30: Representations of orders, lattices, algebras over commutative rings [See also 16Hxx]
- 16G50: Cohen-Macaulay modules
- 16G60: Representation type (finite, tame, wild, etc.)
- 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
- 16G99: None of the above, but in this section

- 16Hxx: Algebras and orders {For arithmetic aspects, see 11R52, 11R54, 11S45; for representation theory, see 16G30}
- 16H05: Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
- 16H10: Orders in separable algebras
- 16H15: Commutative orders
- 16H20: Lattices over orders
- 16H99: None of the above, but in this section

- 16Kxx: Division rings and semisimple Artin rings [See also 12E15, 15A30]
- 16Lxx: Local rings and generalizations
- 16L30: Noncommutative local and semilocal rings, perfect rings
- 16L60: Quasi-Frobenius rings [See also 16D50]
- 16L99: None of the above, but in this section

- 16Nxx: Radicals and radical properties of rings
- 16Pxx: Chain conditions, growth conditions, and other forms of finiteness
- 16P10: Finite rings and finite-dimensional algebras {For semisimple, see 16K20; for commutative, see 11Txx, 13Mxx}
- 16P20: Artinian rings and modules
- 16P40: Noetherian rings and modules
- 16P50: Localization and Noetherian rings [See also 16U20]
- 16P60: Chain conditions on annihilators and summands: Goldie-type conditions [See also 16U20], Krull dimension
- 16P70: Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence
- 16P90: Growth rate, Gelfand-Kirillov dimension
- 16P99: None of the above, but in this section

- 16Rxx: Rings with polynomial identity
- 16R10: $T$-ideals, identities, varieties of rings and algebras
- 16R20: Semiprime p.i. rings, rings embeddable in matrices over commutative rings
- 16R30: Trace rings and invariant theory
- 16R40: Identities other than those of matrices over commutative rings
- 16R50: Other kinds of identities (generalized polynomial, rational, involution)
- 16R60: Functional identities
- 16R99: None of the above, but in this section

- 16Sxx: Rings and algebras arising under various constructions
- 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
- 16S15: Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
- 16S20: Centralizing and normalizing extensions
- 16S30: Universal enveloping algebras of Lie algebras [See mainly 17B35]
- 16S32: Rings of differential operators [See also 13N10, 32C38]
- 16S34: Group rings [See also 20C05, 20C07], Laurent polynomial rings
- 16S35: Twisted and skew group rings, crossed products
- 16S36: Ordinary and skew polynomial rings and semigroup rings [See also 20M25]
- 16S37: Quadratic and Koszul algebras
- 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22]
- 16S40: Smash products of general Hopf actions [See also 16T05]
- 16S50: Endomorphism rings; matrix rings [See also 15-XX]
- 16S60: Rings of functions, subdirect products, sheaves of rings
- 16S70: Extensions of rings by ideals
- 16S80: Deformations of rings [See also 13D10, 14D15]
- 16S85: Rings of fractions and localizations [See also 13B30]
- 16S90: Torsion theories; radicals on module categories [See also 13D30, 18E40] {For radicals of rings, see 16Nxx}
- 16S99: None of the above, but in this section

- 16Txx: Hopf algebras, quantum groups and related topics
- 16T05: Hopf algebras and their applications [See also 16S40, 57T05]
- 16T10: Bialgebras
- 16T15: Coalgebras and comodules; corings
- 16T20: Ring-theoretic aspects of quantum groups [See also 17B37, 20G42, 81R50]
- 16T25: Yang-Baxter equations
- 16T30: Connections with combinatorics
- 16T99: None of the above, but in this section

- 16Uxx: Conditions on elements
- 16U10: Integral domains
- 16U20: Ore rings, multiplicative sets, Ore localization
- 16U30: Divisibility, noncommutative UFDs
- 16U60: Units, groups of units
- 16U70: Center, normalizer (invariant elements)
- 16U80: Generalizations of commutativity
- 16U99: None of the above, but in this section

- 16Wxx: Rings and algebras with additional structure
- 16W10: Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx]
- 16W20: Automorphisms and endomorphisms
- 16W22: Actions of groups and semigroups; invariant theory
- 16W25: Derivations, actions of Lie algebras
- 16W50: Graded rings and modules
- 16W55: “Super” (or “skew”) structure [See also 17A70, 17Bxx, 17C70] {For exterior algebras, see 15A75; for Clifford algebras, see 11E88, 15A66}
- 16W60: Valuations, completions, formal power series and related constructions [See also 13Jxx]
- 16W70: Filtered rings; filtrational and graded techniques
- 16W80: Topological and ordered rings and modules [See also 06F25, 13Jxx]
- 16W99: None of the above, but in this section

- 16Yxx: Generalizations {For nonassociative rings, see 17-XX}
- 16Zxx: Computational aspects of associative rings
- 16Z05: Computational aspects of associative rings [See also 68W30]
- 16Z99: None of the above, but in this section

- 17-XX: Nonassociative rings and algebras
- 17-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 17-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 17-02: Research exposition (monographs, survey articles)
- 17-03: Historical (must also be assigned at least one classification number from Section 01)
- 17-04: Explicit machine computation and programs (not the theory of computation or programming)
- 17-06: Proceedings, conferences, collections, etc.
- 17-08: Computational methods
- 17Axx: General nonassociative rings
- 17A01: General theory
- 17A05: Power-associative rings
- 17A15: Noncommutative Jordan algebras
- 17A20: Flexible algebras
- 17A30: Algebras satisfying other identities
- 17A32: Leibniz algebras
- 17A35: Division algebras
- 17A36: Automorphisms, derivations, other operators
- 17A40: Ternary compositions
- 17A42: Other $n$-ary compositions $(n \ge 3)$
- 17A45: Quadratic algebras (but not quadratic Jordan algebras)
- 17A50: Free algebras
- 17A60: Structure theory
- 17A65: Radical theory
- 17A70: Superalgebras
- 17A75: Composition algebras
- 17A80: Valued algebras
- 17A99: None of the above, but in this section

- 17Bxx: Lie algebras and Lie superalgebras {For Lie groups, see 22Exx}
- 17B01: Identities, free Lie (super)algebras
- 17B05: Structure theory
- 17B08: Coadjoint orbits; nilpotent varieties
- 17B10: Representations, algebraic theory (weights)
- 17B15: Representations, analytic theory
- 17B20: Simple, semisimple, reductive (super)algebras
- 17B22: Root systems
- 17B25: Exceptional (super)algebras
- 17B30: Solvable, nilpotent (super)algebras
- 17B35: Universal enveloping (super)algebras [See also 16S30]
- 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
- 17B40: Automorphisms, derivations, other operators
- 17B45: Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
- 17B50: Modular Lie (super)algebras
- 17B55: Homological methods in Lie (super)algebras
- 17B56: Cohomology of Lie (super)algebras
- 17B60: Lie (super)algebras associated with other structures (associative, Jordan, etc.) [See also 16W10, 17C40, 17C50]
- 17B62: Lie bialgebras; Lie coalgebras
- 17B63: Poisson algebras
- 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65]
- 17B66: Lie algebras of vector fields and related (super) algebras
- 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
- 17B68: Virasoro and related algebras
- 17B69: Vertex operators; vertex operator algebras and related structures
- 17B70: Graded Lie (super)algebras
- 17B75: Color Lie (super)algebras
- 17B80: Applications to integrable systems
- 17B81: Applications to physics
- 17B99: None of the above, but in this section

- 17Cxx: Jordan algebras (algebras, triples and pairs)
- 17C05: Identities and free Jordan structures
- 17C10: Structure theory
- 17C17: Radicals
- 17C20: Simple, semisimple algebras
- 17C27: Idempotents, Peirce decompositions
- 17C30: Associated groups, automorphisms
- 17C36: Associated manifolds
- 17C37: Associated geometries
- 17C40: Exceptional Jordan structures
- 17C50: Jordan structures associated with other structures [See also 16W10]
- 17C55: Finite-dimensional structures
- 17C60: Division algebras
- 17C65: Jordan structures on Banach spaces and algebras [See also 46H70, 46L70]
- 17C70: Super structures
- 17C90: Applications to physics
- 17C99: None of the above, but in this section

- 17Dxx: Other nonassociative rings and algebras
- 17D05: Alternative rings
- 17D10: Mal'cev (Mal'tsev) rings and algebras
- 17D15: Right alternative rings
- 17D20: $(\gamma, \delta)$-rings, including $(1,-1)$-rings
- 17D25: Lie-admissible algebras
- 17D92: Genetic algebras
- 17D99: None of the above, but in this section

- 18-XX: Category theory; homological algebra {For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topology}
- 18-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 18-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 18-02: Research exposition (monographs, survey articles)
- 18-03: Historical (must also be assigned at least one classification number from Section 01)
- 18-04: Explicit machine computation and programs (not the theory of computation or programming)
- 18-06: Proceedings, conferences, collections, etc.
- 18Axx: General theory of categories and functors
- 18A05: Definitions, generalizations
- 18A10: Graphs, diagram schemes, precategories [See especially 20L05]
- 18A15: Foundations, relations to logic and deductive systems [See also 03-XX]
- 18A20: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
- 18A22: Special properties of functors (faithful, full, etc.)
- 18A23: Natural morphisms, dinatural morphisms
- 18A25: Functor categories, comma categories
- 18A30: Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
- 18A32: Factorization of morphisms, substructures, quotient structures, congruences, amalgams
- 18A35: Categories admitting limits (complete categories), functors preserving limits, completions
- 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
- 18A99: None of the above, but in this section

- 18Bxx: Special categories
- 18B05: Category of sets, characterizations [See also 03-XX]
- 18B10: Category of relations, additive relations
- 18B15: Embedding theorems, universal categories [See also 18E20]
- 18B20: Categories of machines, automata, operative categories [See also 03D05, 68Qxx]
- 18B25: Topoi [See also 03G30]
- 18B30: Categories of topological spaces and continuous mappings [See also 54-XX]
- 18B35: Preorders, orders and lattices (viewed as categories) [See also 06-XX]
- 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx]
- 18B99: None of the above, but in this section

- 18Cxx: Categories and theories
- 18C05: Equational categories [See also 03C05, 08C05]
- 18C10: Theories (e.g. algebraic theories), structure, and semantics [See also 03G30]
- 18C15: Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples [See also 18Gxx]
- 18C20: Algebras and Kleisli categories associated with monads
- 18C30: Sketches and generalizations
- 18C35: Accessible and locally presentable categories
- 18C50: Categorical semantics of formal languages [See also 68Q55, 68Q65]
- 18C99: None of the above, but in this section

- 18Dxx: Categories with structure
- 18D05: Double categories, $2$-categories, bicategories and generalizations
- 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]
- 18D15: Closed categories (closed monoidal and Cartesian closed categories, etc.)
- 18D20: Enriched categories (over closed or monoidal categories)
- 18D25: Strong functors, strong adjunctions
- 18D30: Fibered categories
- 18D35: Structured objects in a category (group objects, etc.)
- 18D50: Operads [See also 55P48]
- 18D99: None of the above, but in this section

- 18Exx: Abelian categories
- 18E05: Preadditive, additive categories
- 18E10: Exact categories, abelian categories
- 18E15: Grothendieck categories
- 18E20: Embedding theorems [See also 18B15]
- 18E25: Derived functors and satellites
- 18E30: Derived categories, triangulated categories
- 18E35: Localization of categories
- 18E40: Torsion theories, radicals [See also 13D30, 16S90]
- 18E99: None of the above, but in this section

- 18Fxx: Categories and geometry
- 18F05: Local categories and functors
- 18F10: Grothendieck topologies [See also 14F20]
- 18F15: Abstract manifolds and fiber bundles [See also 55Rxx, 57Pxx]
- 18F20: Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30]
- 18F25: Algebraic $K$-theory and $L$-theory [See also 11Exx, 11R70, 11S70, 12-XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67]
- 18F30: Grothendieck groups [See also 13D15, 16E20, 19Axx]
- 18F99: None of the above, but in this section

- 18Gxx: Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx]
- 18G05: Projectives and injectives [See also 13C10, 13C11, 16D40, 16D50]
- 18G10: Resolutions; derived functors [See also 13D02, 16E05, 18E25]
- 18G15: Ext and Tor, generalizations, Künneth formula [See also 55U25]
- 18G20: Homological dimension [See also 13D05, 16E10]
- 18G25: Relative homological algebra, projective classes
- 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10]
- 18G35: Chain complexes [See also 18E30, 55U15]
- 18G40: Spectral sequences, hypercohomology [See also 55Txx]
- 18G50: Nonabelian homological algebra
- 18G55: Homotopical algebra
- 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]
- 18G99: None of the above, but in this section

- 19-XX: $K$-theory [See also 16E20, 18F25]
- 19-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 19-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 19-02: Research exposition (monographs, survey articles)
- 19-03: Historical (must also be assigned at least one classification number from Section 01)
- 19-04: Explicit machine computation and programs (not the theory of computation or programming)
- 19-06: Proceedings, conferences, collections, etc.
- 19Axx: Grothendieck groups and $K_0$ [See also 13D15, 18F30]
- 19A13: Stability for projective modules [See also 13C10]
- 19A15: Efficient generation
- 19A22: Frobenius induction, Burnside and representation rings
- 19A31: $K_0$ of group rings and orders
- 19A49: $K_0$ of other rings
- 19A99: None of the above, but in this section

- 19Bxx: Whitehead groups and $K_1$
- 19Cxx: Steinberg groups and $K_2$
- 19C09: Central extensions and Schur multipliers
- 19C20: Symbols, presentations and stability of $K_2$
- 19C30: $K_2$ and the Brauer group
- 19C40: Excision for $K_2$
- 19C99: None of the above, but in this section

- 19Dxx: Higher algebraic $K$-theory
- 19D06: $Q$- and plus-constructions
- 19D10: Algebraic $K$-theory of spaces
- 19D23: Symmetric monoidal categories [See also 18D10]
- 19D25: Karoubi-Villamayor-Gersten $K$-theory
- 19D35: Negative $K$-theory, NK and Nil
- 19D45: Higher symbols, Milnor $K$-theory
- 19D50: Computations of higher $K$-theory of rings [See also 13D15, 16E20]
- 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]
- 19D99: None of the above, but in this section

- 19Exx: $K$-theory in geometry
- 19Fxx: $K$-theory in number theory [See also 11R70, 11S70]
- 19Gxx: $K$-theory of forms [See also 11Exx]
- 19Jxx: Obstructions from topology
- 19J05: Finiteness and other obstructions in $K_0$
- 19J10: Whitehead (and related) torsion
- 19J25: Surgery obstructions [See also 57R67]
- 19J35: Obstructions to group actions
- 19J99: None of the above, but in this section

- 19Kxx: $K$-theory and operator algebras [See mainly 46L80, and also 46M20]
- 19Lxx: Topological $K$-theory [See also 55N15, 55R50, 55S25]
- 19L10: Riemann-Roch theorems, Chern characters
- 19L20: $J$-homomorphism, Adams operations [See also 55Q50]
- 19L41: Connective $K$-theory, cobordism [See also 55N22]
- 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]
- 19L50: Twisted $K$-theory; differential $K$-theory
- 19L64: Computations, geometric applications
- 19L99: None of the above, but in this section

- 19Mxx: Miscellaneous applications of $K$-theory
- 19M05: Miscellaneous applications of $K$-theory
- 19M99: None of the above, but in this section

- 20-XX: Group theory and generalizations
- 20-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 20-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 20-02: Research exposition (monographs, survey articles)
- 20-03: Historical (must also be assigned at least one classification number from Section 01)
- 20-04: Explicit machine computation and programs (not the theory of computation or programming)
- 20-06: Proceedings, conferences, collections, etc.
- 20Axx: Foundations
- 20A05: Axiomatics and elementary properties
- 20A10: Metamathematical considerations {For word problems, see 20F10}
- 20A15: Applications of logic to group theory
- 20A99: None of the above, but in this section

- 20Bxx: Permutation groups
- 20B05: General theory for finite groups
- 20B07: General theory for infinite groups
- 20B10: Characterization theorems
- 20B15: Primitive groups
- 20B20: Multiply transitive finite groups
- 20B22: Multiply transitive infinite groups
- 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX]
- 20B27: Infinite automorphism groups [See also 12F10]
- 20B30: Symmetric groups
- 20B35: Subgroups of symmetric groups
- 20B40: Computational methods
- 20B99: None of the above, but in this section

- 20Cxx: Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)]
- 20C05: Group rings of finite groups and their modules [See also 16S34]
- 20C07: Group rings of infinite groups and their modules [See also 16S34]
- 20C08: Hecke algebras and their representations
- 20C10: Integral representations of finite groups
- 20C11: $p$-adic representations of finite groups
- 20C12: Integral representations of infinite groups
- 20C15: Ordinary representations and characters
- 20C20: Modular representations and characters
- 20C25: Projective representations and multipliers
- 20C30: Representations of finite symmetric groups
- 20C32: Representations of infinite symmetric groups
- 20C33: Representations of finite groups of Lie type
- 20C34: Representations of sporadic groups
- 20C35: Applications of group representations to physics
- 20C40: Computational methods
- 20C99: None of the above, but in this section

- 20Dxx: Abstract finite groups
- 20D05: Finite simple groups and their classification
- 20D06: Simple groups: alternating groups and groups of Lie type [See also 20Gxx]
- 20D08: Simple groups: sporadic groups
- 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17]
- 20D15: Nilpotent groups, $p$-groups
- 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
- 20D25: Special subgroups (Frattini, Fitting, etc.)
- 20D30: Series and lattices of subgroups
- 20D35: Subnormal subgroups
- 20D40: Products of subgroups
- 20D45: Automorphisms
- 20D60: Arithmetic and combinatorial problems
- 20D99: None of the above, but in this section

- 20Exx: Structure and classification of infinite or finite groups
- 20E05: Free nonabelian groups
- 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
- 20E07: Subgroup theorems; subgroup growth
- 20E08: Groups acting on trees [See also 20F65]
- 20E10: Quasivarieties and varieties of groups
- 20E15: Chains and lattices of subgroups, subnormal subgroups [See also 20F22]
- 20E18: Limits, profinite groups
- 20E22: Extensions, wreath products, and other compositions [See also 20J05]
- 20E25: Local properties
- 20E26: Residual properties and generalizations; residually finite groups
- 20E28: Maximal subgroups
- 20E32: Simple groups [See also 20D05]
- 20E34: General structure theorems
- 20E36: Automorphisms of infinite groups [For automorphisms of finite groups, see 20D45]
- 20E42: Groups with a $BN$-pair; buildings [See also 51E24]
- 20E45: Conjugacy classes
- 20E99: None of the above, but in this section

- 20Fxx: Special aspects of infinite or finite groups
- 20F05: Generators, relations, and presentations
- 20F06: Cancellation theory; application of van Kampen diagrams [See also 57M05]
- 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70]
- 20F11: Groups of finite Morley rank [See also 03C45, 03C60]
- 20F12: Commutator calculus
- 20F14: Derived series, central series, and generalizations
- 20F16: Solvable groups, supersolvable groups [See also 20D10]
- 20F17: Formations of groups, Fitting classes [See also 20D10]
- 20F18: Nilpotent groups [See also 20D15]
- 20F19: Generalizations of solvable and nilpotent groups
- 20F22: Other classes of groups defined by subgroup chains
- 20F24: FC-groups and their generalizations
- 20F28: Automorphism groups of groups [See also 20E36]
- 20F29: Representations of groups as automorphism groups of algebraic systems
- 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx]
- 20F36: Braid groups; Artin groups
- 20F38: Other groups related to topology or analysis
- 20F40: Associated Lie structures
- 20F45: Engel conditions
- 20F50: Periodic groups; locally finite groups
- 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
- 20F60: Ordered groups [See mainly 06F15]
- 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
- 20F67: Hyperbolic groups and nonpositively curved groups
- 20F69: Asymptotic properties of groups
- 20F70: Algebraic geometry over groups; equations over groups
- 20F99: None of the above, but in this section

- 20Gxx: Linear algebraic groups and related topics {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55}
- 20G05: Representation theory
- 20G07: Structure theory
- 20G10: Cohomology theory
- 20G15: Linear algebraic groups over arbitrary fields
- 20G20: Linear algebraic groups over the reals, the complexes, the quaternions
- 20G25: Linear algebraic groups over local fields and their integers
- 20G30: Linear algebraic groups over global fields and their integers
- 20G35: Linear algebraic groups over adèles and other rings and schemes
- 20G40: Linear algebraic groups over finite fields
- 20G41: Exceptional groups
- 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]
- 20G43: Schur and $q$-Schur algebras
- 20G44: Kac-Moody groups
- 20G45: Applications to physics
- 20G99: None of the above, but in this section

- 20Hxx: Other groups of matrices [See also 15A30]
- 20H05: Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20]
- 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
- 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25]
- 20H20: Other matrix groups over fields
- 20H25: Other matrix groups over rings
- 20H30: Other matrix groups over finite fields
- 20H99: None of the above, but in this section

- 20Jxx: Connections with homological algebra and category theory
- 20J05: Homological methods in group theory
- 20J06: Cohomology of groups
- 20J15: Category of groups
- 20J99: None of the above, but in this section

- 20Kxx: Abelian groups
- 20K01: Finite abelian groups [For sumsets, see 11B13 and 11P70]
- 20K10: Torsion groups, primary groups and generalized primary groups
- 20K15: Torsion-free groups, finite rank
- 20K20: Torsion-free groups, infinite rank
- 20K21: Mixed groups
- 20K25: Direct sums, direct products, etc.
- 20K27: Subgroups
- 20K30: Automorphisms, homomorphisms, endomorphisms, etc.
- 20K35: Extensions
- 20K40: Homological and categorical methods
- 20K45: Topological methods [See also 22A05, 22B05]
- 20K99: None of the above, but in this section

- 20Lxx: Groupoids (i.e. small categories in which all morphisms are isomorphisms) {For sets with a single binary operation, see 20N02; for topological groupoids, see 22A22, 58H05}
- 20Mxx: Semigroups
- 20M05: Free semigroups, generators and relations, word problems [See also 03D40, 08A50, 20F10]
- 20M07: Varieties and pseudovarieties of semigroups
- 20M10: General structure theory
- 20M11: Radical theory
- 20M12: Ideal theory
- 20M13: Arithmetic theory of monoids
- 20M14: Commutative semigroups
- 20M15: Mappings of semigroups
- 20M17: Regular semigroups
- 20M18: Inverse semigroups
- 20M19: Orthodox semigroups
- 20M20: Semigroups of transformations, etc. [See also 47D03, 47H20, 54H15]
- 20M25: Semigroup rings, multiplicative semigroups of rings [See also 16S36, 16Y60]
- 20M30: Representation of semigroups; actions of semigroups on sets
- 20M32: Algebraic monoids
- 20M35: Semigroups in automata theory, linguistics, etc. [See also 03D05, 68Q70, 68T50]
- 20M50: Connections of semigroups with homological algebra and category theory
- 20M99: None of the above, but in this section

- 20Nxx: Other generalizations of groups
- 20Pxx: Probabilistic methods in group theory [See also 60Bxx]
- 20P05: Probabilistic methods in group theory [See also 60Bxx]
- 20P99: None of the above, but in this section

- 22-XX: Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX}
- 22-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 22-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 22-02: Research exposition (monographs, survey articles)
- 22-03: Historical (must also be assigned at least one classification number from Section 01)
- 22-04: Explicit machine computation and programs (not the theory of computation or programming)
- 22-06: Proceedings, conferences, collections, etc.
- 22Axx: Topological and differentiable algebraic systems {For topological rings and fields, see 12Jxx, 13Jxx, 16W80}
- 22A05: Structure of general topological groups
- 22A10: Analysis on general topological groups
- 22A15: Structure of topological semigroups
- 22A20: Analysis on topological semigroups
- 22A22: Topological groupoids (including differentiable and Lie groupoids) [See also 58H05]
- 22A25: Representations of general topological groups and semigroups
- 22A26: Topological semilattices, lattices and applications [See also 06B30, 06B35, 06F30]
- 22A30: Other topological algebraic systems and their representations
- 22A99: None of the above, but in this section

- 22Bxx: Locally compact abelian groups (LCA groups)
- 22B05: General properties and structure of LCA groups
- 22B10: Structure of group algebras of LCA groups
- 22B99: None of the above, but in this section

- 22Cxx: Compact groups
- 22C05: Compact groups
- 22C99: None of the above, but in this section

- 22Dxx: Locally compact groups and their algebras
- 22D05: General properties and structure of locally compact groups
- 22D10: Unitary representations of locally compact groups
- 22D12: Other representations of locally compact groups
- 22D15: Group algebras of locally compact groups
- 22D20: Representations of group algebras
- 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
- 22D30: Induced representations
- 22D35: Duality theorems
- 22D40: Ergodic theory on groups [See also 28Dxx]
- 22D45: Automorphism groups of locally compact groups
- 22D99: None of the above, but in this section

- 22Exx: Lie groups {For the topology of Lie groups and homogeneous spaces, see 57Sxx, 57Txx; for analysis thereon, see 43A80, 43A85, 43A90}
- 22E05: Local Lie groups [See also 34-XX, 35-XX, 58H05]
- 22E10: General properties and structure of complex Lie groups [See also 32M05]
- 22E15: General properties and structure of real Lie groups
- 22E20: General properties and structure of other Lie groups
- 22E25: Nilpotent and solvable Lie groups
- 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
- 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
- 22E35: Analysis on $p$-adic Lie groups
- 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
- 22E41: Continuous cohomology [See also 57R32, 57Txx, 58H10]
- 22E43: Structure and representation of the Lorentz group
- 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
- 22E46: Semisimple Lie groups and their representations
- 22E47: Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10]
- 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]
- 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings [See also 20G05]
- 22E57: Geometric Langlands program: representation-theoretic aspects [See also 14D24]
- 22E60: Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx}
- 22E65: Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05]
- 22E66: Analysis on and representations of infinite-dimensional Lie groups
- 22E67: Loop groups and related constructions, group-theoretic treatment [See also 58D05]
- 22E70: Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10]
- 22E99: None of the above, but in this section

- 22Fxx: Noncompact transformation groups
- 22F05: General theory of group and pseudogroup actions {For topological properties of spaces with an action, see 57S20}
- 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx]
- 22F30: Homogeneous spaces {For general actions on manifolds or preserving geometrical structures, see 57M60, 57Sxx; for discrete subgroups of Lie groups, see especially 22E40}
- 22F50: Groups as automorphisms of other structures
- 22F99: None of the above, but in this section

- 26-XX: Real functions [See also 54C30]
- 26-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 26-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 26-02: Research exposition (monographs, survey articles)
- 26-03: Historical (must also be assigned at least one classification number from Section 01)
- 26-04: Explicit machine computation and programs (not the theory of computation or programming)
- 26-06: Proceedings, conferences, collections, etc.
- 26Axx: Functions of one variable
- 26A03: Foundations: limits and generalizations, elementary topology of the line
- 26A06: One-variable calculus
- 26A09: Elementary functions
- 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48]
- 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
- 26A16: Lipschitz (Hölder) classes
- 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25]
- 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]
- 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]
- 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
- 26A30: Singular functions, Cantor functions, functions with other special properties
- 26A33: Fractional derivatives and integrals
- 26A36: Antidifferentiation
- 26A39: Denjoy and Perron integrals, other special integrals
- 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]
- 26A45: Functions of bounded variation, generalizations
- 26A46: Absolutely continuous functions
- 26A48: Monotonic functions, generalizations
- 26A51: Convexity, generalizations
- 26A99: None of the above, but in this section

- 26Bxx: Functions of several variables
- 26B05: Continuity and differentiation questions
- 26B10: Implicit function theorems, Jacobians, transformations with several variables
- 26B12: Calculus of vector functions
- 26B15: Integration: length, area, volume [See also 28A75, 51M25]
- 26B20: Integral formulas (Stokes, Gauss, Green, etc.)
- 26B25: Convexity, generalizations
- 26B30: Absolutely continuous functions, functions of bounded variation
- 26B35: Special properties of functions of several variables, Hölder conditions, etc.
- 26B40: Representation and superposition of functions
- 26B99: None of the above, but in this section

- 26Cxx: Polynomials, rational functions
- 26Dxx: Inequalities {For maximal function inequalities, see 42B25; for functional inequalities, see 39B72; for probabilistic inequalities, see 60E15}
- 26D05: Inequalities for trigonometric functions and polynomials
- 26D07: Inequalities involving other types of functions
- 26D10: Inequalities involving derivatives and differential and integral operators
- 26D15: Inequalities for sums, series and integrals
- 26D20: Other analytical inequalities
- 26D99: None of the above, but in this section

- 26Exx: Miscellaneous topics [See also 58Cxx]
- 26E05: Real-analytic functions [See also 32B05, 32C05]
- 26E10: $C^\infty$-functions, quasi-analytic functions [See also 58C25]
- 26E15: Calculus of functions on infinite-dimensional spaces [See also 46G05, 58Cxx]
- 26E20: Calculus of functions taking values in infinite-dimensional spaces [See also 46E40, 46G10, 58Cxx]
- 26E25: Set-valued functions [See also 28B20, 49J53, 54C60] {For nonsmooth analysis, see 49J52, 58Cxx, 90Cxx}
- 26E30: Non-Archimedean analysis [See also 12J25]
- 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05]
- 26E40: Constructive real analysis [See also 03F60]
- 26E50: Fuzzy real analysis [See also 03E72, 28E10]
- 26E60: Means [See also 47A64]
- 26E70: Real analysis on time scales or measure chains {For dynamic equations on time scales or measure chains see 34N05}
- 26E99: None of the above, but in this section

- 28-XX: Measure and integration {For analysis on manifolds, see 58-XX}
- 28-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 28-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 28-02: Research exposition (monographs, survey articles)
- 28-03: Historical (must also be assigned at least one classification number from Section 01)
- 28-04: Explicit machine computation and programs (not the theory of computation or programming)
- 28-06: Proceedings, conferences, collections, etc.
- 28Axx: Classical measure theory
- 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
- 28A10: Real- or complex-valued set functions
- 28A12: Contents, measures, outer measures, capacities
- 28A15: Abstract differentiation theory, differentiation of set functions [See also 26A24]
- 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
- 28A25: Integration with respect to measures and other set functions
- 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]
- 28A35: Measures and integrals in product spaces
- 28A50: Integration and disintegration of measures
- 28A51: Lifting theory [See also 46G15]
- 28A60: Measures on Boolean rings, measure algebras [See also 54H10]
- 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
- 28A78: Hausdorff and packing measures
- 28A80: Fractals [See also 37Fxx]
- 28A99: None of the above, but in this section

- 28Bxx: Set functions, measures and integrals with values in abstract spaces
- 28B05: Vector-valued set functions, measures and integrals [See also 46G10]
- 28B10: Group- or semigroup-valued set functions, measures and integrals
- 28B15: Set functions, measures and integrals with values in ordered spaces
- 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14]
- 28B99: None of the above, but in this section

- 28Cxx: Set functions and measures on spaces with additional structure [See also 46G12, 58C35, 58D20]
- 28C05: Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
- 28C10: Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]
- 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
- 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11]
- 28C99: None of the above, but in this section

- 28Dxx: Measure-theoretic ergodic theory [See also 11K50, 11K55, 22D40, 37Axx, 47A35, 54H20, 60Fxx, 60G10]
- 28D05: Measure-preserving transformations
- 28D10: One-parameter continuous families of measure-preserving transformations
- 28D15: General groups of measure-preserving transformations
- 28D20: Entropy and other invariants
- 28D99: None of the above, but in this section

- 28Exx: Miscellaneous topics in measure theory

- 30-XX: Functions of a complex variable {For analysis on manifolds, see 58-XX}
- 30-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 30-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 30-02: Research exposition (monographs, survey articles)
- 30-03: Historical (must also be assigned at least one classification number from Section 01)
- 30-04: Explicit machine computation and programs (not the theory of computation or programming)
- 30-06: Proceedings, conferences, collections, etc.
- 30Axx: General properties
- 30A05: Monogenic properties of complex functions (including polygenic and areolar monogenic functions)
- 30A10: Inequalities in the complex domain
- 30A99: None of the above, but in this section

- 30Bxx: Series expansions
- 30B10: Power series (including lacunary series)
- 30B20: Random power series
- 30B30: Boundary behavior of power series, over-convergence
- 30B40: Analytic continuation
- 30B50: Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX]
- 30B60: Completeness problems, closure of a system of functions
- 30B70: Continued fractions [See also 11A55, 40A15]
- 30B99: None of the above, but in this section

- 30Cxx: Geometric function theory
- 30C10: Polynomials
- 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
- 30C20: Conformal mappings of special domains
- 30C25: Covering theorems in conformal mapping theory
- 30C30: Numerical methods in conformal mapping theory [See also 65E05]
- 30C35: General theory of conformal mappings
- 30C40: Kernel functions and applications
- 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
- 30C50: Coefficient problems for univalent and multivalent functions
- 30C55: General theory of univalent and multivalent functions
- 30C62: Quasiconformal mappings in the plane
- 30C65: Quasiconformal mappings in ${\bf R}^n$, other generalizations
- 30C70: Extremal problems for conformal and quasiconformal mappings, variational methods
- 30C75: Extremal problems for conformal and quasiconformal mappings, other methods
- 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
- 30C85: Capacity and harmonic measure in the complex plane [See also 31A15]
- 30C99: None of the above, but in this section

- 30Dxx: Entire and meromorphic functions, and related topics
- 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]
- 30D10: Representations of entire functions by series and integrals
- 30D15: Special classes of entire functions and growth estimates
- 30D20: Entire functions, general theory
- 30D30: Meromorphic functions, general theory
- 30D35: Distribution of values, Nevanlinna theory
- 30D40: Cluster sets, prime ends, boundary behavior
- 30D45: Bloch functions, normal functions, normal families
- 30D60: Quasi-analytic and other classes of functions
- 30D99: None of the above, but in this section

- 30Exx: Miscellaneous topics of analysis in the complex domain
- 30E05: Moment problems, interpolation problems
- 30E10: Approximation in the complex domain
- 30E15: Asymptotic representations in the complex domain
- 30E20: Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx]
- 30E25: Boundary value problems [See also 45Exx]
- 30E99: None of the above, but in this section

- 30Fxx: Riemann surfaces
- 30F10: Compact Riemann surfaces and uniformization [See also 14H15, 32G15]
- 30F15: Harmonic functions on Riemann surfaces
- 30F20: Classification theory of Riemann surfaces
- 30F25: Ideal boundary theory
- 30F30: Differentials on Riemann surfaces
- 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]
- 30F40: Kleinian groups [See also 20H10]
- 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
- 30F50: Klein surfaces
- 30F60: Teichmüller theory [See also 32G15]
- 30F99: None of the above, but in this section

- 30Gxx: Generalized function theory
- 30G06: Non-Archimedean function theory [See also 12J25]; nonstandard function theory [See also 03H05]
- 30G12: Finely holomorphic functions and topological function theory
- 30G20: Generalizations of Bers or Vekua type (pseudoanalytic, $p$-analytic, etc.)
- 30G25: Discrete analytic functions
- 30G30: Other generalizations of analytic functions (including abstract-valued functions)
- 30G35: Functions of hypercomplex variables and generalized variables
- 30G99: None of the above, but in this section

- 30Hxx: Spaces and algebras of analytic functions
- 30H05: Bounded analytic functions
- 30H10: Hardy spaces
- 30H15: Nevanlinna class and Smirnov class
- 30H20: Bergman spaces, Fock spaces
- 30H25: Besov spaces and $Q_p$-spaces
- 30H30: Bloch spaces
- 30H35: BMO-spaces
- 30H50: Algebras of analytic functions
- 30H80: Corona theorems
- 30H99: None of the above, but in this section

- 30Jxx: Function theory on the disc
- 30J05: Inner functions
- 30J10: Blaschke products
- 30J15: Singular inner functions
- 30J99: None of the above, but in this section

- 30Kxx: Universal holomorphic functions
- 30K05: Universal Taylor series
- 30K10: Universal Dirichlet series
- 30K15: Bounded universal functions
- 30K20: Compositional universality
- 30K99: None of the above, but in this section

- 30Lxx: Analysis on metric spaces
- 30L05: Geometric embeddings of metric spaces
- 30L10: Quasiconformal mappings in metric spaces
- 30L99: None of the above, but in this section

- 31-XX: Potential theory {For probabilistic potential theory, see 60J45}
- 31-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 31-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 31-02: Research exposition (monographs, survey articles)
- 31-03: Historical (must also be assigned at least one classification number from Section 01)
- 31-04: Explicit machine computation and programs (not the theory of computation or programming)
- 31-06: Proceedings, conferences, collections, etc.
- 31Axx: Two-dimensional theory
- 31A05: Harmonic, subharmonic, superharmonic functions
- 31A10: Integral representations, integral operators, integral equations methods
- 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85]
- 31A20: Boundary behavior (theorems of Fatou type, etc.)
- 31A25: Boundary value and inverse problems
- 31A30: Biharmonic, polyharmonic functions and equations, Poisson's equation
- 31A35: Connections with differential equations
- 31A99: None of the above, but in this section

- 31Bxx: Higher-dimensional theory
- 31B05: Harmonic, subharmonic, superharmonic functions
- 31B10: Integral representations, integral operators, integral equations methods
- 31B15: Potentials and capacities, extremal length
- 31B20: Boundary value and inverse problems
- 31B25: Boundary behavior
- 31B30: Biharmonic and polyharmonic equations and functions
- 31B35: Connections with differential equations
- 31B99: None of the above, but in this section

- 31Cxx: Other generalizations
- 31C05: Harmonic, subharmonic, superharmonic functions
- 31C10: Pluriharmonic and plurisubharmonic functions [See also 32U05]
- 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]
- 31C15: Potentials and capacities
- 31C20: Discrete potential theory and numerical methods
- 31C25: Dirichlet spaces
- 31C35: Martin boundary theory [See also 60J50]
- 31C40: Fine potential theory
- 31C45: Other generalizations (nonlinear potential theory, etc.)
- 31C99: None of the above, but in this section

- 31Dxx: Axiomatic potential theory
- 31D05: Axiomatic potential theory
- 31D99: None of the above, but in this section

- 31Exx: Potential theory on metric spaces
- 31E05: Potential theory on metric spaces
- 31E99: None of the above, but in this section

- 32-XX: Several complex variables and analytic spaces {For infinite-dimensional holomorphy, see 46G20, 58B12}
- 32-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 32-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 32-02: Research exposition (monographs, survey articles)
- 32-03: Historical (must also be assigned at least one classification number from Section 01)
- 32-04: Explicit machine computation and programs (not the theory of computation or programming)
- 32-06: Proceedings, conferences, collections, etc.
- 32Axx: Holomorphic functions of several complex variables
- 32A05: Power series, series of functions
- 32A07: Special domains (Reinhardt, Hartogs, circular, tube)
- 32A10: Holomorphic functions
- 32A12: Multifunctions
- 32A15: Entire functions
- 32A17: Special families of functions
- 32A18: Bloch functions, normal functions
- 32A19: Normal families of functions, mappings
- 32A20: Meromorphic functions
- 32A22: Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30}
- 32A25: Integral representations; canonical kernels (Szegő, Bergman, etc.)
- 32A26: Integral representations, constructed kernels (e.g. Cauchy, Fantappiè-type kernels)
- 32A27: Local theory of residues [See also 32C30]
- 32A30: Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) {For functions of several hypercomplex variables, see 30G35}
- 32A35: $H^p$-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15]
- 32A36: Bergman spaces
- 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx]
- 32A38: Algebras of holomorphic functions [See also 30H05, 46J10, 46J15]
- 32A40: Boundary behavior of holomorphic functions
- 32A45: Hyperfunctions [See also 46F15]
- 32A50: Harmonic analysis of several complex variables [See mainly 43-XX]
- 32A55: Singular integrals
- 32A60: Zero sets of holomorphic functions
- 32A65: Banach algebra techniques [See mainly 46Jxx]
- 32A70: Functional analysis techniques [See mainly 46Exx]
- 32A99: None of the above, but in this section

- 32Bxx: Local analytic geometry [See also 13-XX and 14-XX]
- 32B05: Analytic algebras and generalizations, preparation theorems
- 32B10: Germs of analytic sets, local parametrization
- 32B15: Analytic subsets of affine space
- 32B20: Semi-analytic sets and subanalytic sets [See also 14P15]
- 32B25: Triangulation and related questions
- 32B99: None of the above, but in this section

- 32Cxx: Analytic spaces
- 32C05: Real-analytic manifolds, real-analytic spaces [See also 14Pxx, 58A07]
- 32C07: Real-analytic sets, complex Nash functions [See also 14P15, 14P20]
- 32C09: Embedding of real analytic manifolds
- 32C11: Complex supergeometry [See also 14A22, 14M30, 58A50]
- 32C15: Complex spaces
- 32C18: Topology of analytic spaces
- 32C20: Normal analytic spaces
- 32C22: Embedding of analytic spaces
- 32C25: Analytic subsets and submanifolds
- 32C30: Integration on analytic sets and spaces, currents {For local theory, see 32A25 or 32A27}
- 32C35: Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30]
- 32C36: Local cohomology of analytic spaces
- 32C37: Duality theorems
- 32C38: Sheaves of differential operators and their modules, $D$-modules [See also 14F10, 16S32, 35A27, 58J15]
- 32C55: The Levi problem in complex spaces; generalizations
- 32C81: Applications to physics
- 32C99: None of the above, but in this section

- 32Dxx: Analytic continuation
- 32D05: Domains of holomorphy
- 32D10: Envelopes of holomorphy
- 32D15: Continuation of analytic objects
- 32D20: Removable singularities
- 32D26: Riemann domains
- 32D99: None of the above, but in this section

- 32Exx: Holomorphic convexity
- 32E05: Holomorphically convex complex spaces, reduction theory
- 32E10: Stein spaces, Stein manifolds
- 32E20: Polynomial convexity
- 32E30: Holomorphic and polynomial approximation, Runge pairs, interpolation
- 32E35: Global boundary behavior of holomorphic functions
- 32E40: The Levi problem
- 32E99: None of the above, but in this section

- 32Fxx: Geometric convexity
- 32F10: $q$-convexity, $q$-concavity
- 32F17: Other notions of convexity
- 32F18: Finite-type conditions
- 32F27: Topological consequences of geometric convexity
- 32F32: Analytical consequences of geometric convexity (vanishing theorems, etc.)
- 32F45: Invariant metrics and pseudodistances
- 32F99: None of the above, but in this section

- 32Gxx: Deformations of analytic structures
- 32G05: Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15]
- 32G07: Deformations of special (e.g. CR) structures
- 32G08: Deformations of fiber bundles
- 32G10: Deformations of submanifolds and subspaces
- 32G13: Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15]
- 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
- 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]
- 32G34: Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) [See also 34Mxx]
- 32G81: Applications to physics
- 32G99: None of the above, but in this section

- 32Hxx: Holomorphic mappings and correspondences
- 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
- 32H04: Meromorphic mappings
- 32H12: Boundary uniqueness of mappings
- 32H25: Picard-type theorems and generalizations {For function-theoretic properties, see 32A22}
- 32H30: Value distribution theory in higher dimensions {For function-theoretic properties, see 32A22}
- 32H35: Proper mappings, finiteness theorems
- 32H40: Boundary regularity of mappings
- 32H50: Iteration problems
- 32H99: None of the above, but in this section

- 32Jxx: Compact analytic spaces {For Riemann surfaces, see 14Hxx, 30Fxx; for algebraic theory, see 14Jxx}
- 32J05: Compactification of analytic spaces
- 32J10: Algebraic dependence theorems
- 32J15: Compact surfaces
- 32J17: Compact $3$-folds
- 32J18: Compact $n$-folds
- 32J25: Transcendental methods of algebraic geometry [See also 14C30]
- 32J27: Compact Kähler manifolds: generalizations, classification
- 32J81: Applications to physics
- 32J99: None of the above, but in this section

- 32Kxx: Generalizations of analytic spaces (should also be assigned at least one other classification number from Section 32 describing the type of problem)
- 32Lxx: Holomorphic fiber spaces [See also 55Rxx]
- 32L05: Holomorphic bundles and generalizations
- 32L10: Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30]
- 32L15: Bundle convexity [See also 32F10]
- 32L20: Vanishing theorems
- 32L25: Twistor theory, double fibrations [See also 53C28]
- 32L81: Applications to physics
- 32L99: None of the above, but in this section

- 32Mxx: Complex spaces with a group of automorphisms
- 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]
- 32M10: Homogeneous complex manifolds [See also 14M17, 57T15]
- 32M12: Almost homogeneous manifolds and spaces [See also 14M17]
- 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
- 32M17: Automorphism groups of ${\bf C}^n$ and affine manifolds
- 32M25: Complex vector fields
- 32M99: None of the above, but in this section

- 32Nxx: Automorphic functions [See also 11Fxx, 20H10, 22E40, 30F35]
- 32N05: General theory of automorphic functions of several complex variables
- 32N10: Automorphic forms
- 32N15: Automorphic functions in symmetric domains
- 32N99: None of the above, but in this section

- 32Pxx: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)
- 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)
- 32P99: None of the above, but in this section

- 32Qxx: Complex manifolds
- 32Q05: Negative curvature manifolds
- 32Q10: Positive curvature manifolds
- 32Q15: Kähler manifolds
- 32Q20: Kähler-Einstein manifolds [See also 53Cxx]
- 32Q25: Calabi-Yau theory [See also 14J30]
- 32Q26: Notions of stability
- 32Q28: Stein manifolds
- 32Q30: Uniformization
- 32Q35: Complex manifolds as subdomains of Euclidean space
- 32Q40: Embedding theorems
- 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds
- 32Q55: Topological aspects of complex manifolds
- 32Q57: Classification theorems
- 32Q60: Almost complex manifolds
- 32Q65: Pseudoholomorphic curves
- 32Q99: None of the above, but in this section

- 32Sxx: Singularities [See also 58Kxx]
- 32S05: Local singularities [See also 14J17]
- 32S10: Invariants of analytic local rings
- 32S15: Equisingularity (topological and analytic) [See also 14E15]
- 32S20: Global theory of singularities; cohomological properties [See also 14E15]
- 32S22: Relations with arrangements of hyperplanes [See also 52C35]
- 32S25: Surface and hypersurface singularities [See also 14J17]
- 32S30: Deformations of singularities; vanishing cycles [See also 14B07]
- 32S35: Mixed Hodge theory of singular varieties [See also 14C30, 14D07]
- 32S40: Monodromy; relations with differential equations and $D$-modules
- 32S45: Modifications; resolution of singularities [See also 14E15]
- 32S50: Topological aspects: Lefschetz theorems, topological classification, invariants
- 32S55: Milnor fibration; relations with knot theory [See also 57M25, 57Q45]
- 32S60: Stratifications; constructible sheaves; intersection cohomology [See also 58Kxx]
- 32S65: Singularities of holomorphic vector fields and foliations
- 32S70: Other operations on singularities
- 32S99: None of the above, but in this section

- 32Txx: Pseudoconvex domains
- 32T05: Domains of holomorphy
- 32T15: Strongly pseudoconvex domains
- 32T20: Worm domains
- 32T25: Finite type domains
- 32T27: Geometric and analytic invariants on weakly pseudoconvex boundaries
- 32T35: Exhaustion functions
- 32T40: Peak functions
- 32T99: None of the above, but in this section

- 32Uxx: Pluripotential theory
- 32U05: Plurisubharmonic functions and generalizations [See also 31C10]
- 32U10: Plurisubharmonic exhaustion functions
- 32U15: General pluripotential theory
- 32U20: Capacity theory and generalizations
- 32U25: Lelong numbers
- 32U30: Removable sets
- 32U35: Pluricomplex Green functions
- 32U40: Currents
- 32U99: None of the above, but in this section

- 32Vxx: CR manifolds
- 32V05: CR structures, CR operators, and generalizations
- 32V10: CR functions
- 32V15: CR manifolds as boundaries of domains
- 32V20: Analysis on CR manifolds
- 32V25: Extension of functions and other analytic objects from CR manifolds
- 32V30: Embeddings of CR manifolds
- 32V35: Finite type conditions on CR manifolds
- 32V40: Real submanifolds in complex manifolds
- 32V99: None of the above, but in this section

- 32Wxx: Differential operators in several variables
- 32W05: $\overline\partial$ and $\overline\partial$-Neumann operators
- 32W10: $\overline\partial_b$ and $\overline\partial_b$-Neumann operators
- 32W20: Complex Monge-Ampère operators
- 32W25: Pseudodifferential operators in several complex variables
- 32W30: Heat kernels in several complex variables
- 32W50: Other partial differential equations of complex analysis
- 32W99: None of the above, but in this section

- 33-XX: Special functions (33-XX deals with the properties of functions as functions) {For orthogonal functions, see 42Cxx; for aspects of combinatorics see 05Axx; for number-theoretic aspects see 11-XX; for representation theory see 22Exx}
- 33-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 33-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 33-02: Research exposition (monographs, survey articles)
- 33-03: Historical (must also be assigned at least one classification number from Section 01)
- 33-04: Explicit machine computation and programs (not the theory of computation or programming)
- 33-06: Proceedings, conferences, collections, etc.
- 33Bxx: Elementary classical functions
- 33B10: Exponential and trigonometric functions
- 33B15: Gamma, beta and polygamma functions
- 33B20: Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
- 33B30: Higher logarithm functions
- 33B99: None of the above, but in this section

- 33Cxx: Hypergeometric functions
- 33C05: Classical hypergeometric functions, ${}_2F_1$
- 33C10: Bessel and Airy functions, cylinder functions, ${}_0F_1$
- 33C15: Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
- 33C20: Generalized hypergeometric series, ${}_pF_q$
- 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
- 33C47: Other special orthogonal polynomials and functions
- 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
- 33C52: Orthogonal polynomials and functions associated with root systems
- 33C55: Spherical harmonics
- 33C60: Hypergeometric integrals and functions defined by them ($E$, $G$, $H$ and $I$ functions)
- 33C65: Appell, Horn and Lauricella functions
- 33C67: Hypergeometric functions associated with root systems
- 33C70: Other hypergeometric functions and integrals in several variables
- 33C75: Elliptic integrals as hypergeometric functions
- 33C80: Connections with groups and algebras, and related topics
- 33C90: Applications
- 33C99: None of the above, but in this section

- 33Dxx: Basic hypergeometric functions
- 33D05: $q$-gamma functions, $q$-beta functions and integrals
- 33D15: Basic hypergeometric functions in one variable, ${}_r\phi_s$
- 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
- 33D50: Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable
- 33D52: Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
- 33D60: Basic hypergeometric integrals and functions defined by them
- 33D65: Bibasic functions and multiple bases
- 33D67: Basic hypergeometric functions associated with root systems
- 33D70: Other basic hypergeometric functions and integrals in several variables
- 33D80: Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics
- 33D90: Applications
- 33D99: None of the above, but in this section

- 33Exx: Other special functions
- 33E05: Elliptic functions and integrals
- 33E10: Lamé, Mathieu, and spheroidal wave functions
- 33E12: Mittag-Leffler functions and generalizations
- 33E15: Other wave functions
- 33E17: Painlevé-type functions
- 33E20: Other functions defined by series and integrals
- 33E30: Other functions coming from differential, difference and integral equations
- 33E50: Special functions in characteristic $p$ (gamma functions, etc.)
- 33E99: None of the above, but in this section

- 33Fxx: Computational aspects

- 34-XX: Ordinary differential equations
- 34-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 34-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 34-02: Research exposition (monographs, survey articles)
- 34-03: Historical (must also be assigned at least one classification number from Section 01)
- 34-04: Explicit machine computation and programs (not the theory of computation or programming)
- 34-06: Proceedings, conferences, collections, etc.
- 34Axx: General theory
- 34A05: Explicit solutions and reductions
- 34A07: Fuzzy differential equations
- 34A08: Fractional differential equations
- 34A09: Implicit equations, differential-algebraic equations [See also 65L80]
- 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
- 34A25: Analytical theory: series, transformations, transforms, operational calculus, etc. [See also 44-XX]
- 34A26: Geometric methods in differential equations
- 34A30: Linear equations and systems, general
- 34A33: Lattice differential equations
- 34A34: Nonlinear equations and systems, general
- 34A35: Differential equations of infinite order
- 34A36: Discontinuous equations
- 34A37: Differential equations with impulses
- 34A38: Hybrid systems
- 34A40: Differential inequalities [See also 26D20]
- 34A45: Theoretical approximation of solutions {For numerical analysis, see 65Lxx}
- 34A55: Inverse problems
- 34A60: Differential inclusions [See also 49J21, 49K21]
- 34A99: None of the above, but in this section

- 34Bxx: Boundary value problems {For ordinary differential operators, see 34Lxx}
- 34B05: Linear boundary value problems
- 34B07: Linear boundary value problems with nonlinear dependence on the spectral parameter
- 34B08: Parameter dependent boundary value problems
- 34B09: Boundary eigenvalue problems
- 34B10: Nonlocal and multipoint boundary value problems
- 34B15: Nonlinear boundary value problems
- 34B16: Singular nonlinear boundary value problems
- 34B18: Positive solutions of nonlinear boundary value problems
- 34B20: Weyl theory and its generalizations
- 34B24: Sturm-Liouville theory [See also 34Lxx]
- 34B27: Green functions
- 34B30: Special equations (Mathieu, Hill, Bessel, etc.)
- 34B37: Boundary value problems with impulses
- 34B40: Boundary value problems on infinite intervals
- 34B45: Boundary value problems on graphs and networks
- 34B60: Applications
- 34B99: None of the above, but in this section

- 34Cxx: Qualitative theory [See also 37-XX]
- 34C05: Location of integral curves, singular points, limit cycles
- 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
- 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.)
- 34C10: Oscillation theory, zeros, disconjugacy and comparison theory
- 34C11: Growth, boundedness
- 34C12: Monotone systems
- 34C14: Symmetries, invariants
- 34C15: Nonlinear oscillations, coupled oscillators
- 34C20: Transformation and reduction of equations and systems, normal forms
- 34C23: Bifurcation [See also 37Gxx]
- 34C25: Periodic solutions
- 34C26: Relaxation oscillations
- 34C27: Almost and pseudo-almost periodic solutions
- 34C28: Complex behavior, chaotic systems [See also 37Dxx]
- 34C29: Averaging method
- 34C37: Homoclinic and heteroclinic solutions
- 34C40: Equations and systems on manifolds
- 34C41: Equivalence, asymptotic equivalence
- 34C45: Invariant manifolds
- 34C46: Multifrequency systems
- 34C55: Hysteresis
- 34C60: Qualitative investigation and simulation of models
- 34C99: None of the above, but in this section

- 34Dxx: Stability theory [See also 37C75, 93Dxx]
- 34D05: Asymptotic properties
- 34D06: Synchronization
- 34D08: Characteristic and Lyapunov exponents
- 34D09: Dichotomy, trichotomy
- 34D10: Perturbations
- 34D15: Singular perturbations
- 34D20: Stability
- 34D23: Global stability
- 34D30: Structural stability and analogous concepts [See also 37C20]
- 34D35: Stability of manifolds of solutions
- 34D45: Attractors [See also 37C70, 37D45]
- 34D99: None of the above, but in this section

- 34Exx: Asymptotic theory
- 34E05: Asymptotic expansions
- 34E10: Perturbations, asymptotics
- 34E13: Multiple scale methods
- 34E15: Singular perturbations, general theory
- 34E17: Canard solutions
- 34E18: Methods of nonstandard analysis
- 34E20: Singular perturbations, turning point theory, WKB methods
- 34E99: None of the above, but in this section

- 34Fxx: Equations and systems with randomness [See also 34K50, 60H10, 93E03]
- 34Gxx: Differential equations in abstract spaces [See also 34Lxx, 37Kxx, 47Dxx, 47Hxx, 47Jxx, 58D25]
- 34Hxx: Control problems [See also 49J15, 49K15, 93C15]
- 34Kxx: Functional-differential and differential-difference equations [See also 37-XX]
- 34K05: General theory
- 34K06: Linear functional-differential equations
- 34K07: Theoretical approximation of solutions
- 34K08: Spectral theory of functional-differential operators
- 34K09: Functional-differential inclusions
- 34K10: Boundary value problems
- 34K11: Oscillation theory
- 34K12: Growth, boundedness, comparison of solutions
- 34K13: Periodic solutions
- 34K14: Almost and pseudo-periodic solutions
- 34K17: Transformation and reduction of equations and systems, normal forms
- 34K18: Bifurcation theory
- 34K19: Invariant manifolds
- 34K20: Stability theory
- 34K21: Stationary solutions
- 34K23: Complex (chaotic) behavior of solutions
- 34K25: Asymptotic theory
- 34K26: Singular perturbations
- 34K27: Perturbations
- 34K28: Numerical approximation of solutions
- 34K29: Inverse problems
- 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]
- 34K31: Lattice functional-differential equations
- 34K32: Implicit equations
- 34K33: Averaging
- 34K34: Hybrid systems
- 34K35: Control problems [See also 49J21, 49K21, 93C23]
- 34K36: Fuzzy functional-differential equations
- 34K37: Functional-differential equations with fractional derivatives
- 34K38: Functional-differential inequalities
- 34K40: Neutral equations
- 34K45: Equations with impulses
- 34K50: Stochastic functional-differential equations [See also 60Hxx]
- 34K60: Qualitative investigation and simulation of models
- 34K99: None of the above, but in this section

- 34Lxx: Ordinary differential operators [See also 47E05]
- 34L05: General spectral theory
- 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions
- 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds
- 34L16: Numerical approximation of eigenvalues and of other parts of the spectrum
- 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
- 34L25: Scattering theory, inverse scattering
- 34L30: Nonlinear ordinary differential operators
- 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.)
- 34L99: None of the above, but in this section

- 34Mxx: Differential equations in the complex domain [See also 30Dxx, 32G34]
- 34M03: Linear equations and systems
- 34M05: Entire and meromorphic solutions
- 34M10: Oscillation, growth of solutions
- 34M15: Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical)
- 34M25: Formal solutions, transform techniques
- 34M30: Asymptotics, summation methods
- 34M35: Singularities, monodromy, local behavior of solutions, normal forms
- 34M40: Stokes phenomena and connection problems (linear and nonlinear)
- 34M45: Differential equations on complex manifolds
- 34M50: Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.)
- 34M55: Painlevé and other special equations; classification, hierarchies;
- 34M56: Isomonodromic deformations
- 34M60: Singular perturbation problems in the complex domain (complex WKB, turning points, steepest descent) [See also 34E20]
- 34M99: None of the above, but in this section

- 34Nxx: Dynamic equations on time scales or measure chains {For real analysis on time scales see 26E70}
- 34N05: Dynamic equations on time scales or measure chains {For real analysis on time scales or measure chains, see 26E70}
- 34N99: None of the above, but in this section

- 35-XX: Partial differential equations
- 35-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 35-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 35-02: Research exposition (monographs, survey articles)
- 35-03: Historical (must also be assigned at least one classification number from Section 01)
- 35-04: Explicit machine computation and programs (not the theory of computation or programming)
- 35-06: Proceedings, conferences, collections, etc.
- 35Axx: General topics
- 35A01: Existence problems: global existence, local existence, non-existence
- 35A02: Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness
- 35A08: Fundamental solutions
- 35A09: Classical solutions
- 35A10: Cauchy-Kovalevskaya theorems
- 35A15: Variational methods
- 35A16: Topological and monotonicity methods
- 35A17: Parametrices
- 35A18: Wave front sets
- 35A20: Analytic methods, singularities
- 35A21: Propagation of singularities
- 35A22: Transform methods (e.g. integral transforms)
- 35A23: Inequalities involving derivatives and differential and integral operators, inequalities for integrals
- 35A24: Methods of ordinary differential equations
- 35A25: Other special methods
- 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15]
- 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72]
- 35A35: Theoretical approximation to solutions {For numerical analysis, see 65Mxx, 65Nxx}
- 35A99: None of the above, but in this section

- 35Bxx: Qualitative properties of solutions
- 35B05: Oscillation, zeros of solutions, mean value theorems, etc.
- 35B06: Symmetries, invariants, etc.
- 35B07: Axially symmetric solutions
- 35B08: Entire solutions
- 35B09: Positive solutions
- 35B10: Periodic solutions
- 35B15: Almost and pseudo-almost periodic solutions
- 35B20: Perturbations
- 35B25: Singular perturbations
- 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]
- 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]
- 35B32: Bifurcation [See also 37Gxx, 37K50]
- 35B33: Critical exponents
- 35B34: Resonances
- 35B35: Stability
- 35B36: Pattern formation
- 35B38: Critical points
- 35B40: Asymptotic behavior of solutions
- 35B41: Attractors
- 35B42: Inertial manifolds
- 35B44: Blow-up
- 35B45: A priori estimates
- 35B50: Maximum principles
- 35B51: Comparison principles
- 35B53: Liouville theorems, Phragmén-Lindelöf theorems
- 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx]
- 35B65: Smoothness and regularity of solutions
- 35B99: None of the above, but in this section

- 35Cxx: Representations of solutions
- 35C05: Solutions in closed form
- 35C06: Self-similar solutions
- 35C07: Traveling wave solutions
- 35C08: Soliton solutions
- 35C09: Trigonometric solutions
- 35C10: Series solutions
- 35C11: Polynomial solutions
- 35C15: Integral representations of solutions
- 35C20: Asymptotic expansions
- 35C99: None of the above, but in this section

- 35Dxx: Generalized solutions
- 35D30: Weak solutions
- 35D35: Strong solutions
- 35D40: Viscosity solutions
- 35D99: None of the above, but in this section

- 35Exx: Equations and systems with constant coefficients [See also 35N05]
- 35E05: Fundamental solutions
- 35E10: Convexity properties
- 35E15: Initial value problems
- 35E20: General theory
- 35E99: None of the above, but in this section

- 35Fxx: General first-order equations and systems
- 35F05: Linear first-order equations
- 35F10: Initial value problems for linear first-order equations
- 35F15: Boundary value problems for linear first-order equations
- 35F16: Initial-boundary value problems for linear first-order equations
- 35F20: Nonlinear first-order equations
- 35F21: Hamilton-Jacobi equations
- 35F25: Initial value problems for nonlinear first-order equations
- 35F30: Boundary value problems for nonlinear first-order equations
- 35F31: Initial-boundary value problems for nonlinear first-order equations
- 35F35: Linear first-order systems
- 35F40: Initial value problems for linear first-order systems
- 35F45: Boundary value problems for linear first-order systems
- 35F46: Initial-boundary value problems for linear first-order systems
- 35F50: Nonlinear first-order systems
- 35F55: Initial value problems for nonlinear first-order systems
- 35F60: Boundary value problems for nonlinear first-order systems
- 35F61: Initial-boundary value problems for nonlinear first-order systems
- 35F99: None of the above, but in this section

- 35Gxx: General higher-order equations and systems
- 35G05: Linear higher-order equations
- 35G10: Initial value problems for linear higher-order equations
- 35G15: Boundary value problems for linear higher-order equations
- 35G16: Initial-boundary value problems for linear higher-order equations
- 35G20: Nonlinear higher-order equations
- 35G25: Initial value problems for nonlinear higher-order equations
- 35G30: Boundary value problems for nonlinear higher-order equations
- 35G31: Initial-boundary value problems for nonlinear higher-order equations
- 35G35: Linear higher-order systems
- 35G40: Initial value problems for linear higher-order systems
- 35G45: Boundary value problems for linear higher-order systems
- 35G46: Initial-boundary value problems for linear higher-order systems
- 35G50: Nonlinear higher-order systems
- 35G55: Initial value problems for nonlinear higher-order systems
- 35G60: Boundary value problems for nonlinear higher-order systems
- 35G61: Initial-boundary value problems for nonlinear higher-order systems
- 35G99: None of the above, but in this section

- 35Hxx: Close-to-elliptic equations and systems
- 35H10: Hypoelliptic equations
- 35H20: Subelliptic equations
- 35H30: Quasi-elliptic equations
- 35H99: None of the above, but in this section

- 35Jxx: Elliptic equations and systems [See also 58J10, 58J20]
- 35J05: Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation [See also 31Axx, 31Bxx]
- 35J08: Green's functions
- 35J10: Schrödinger operator [See also 35Pxx]
- 35J15: Second-order elliptic equations
- 35J20: Variational methods for second-order elliptic equations
- 35J25: Boundary value problems for second-order elliptic equations
- 35J30: Higher-order elliptic equations [See also 31A30, 31B30]
- 35J35: Variational methods for higher-order elliptic equations
- 35J40: Boundary value problems for higher-order elliptic equations
- 35J46: First-order elliptic systems
- 35J47: Second-order elliptic systems
- 35J48: Higher-order elliptic systems
- 35J50: Variational methods for elliptic systems
- 35J56: Boundary value problems for first-order elliptic systems
- 35J57: Boundary value problems for second-order elliptic systems
- 35J58: Boundary value problems for higher-order elliptic systems
- 35J60: Nonlinear elliptic equations
- 35J61: Semilinear elliptic equations
- 35J62: Quasilinear elliptic equations
- 35J65: Nonlinear boundary value problems for linear elliptic equations
- 35J66: Nonlinear boundary value problems for nonlinear elliptic equations
- 35J67: Boundary values of solutions to elliptic equations
- 35J70: Degenerate elliptic equations
- 35J75: Singular elliptic equations
- 35J86: Linear elliptic unilateral problems and linear elliptic variational inequalities [See also 35R35, 49J40]
- 35J87: Nonlinear elliptic unilateral problems and nonlinear elliptic variational inequalities [See also 35R35, 49J40]
- 35J88: Systems of elliptic variational inequalities [See also 35R35, 49J40]
- 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
- 35J92: Quasilinear elliptic equations with $p$-Laplacian
- 35J93: Quasilinear elliptic equations with mean curvature operator
- 35J96: Elliptic Monge-Ampère equations
- 35J99: None of the above, but in this section

- 35Kxx: Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35]
- 35K05: Heat equation
- 35K08: Heat kernel
- 35K10: Second-order parabolic equations
- 35K15: Initial value problems for second-order parabolic equations
- 35K20: Initial-boundary value problems for second-order parabolic equations
- 35K25: Higher-order parabolic equations
- 35K30: Initial value problems for higher-order parabolic equations
- 35K35: Initial-boundary value problems for higher-order parabolic equations
- 35K40: Second-order parabolic systems
- 35K41: Higher-order parabolic systems
- 35K45: Initial value problems for second-order parabolic systems
- 35K46: Initial value problems for higher-order parabolic systems
- 35K51: Initial-boundary value problems for second-order parabolic systems
- 35K52: Initial-boundary value problems for higher-order parabolic systems
- 35K55: Nonlinear parabolic equations
- 35K57: Reaction-diffusion equations
- 35K58: Semilinear parabolic equations
- 35K59: Quasilinear parabolic equations
- 35K60: Nonlinear initial value problems for linear parabolic equations
- 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations
- 35K65: Degenerate parabolic equations
- 35K67: Singular parabolic equations
- 35K70: Ultraparabolic equations, pseudoparabolic equations, etc.
- 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]
- 35K86: Nonlinear parabolic unilateral problems and nonlinear parabolic variational inequalities [See also 35R35, 49J40]
- 35K87: Systems of parabolic variational inequalities [See also 35R35, 49J40]
- 35K90: Abstract parabolic equations
- 35K91: Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
- 35K92: Quasilinear parabolic equations with $p$-Laplacian
- 35K93: Quasilinear parabolic equations with mean curvature operator
- 35K96: Parabolic Monge-Ampère equations
- 35K99: None of the above, but in this section

- 35Lxx: Hyperbolic equations and systems [See also 58J45]
- 35L02: First-order hyperbolic equations
- 35L03: Initial value problems for first-order hyperbolic equations
- 35L04: Initial-boundary value problems for first-order hyperbolic equations
- 35L05: Wave equation
- 35L10: Second-order hyperbolic equations
- 35L15: Initial value problems for second-order hyperbolic equations
- 35L20: Initial-boundary value problems for second-order hyperbolic equations
- 35L25: Higher-order hyperbolic equations
- 35L30: Initial value problems for higher-order hyperbolic equations
- 35L35: Initial-boundary value problems for higher-order hyperbolic equations
- 35L40: First-order hyperbolic systems
- 35L45: Initial value problems for first-order hyperbolic systems
- 35L50: Initial-boundary value problems for first-order hyperbolic systems
- 35L51: Second-order hyperbolic systems
- 35L52: Initial value problems for second-order hyperbolic systems
- 35L53: Initial-boundary value problems for second-order hyperbolic systems
- 35L55: Higher-order hyperbolic systems
- 35L56: Initial value problems for higher-order hyperbolic systems
- 35L57: Initial-boundary value problems for higher-order hyperbolic systems
- 35L60: Nonlinear first-order hyperbolic equations
- 35L65: Conservation laws
- 35L67: Shocks and singularities [See also 58Kxx, 76L05]
- 35L70: Nonlinear second-order hyperbolic equations
- 35L71: Semilinear second-order hyperbolic equations
- 35L72: Quasilinear second-order hyperbolic equations
- 35L75: Nonlinear higher-order hyperbolic equations
- 35L76: Semilinear higher-order hyperbolic equations
- 35L77: Quasilinear higher-order hyperbolic equations
- 35L80: Degenerate hyperbolic equations
- 35L81: Singular hyperbolic equations
- 35L82: Pseudohyperbolic equations
- 35L85: Linear hyperbolic unilateral problems and linear hyperbolic variational inequalities [See also 35R35, 49J40]
- 35L86: Nonlinear hyperbolic unilateral problems and nonlinear hyperbolic variational inequalities [See also 35R35, 49J40]
- 35L87: Unilateral problems and variational inequalities for hyperbolic systems [See also 35R35, 49J40]
- 35L90: Abstract hyperbolic equations
- 35L99: None of the above, but in this section

- 35Mxx: Equations and systems of special type (mixed, composite, etc.)
- 35M10: Equations of mixed type
- 35M11: Initial value problems for equations of mixed type
- 35M12: Boundary value problems for equations of mixed type
- 35M13: Initial-boundary value problems for equations of mixed type
- 35M30: Systems of mixed type
- 35M31: Initial value problems for systems of mixed type
- 35M32: Boundary value problems for systems of mixed type
- 35M33: Initial-boundary value problems for systems of mixed type
- 35M85: Linear unilateral problems and variational inequalities of mixed type [See also 35R35, 49J40]
- 35M86: Nonlinear unilateral problems and nonlinear variational inequalities of mixed type [See also 35R35, 49J40]
- 35M87: Systems of variational inequalities of mixed type [See also 35R35, 49J40]
- 35M99: None of the above, but in this section

- 35Nxx: Overdetermined systems [See also 58Hxx, 58J10, 58J15]
- 35N05: Overdetermined systems with constant coefficients
- 35N10: Overdetermined systems with variable coefficients
- 35N15: $\overline\partial$-Neumann problem and generalizations; formal complexes [See also 32W05, 32W10, 58J10]
- 35N20: Overdetermined initial value problems
- 35N25: Overdetermined boundary value problems
- 35N30: Overdetermined initial-boundary value problems
- 35N99: None of the above, but in this section

- 35Pxx: Spectral theory and eigenvalue problems [See also 47Axx, 47Bxx, 47F05]
- 35P05: General topics in linear spectral theory
- 35P10: Completeness of eigenfunctions, eigenfunction expansions
- 35P15: Estimation of eigenvalues, upper and lower bounds
- 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
- 35P25: Scattering theory [See also 47A40]
- 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
- 35P99: None of the above, but in this section

- 35Qxx: Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05]
- 35Q05: Euler-Poisson-Darboux equations
- 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20]
- 35Q20: Boltzmann equations
- 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
- 35Q31: Euler equations [See also 76D05, 76D07, 76N10]
- 35Q35: PDEs in connection with fluid mechanics
- 35Q40: PDEs in connection with quantum mechanics
- 35Q41: Time-dependent Schrödinger equations, Dirac equations
- 35Q51: Soliton-like equations [See also 37K40]
- 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
- 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
- 35Q56: Ginzburg-Landau equations
- 35Q60: PDEs in connection with optics and electromagnetic theory
- 35Q61: Maxwell equations
- 35Q62: PDEs in connection with statistics
- 35Q68: PDEs in connection with computer science
- 35Q70: PDEs in connection with mechanics of particles and systems
- 35Q74: PDEs in connection with mechanics of deformable solids
- 35Q75: PDEs in connection with relativity and gravitational theory
- 35Q76: Einstein equations
- 35Q79: PDEs in connection with classical thermodynamics and heat transfer
- 35Q82: PDEs in connection with statistical mechanics
- 35Q83: Vlasov-like equations
- 35Q84: Fokker-Planck equations
- 35Q85: PDEs in connection with astronomy and astrophysics
- 35Q86: PDEs in connection with geophysics
- 35Q90: PDEs in connection with mathematical programming
- 35Q91: PDEs in connection with game theory, economics, social and behavioral sciences
- 35Q92: PDEs in connection with biology and other natural sciences
- 35Q93: PDEs in connection with control and optimization
- 35Q94: PDEs in connection with information and communication
- 35Q99: None of the above, but in this section

- 35Rxx: Miscellaneous topics {For equations on manifolds, see 58Jxx; for manifolds of solutions, see 58Bxx; for stochastic PDE, see also 60H15}
- 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx]
- 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)
- 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc.
- 35R05: Partial differential equations with discontinuous coefficients or data
- 35R06: Partial differential equations with measure
- 35R09: Integro-partial differential equations [See also 45Kxx]
- 35R10: Partial functional-differential equations
- 35R11: Fractional partial differential equations
- 35R12: Impulsive partial differential equations
- 35R13: Fuzzy partial differential equations
- 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25]
- 35R20: Partial operator-differential equations (i.e., PDE on finite-dimensional spaces for abstract space valued functions) [See also 34Gxx, 47A50, 47D03, 47D06, 47D09, 47H20, 47Jxx]
- 35R25: Improperly posed problems
- 35R30: Inverse problems
- 35R35: Free boundary problems
- 35R37: Moving boundary problems
- 35R45: Partial differential inequalities
- 35R50: Partial differential equations of infinite order
- 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]
- 35R70: Partial differential equations with multivalued right-hand sides
- 35R99: None of the above, but in this section

- 35Sxx: Pseudodifferential operators and other generalizations of partial differential operators [See also 47G30, 58J40]
- 35S05: Pseudodifferential operators
- 35S10: Initial value problems for pseudodifferential operators
- 35S11: Initial-boundary value problems for pseudodifferential operators
- 35S15: Boundary value problems for pseudodifferential operators
- 35S30: Fourier integral operators
- 35S35: Topological aspects: intersection cohomology, stratified sets, etc. [See also 32C38, 32S40, 32S60, 58J15]
- 35S50: Paradifferential operators
- 35S99: None of the above, but in this section

- 37-XX: Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
- 37-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 37-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 37-02: Research exposition (monographs, survey articles)
- 37-03: Historical (must also be assigned at least one classification number from Section 01)
- 37-04: Explicit machine computation and programs (not the theory of computation or programming)
- 37-06: Proceedings, conferences, collections, etc.
- 37Axx: Ergodic theory [See also 28Dxx]
- 37A05: Measure-preserving transformations
- 37A10: One-parameter continuous families of measure-preserving transformations
- 37A15: General groups of measure-preserving transformations [See mainly 22Fxx]
- 37A17: Homogeneous flows [See also 22Fxx]
- 37A20: Orbit equivalence, cocycles, ergodic equivalence relations
- 37A25: Ergodicity, mixing, rates of mixing
- 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}
- 37A35: Entropy and other invariants, isomorphism, classification
- 37A40: Nonsingular (and infinite-measure preserving) transformations
- 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]
- 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
- 37A55: Relations with the theory of $C^*$-algebras [See mainly 46L55]
- 37A60: Dynamical systems in statistical mechanics [See also 82Cxx]
- 37A99: None of the above, but in this section

- 37Bxx: Topological dynamics [See also 54H20]
- 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
- 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]
- 37B15: Cellular automata [See also 68Q80]
- 37B20: Notions of recurrence
- 37B25: Lyapunov functions and stability; attractors, repellers
- 37B30: Index theory, Morse-Conley indices
- 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets
- 37B40: Topological entropy
- 37B45: Continua theory in dynamics
- 37B50: Multi-dimensional shifts of finite type, tiling dynamics
- 37B55: Nonautonomous dynamical systems
- 37B99: None of the above, but in this section

- 37Cxx: Smooth dynamical systems: general theory [See also 34Cxx, 34Dxx]
- 37C05: Smooth mappings and diffeomorphisms
- 37C10: Vector fields, flows, ordinary differential equations
- 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
- 37C20: Generic properties, structural stability
- 37C25: Fixed points, periodic points, fixed-point index theory
- 37C27: Periodic orbits of vector fields and flows
- 37C29: Homoclinic and heteroclinic orbits
- 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
- 37C35: Orbit growth
- 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx]
- 37C45: Dimension theory of dynamical systems
- 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
- 37C55: Periodic and quasiperiodic flows and diffeomorphisms
- 37C60: Nonautonomous smooth dynamical systems [See also 37B55]
- 37C65: Monotone flows
- 37C70: Attractors and repellers, topological structure
- 37C75: Stability theory
- 37C80: Symmetries, equivariant dynamical systems
- 37C85: Dynamics of group actions other than ${\bf Z}$ and ${\bf R}$, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]
- 37C99: None of the above, but in this section

- 37Dxx: Dynamical systems with hyperbolic behavior
- 37D05: Hyperbolic orbits and sets
- 37D10: Invariant manifold theory
- 37D15: Morse-Smale systems
- 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
- 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
- 37D30: Partially hyperbolic systems and dominated splittings
- 37D35: Thermodynamic formalism, variational principles, equilibrium states
- 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
- 37D45: Strange attractors, chaotic dynamics
- 37D50: Hyperbolic systems with singularities (billiards, etc.)
- 37D99: None of the above, but in this section

- 37Exx: Low-dimensional dynamical systems
- 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
- 37E10: Maps of the circle
- 37E15: Combinatorial dynamics (types of periodic orbits)
- 37E20: Universality, renormalization [See also 37F25]
- 37E25: Maps of trees and graphs
- 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
- 37E35: Flows on surfaces
- 37E40: Twist maps
- 37E45: Rotation numbers and vectors
- 37E99: None of the above, but in this section

- 37Fxx: Complex dynamical systems [See also 30D05, 32H50]
- 37F05: Relations and correspondences
- 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
- 37F15: Expanding maps; hyperbolicity; structural stability
- 37F20: Combinatorics and topology
- 37F25: Renormalization
- 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems
- 37F35: Conformal densities and Hausdorff dimension
- 37F40: Geometric limits
- 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
- 37F50: Small divisors, rotation domains and linearization; Fatou and Julia sets
- 37F75: Holomorphic foliations and vector fields [See also 32M25, 32S65, 34Mxx]
- 37F99: None of the above, but in this section

- 37Gxx: Local and nonlocal bifurcation theory [See also 34C23, 34K18]
- 37G05: Normal forms
- 37G10: Bifurcations of singular points
- 37G15: Bifurcations of limit cycles and periodic orbits
- 37G20: Hyperbolic singular points with homoclinic trajectories
- 37G25: Bifurcations connected with nontransversal intersection
- 37G30: Infinite nonwandering sets arising in bifurcations
- 37G35: Attractors and their bifurcations
- 37G40: Symmetries, equivariant bifurcation theory
- 37G99: None of the above, but in this section

- 37Hxx: Random dynamical systems [See also 15B52, 34D08, 34F05, 47B80, 70L05, 82C05, 93Exx]
- 37H05: Foundations, general theory of cocycles, algebraic ergodic theory [See also 37Axx]
- 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]
- 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx]
- 37H20: Bifurcation theory [See also 37Gxx]
- 37H99: None of the above, but in this section

- 37Jxx: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [See also 53Dxx, 70Fxx, 70Hxx]
- 37J05: General theory, relations with symplectic geometry and topology
- 37J10: Symplectic mappings, fixed points
- 37J15: Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
- 37J20: Bifurcation problems
- 37J25: Stability problems
- 37J30: Obstructions to integrability (nonintegrability criteria)
- 37J35: Completely integrable systems, topological structure of phase space, integration methods
- 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
- 37J45: Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
- 37J50: Action-minimizing orbits and measures
- 37J55: Contact systems [See also 53D10]
- 37J60: Nonholonomic dynamical systems [See also 70F25]
- 37J99: None of the above, but in this section

- 37Kxx: Infinite-dimensional Hamiltonian systems [See also 35Axx, 35Qxx]
- 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws
- 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)
- 37K15: Integration of completely integrable systems by inverse spectral and scattering methods
- 37K20: Relations with algebraic geometry, complex analysis, special functions [See also 14H70]
- 37K25: Relations with differential geometry
- 37K30: Relations with infinite-dimensional Lie algebras and other algebraic structures
- 37K35: Lie-Bäcklund and other transformations
- 37K40: Soliton theory, asymptotic behavior of solutions
- 37K45: Stability problems
- 37K50: Bifurcation problems
- 37K55: Perturbations, KAM for infinite-dimensional systems
- 37K60: Lattice dynamics [See also 37L60]
- 37K65: Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
- 37K99: None of the above, but in this section

- 37Lxx: Infinite-dimensional dissipative dynamical systems [See also 35Bxx, 35Qxx]
- 37L05: General theory, nonlinear semigroups, evolution equations
- 37L10: Normal forms, center manifold theory, bifurcation theory
- 37L15: Stability problems
- 37L20: Symmetries
- 37L25: Inertial manifolds and other invariant attracting sets
- 37L30: Attractors and their dimensions, Lyapunov exponents
- 37L40: Invariant measures
- 37L45: Hyperbolicity; Lyapunov functions
- 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems
- 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15]
- 37L60: Lattice dynamics [See also 37K60]
- 37L65: Special approximation methods (nonlinear Galerkin, etc.)
- 37L99: None of the above, but in this section

- 37Mxx: Approximation methods and numerical treatment of dynamical systems [See also 65Pxx]
- 37M05: Simulation
- 37M10: Time series analysis
- 37M15: Symplectic integrators
- 37M20: Computational methods for bifurcation problems
- 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy)
- 37M99: None of the above, but in this section

- 37Nxx: Applications
- 37N05: Dynamical systems in classical and celestial mechanics [See mainly 70Fxx, 70Hxx, 70Kxx]
- 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology [See mainly 76-XX, especially 76D05, 76F20, 86A05, 86A10]
- 37N15: Dynamical systems in solid mechanics [See mainly 74Hxx]
- 37N20: Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
- 37N25: Dynamical systems in biology [See mainly 92-XX, but also 91-XX]
- 37N30: Dynamical systems in numerical analysis
- 37N35: Dynamical systems in control
- 37N40: Dynamical systems in optimization and economics
- 37N99: None of the above, but in this section

- 37Pxx: Arithmetic and non-Archimedean dynamical systems [See also 11S82, 37A45]
- 37P05: Polynomial and rational maps
- 37P10: Analytic and meromorphic maps
- 37P15: Global ground fields
- 37P20: Non-Archimedean local ground fields
- 37P25: Finite ground fields
- 37P30: Height functions; Green functions; invariant measures [See also 11G50, 14G40]
- 37P35: Arithmetic properties of periodic points
- 37P40: Non-Archimedean Fatou and Julia sets
- 37P45: Families and moduli spaces
- 37P50: Dynamical systems on Berkovich spaces
- 37P55: Arithmetic dynamics on general algebraic varieties
- 37P99: None of the above, but in this section

- 39-XX: Difference and functional equations
- 39-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 39-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 39-02: Research exposition (monographs, survey articles)
- 39-03: Historical (must also be assigned at least one classification number from Section 01)
- 39-04: Explicit machine computation and programs (not the theory of computation or programming)
- 39-06: Proceedings, conferences, collections, etc.
- 39Axx: Difference equations {For dynamical systems, see 37-XX; for dynamic equations on time scales, see 34N05}
- 39A05: General theory
- 39A06: Linear equations
- 39A10: Difference equations, additive
- 39A12: Discrete version of topics in analysis
- 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx]
- 39A14: Partial difference equations
- 39A20: Multiplicative and other generalized difference equations, e.g. of Lyness type
- 39A21: Oscillation theory
- 39A22: Growth, boundedness, comparison of solutions
- 39A23: Periodic solutions
- 39A24: Almost periodic solutions
- 39A28: Bifurcation theory
- 39A30: Stability theory
- 39A33: Complex (chaotic) behavior of solutions
- 39A45: Equations in the complex domain
- 39A50: Stochastic difference equations
- 39A60: Applications
- 39A70: Difference operators [See also 47B39]
- 39A99: None of the above, but in this section

- 39Bxx: Functional equations and inequalities [See also 30D05]
- 39B05: General
- 39B12: Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX]
- 39B22: Equations for real functions [See also 26A51, 26B25]
- 39B32: Equations for complex functions [See also 30D05]
- 39B42: Matrix and operator equations [See also 47Jxx]
- 39B52: Equations for functions with more general domains and/or ranges
- 39B55: Orthogonal additivity and other conditional equations
- 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx]
- 39B72: Systems of functional equations and inequalities
- 39B82: Stability, separation, extension, and related topics [See also 46A22]
- 39B99: None of the above, but in this section

- 40-XX: Sequences, series, summability
- 40-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 40-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 40-02: Research exposition (monographs, survey articles)
- 40-03: Historical (must also be assigned at least one classification number from Section 01)
- 40-04: Explicit machine computation and programs (not the theory of computation or programming)
- 40-06: Proceedings, conferences, collections, etc.
- 40Axx: Convergence and divergence of infinite limiting processes
- 40A05: Convergence and divergence of series and sequences
- 40A10: Convergence and divergence of integrals
- 40A15: Convergence and divergence of continued fractions [See also 30B70]
- 40A20: Convergence and divergence of infinite products
- 40A25: Approximation to limiting values (summation of series, etc.) {For the Euler-Maclaurin summation formula, see 65B15}
- 40A30: Convergence and divergence of series and sequences of functions
- 40A35: Ideal and statistical convergence [See also 40G15]
- 40A99: None of the above, but in this section

- 40Bxx: Multiple sequences and series
- 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
- 40B99: None of the above, but in this section

- 40Cxx: General summability methods
- 40C05: Matrix methods
- 40C10: Integral methods
- 40C15: Function-theoretic methods (including power series methods and semicontinuous methods)
- 40C99: None of the above, but in this section

- 40Dxx: Direct theorems on summability
- 40D05: General theorems
- 40D09: Structure of summability fields
- 40D10: Tauberian constants and oscillation limits
- 40D15: Convergence factors and summability factors
- 40D20: Summability and bounded fields of methods
- 40D25: Inclusion and equivalence theorems
- 40D99: None of the above, but in this section

- 40Exx: Inversion theorems
- 40E05: Tauberian theorems, general
- 40E10: Growth estimates
- 40E15: Lacunary inversion theorems
- 40E20: Tauberian constants
- 40E99: None of the above, but in this section

- 40Fxx: Absolute and strong summability (should also be assigned at least one other classification number in Section 40)
- 40F05: Absolute and strong summability (should also be assigned at least one other classification number in Section 40)
- 40F99: None of the above, but in this section

- 40Gxx: Special methods of summability
- 40G05: Cesàro, Euler, Nörlund and Hausdorff methods
- 40G10: Abel, Borel and power series methods
- 40G15: Summability methods using statistical convergence [See also 40A35]
- 40G99: None of the above, but in this section

- 40Hxx: Functional analytic methods in summability
- 40H05: Functional analytic methods in summability
- 40H99: None of the above, but in this section

- 40Jxx: Summability in abstract structures [See also 43A55, 46A35, 46B15]

- 41-XX: Approximations and expansions {For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx}
- 41-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 41-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 41-02: Research exposition (monographs, survey articles)
- 41-03: Historical (must also be assigned at least one classification number from Section 01)
- 41-04: Explicit machine computation and programs (not the theory of computation or programming)
- 41-06: Proceedings, conferences, collections, etc.
- 41Axx: Approximations and expansions {For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx}
- 41A05: Interpolation [See also 42A15 and 65D05]
- 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10}
- 41A15: Spline approximation
- 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities)
- 41A20: Approximation by rational functions
- 41A21: Padé approximation
- 41A25: Rate of convergence, degree of approximation
- 41A27: Inverse theorems
- 41A28: Simultaneous approximation
- 41A29: Approximation with constraints
- 41A30: Approximation by other special function classes
- 41A35: Approximation by operators (in particular, by integral operators)
- 41A36: Approximation by positive operators
- 41A40: Saturation
- 41A44: Best constants
- 41A45: Approximation by arbitrary linear expressions
- 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy
- 41A50: Best approximation, Chebyshev systems
- 41A52: Uniqueness of best approximation
- 41A55: Approximate quadratures
- 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
- 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15]
- 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)
- 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
- 41A80: Remainders in approximation formulas
- 41A99: None of the above, but in this section

- 42-XX: Harmonic analysis on Euclidean spaces
- 42-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 42-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 42-02: Research exposition (monographs, survey articles)
- 42-03: Historical (must also be assigned at least one classification number from Section 01)
- 42-04: Explicit machine computation and programs (not the theory of computation or programming)
- 42-06: Proceedings, conferences, collections, etc.
- 42Axx: Harmonic analysis in one variable
- 42A05: Trigonometric polynomials, inequalities, extremal problems
- 42A10: Trigonometric approximation
- 42A15: Trigonometric interpolation
- 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series {For automorphic theory, see mainly 11F30}
- 42A20: Convergence and absolute convergence of Fourier and trigonometric series
- 42A24: Summability and absolute summability of Fourier and trigonometric series
- 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
- 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
- 42A45: Multipliers
- 42A50: Conjugate functions, conjugate series, singular integrals
- 42A55: Lacunary series of trigonometric and other functions; Riesz products
- 42A61: Probabilistic methods
- 42A63: Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
- 42A65: Completeness of sets of functions
- 42A70: Trigonometric moment problems
- 42A75: Classical almost periodic functions, mean periodic functions [See also 43A60]
- 42A82: Positive definite functions
- 42A85: Convolution, factorization
- 42A99: None of the above, but in this section

- 42Bxx: Harmonic analysis in several variables {For automorphic theory, see mainly 11F30}
- 42B05: Fourier series and coefficients
- 42B08: Summability
- 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
- 42B15: Multipliers
- 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
- 42B25: Maximal functions, Littlewood-Paley theory
- 42B30: $H^p$-spaces
- 42B35: Function spaces arising in harmonic analysis
- 42B37: Harmonic analysis and PDE [See also 35-XX]
- 42B99: None of the above, but in this section

- 42Cxx: Nontrigonometric harmonic analysis
- 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
- 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
- 42C15: General harmonic expansions, frames
- 42C20: Other transformations of harmonic type
- 42C25: Uniqueness and localization for orthogonal series
- 42C30: Completeness of sets of functions
- 42C40: Wavelets and other special systems
- 42C99: None of the above, but in this section

- 43-XX: Abstract harmonic analysis {For other analysis on topological and Lie groups, see 22Exx}
- 43-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 43-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 43-02: Research exposition (monographs, survey articles)
- 43-03: Historical (must also be assigned at least one classification number from Section 01)
- 43-04: Explicit machine computation and programs (not the theory of computation or programming)
- 43-06: Proceedings, conferences, collections, etc.
- 43Axx: Abstract harmonic analysis {For other analysis on topological and Lie groups, see 22Exx}
- 43A05: Measures on groups and semigroups, etc.
- 43A07: Means on groups, semigroups, etc.; amenable groups
- 43A10: Measure algebras on groups, semigroups, etc.
- 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
- 43A17: Analysis on ordered groups, $H^p$-theory
- 43A20: $L^1$-algebras on groups, semigroups, etc.
- 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
- 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
- 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
- 43A32: Other transforms and operators of Fourier type
- 43A35: Positive definite functions on groups, semigroups, etc.
- 43A40: Character groups and dual objects
- 43A45: Spectral synthesis on groups, semigroups, etc.
- 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
- 43A50: Convergence of Fourier series and of inverse transforms
- 43A55: Summability methods on groups, semigroups, etc. [See also 40J05]
- 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
- 43A62: Hypergroups
- 43A65: Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45]
- 43A70: Analysis on specific locally compact and other abelian groups [See also 11R56, 22B05]
- 43A75: Analysis on specific compact groups
- 43A77: Analysis on general compact groups
- 43A80: Analysis on other specific Lie groups [See also 22Exx]
- 43A85: Analysis on homogeneous spaces
- 43A90: Spherical functions [See also 22E45, 22E46, 33C55]
- 43A95: Categorical methods [See also 46Mxx]
- 43A99: None of the above, but in this section

- 44-XX: Integral transforms, operational calculus {For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For numerical methods, see 65R10}
- 44-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 44-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 44-02: Research exposition (monographs, survey articles)
- 44-03: Historical (must also be assigned at least one classification number from Section 01)
- 44-04: Explicit machine computation and programs (not the theory of computation or programming)
- 44-06: Proceedings, conferences, collections, etc.
- 44Axx: Integral transforms, operational calculus {For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For numerical methods, see 65R10}
- 44A05: General transforms [See also 42A38]
- 44A10: Laplace transform
- 44A12: Radon transform [See also 92C55]
- 44A15: Special transforms (Legendre, Hilbert, etc.)
- 44A20: Transforms of special functions
- 44A30: Multiple transforms
- 44A35: Convolution
- 44A40: Calculus of Mikusiński and other operational calculi
- 44A45: Classical operational calculus
- 44A55: Discrete operational calculus
- 44A60: Moment problems
- 44A99: None of the above, but in this section

- 45-XX: Integral equations
- 45-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 45-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 45-02: Research exposition (monographs, survey articles)
- 45-03: Historical (must also be assigned at least one classification number from Section 01)
- 45-04: Explicit machine computation and programs (not the theory of computation or programming)
- 45-06: Proceedings, conferences, collections, etc.
- 45Axx: Linear integral equations
- 45A05: Linear integral equations
- 45A99: None of the above, but in this section

- 45Bxx: Fredholm integral equations
- 45B05: Fredholm integral equations
- 45B99: None of the above, but in this section

- 45Cxx: Eigenvalue problems [See also 34Lxx, 35Pxx, 45P05, 47A75]
- 45Dxx: Volterra integral equations [See also 34A12]
- 45D05: Volterra integral equations [See also 34A12]
- 45D99: None of the above, but in this section

- 45Exx: Singular integral equations [See also 30E20, 30E25, 44A15, 44A35]
- 45Fxx: Systems of linear integral equations
- 45F05: Systems of nonsingular linear integral equations
- 45F10: Dual, triple, etc., integral and series equations
- 45F15: Systems of singular linear integral equations
- 45F99: None of the above, but in this section

- 45Gxx: Nonlinear integral equations [See also 47H30, 47Jxx]
- 45G05: Singular nonlinear integral equations
- 45G10: Other nonlinear integral equations
- 45G15: Systems of nonlinear integral equations
- 45G99: None of the above, but in this section

- 45Hxx: Miscellaneous special kernels [See also 44A15]
- 45H05: Miscellaneous special kernels [See also 44A15]
- 45H99: None of the above, but in this section

- 45Jxx: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20]
- 45Kxx: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
- 45Lxx: Theoretical approximation of solutions {For numerical analysis, see 65Rxx}
- 45L05: Theoretical approximation of solutions {For numerical analysis, see 65Rxx}
- 45L99: None of the above, but in this section

- 45Mxx: Qualitative behavior
- 45M05: Asymptotics
- 45M10: Stability theory
- 45M15: Periodic solutions
- 45M20: Positive solutions
- 45M99: None of the above, but in this section

- 45Nxx: Abstract integral equations, integral equations in abstract spaces
- 45N05: Abstract integral equations, integral equations in abstract spaces
- 45N99: None of the above, but in this section

- 45Pxx: Integral operators [See also 47B38, 47G10]
- 45Qxx: Inverse problems
- 45Q05: Inverse problems
- 45Q99: None of the above, but in this section

- 45Rxx: Random integral equations [See also 60H20]
- 45R05: Random integral equations [See also 60H20]
- 45R99: None of the above, but in this section

- 46-XX: Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx}
- 46-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 46-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 46-02: Research exposition (monographs, survey articles)
- 46-03: Historical (must also be assigned at least one classification number from Section 01)
- 46-04: Explicit machine computation and programs (not the theory of computation or programming)
- 46-06: Proceedings, conferences, collections, etc.
- 46Axx: Topological linear spaces and related structures {For function spaces, see 46Exx}
- 46A03: General theory of locally convex spaces
- 46A04: Locally convex Fréchet spaces and (DF)-spaces
- 46A08: Barrelled spaces, bornological spaces
- 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
- 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40]
- 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
- 46A17: Bornologies and related structures; Mackey convergence, etc.
- 46A19: Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${\bf R}$, etc.)
- 46A20: Duality theory
- 46A22: Theorems of Hahn-Banach type; extension and lifting of functionals and operators [See also 46M10]
- 46A25: Reflexivity and semi-reflexivity [See also 46B10]
- 46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)
- 46A32: Spaces of linear operators; topological tensor products; approximation properties [See also 46B28, 46M05, 47L05, 47L20]
- 46A35: Summability and bases [See also 46B15]
- 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]
- 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45]
- 46A50: Compactness in topological linear spaces; angelic spaces, etc.
- 46A55: Convex sets in topological linear spaces; Choquet theory [See also 52A07]
- 46A61: Graded Fréchet spaces and tame operators
- 46A63: Topological invariants ((DN), ($\Omega$), etc.)
- 46A70: Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
- 46A80: Modular spaces
- 46A99: None of the above, but in this section

- 46Bxx: Normed linear spaces and Banach spaces; Banach lattices {For function spaces, see 46Exx}
- 46B03: Isomorphic theory (including renorming) of Banach spaces
- 46B04: Isometric theory of Banach spaces
- 46B06: Asymptotic theory of Banach spaces [See also 52A23]
- 46B07: Local theory of Banach spaces
- 46B08: Ultraproduct techniques in Banach space theory [See also 46M07]
- 46B09: Probabilistic methods in Banach space theory [See also 60Bxx]
- 46B10: Duality and reflexivity [See also 46A25]
- 46B15: Summability and bases [See also 46A35]
- 46B20: Geometry and structure of normed linear spaces
- 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10]
- 46B25: Classical Banach spaces in the general theory
- 46B26: Nonseparable Banach spaces
- 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]
- 46B40: Ordered normed spaces [See also 46A40, 46B42]
- 46B42: Banach lattices [See also 46A40, 46B40]
- 46B45: Banach sequence spaces [See also 46A45]
- 46B50: Compactness in Banach (or normed) spaces
- 46B70: Interpolation between normed linear spaces [See also 46M35]
- 46B80: Nonlinear classification of Banach spaces; nonlinear quotients
- 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science [See also 05C12, 68Rxx]
- 46B99: None of the above, but in this section

- 46Cxx: Inner product spaces and their generalizations, Hilbert spaces {For function spaces, see 46Exx}
- 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
- 46C07: Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.) [See also 46B70, 46M35]
- 46C15: Characterizations of Hilbert spaces
- 46C20: Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) [See also 47B50]
- 46C50: Generalizations of inner products (semi-inner products, partial inner products, etc.)
- 46C99: None of the above, but in this section

- 46Exx: Linear function spaces and their duals [See also 30H05, 32A38, 46F05] {For function algebras, see 46J10}
- 46E05: Lattices of continuous, differentiable or analytic functions
- 46E10: Topological linear spaces of continuous, differentiable or analytic functions
- 46E15: Banach spaces of continuous, differentiable or analytic functions
- 46E20: Hilbert spaces of continuous, differentiable or analytic functions
- 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
- 46E25: Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15}
- 46E27: Spaces of measures [See also 28A33, 46Gxx]
- 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
- 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
- 46E39: Sobolev (and similar kinds of) spaces of functions of discrete variables
- 46E40: Spaces of vector- and operator-valued functions
- 46E50: Spaces of differentiable or holomorphic functions on infinite-dimensional spaces [See also 46G20, 46G25, 47H60]
- 46E99: None of the above, but in this section

- 46Fxx: Distributions, generalized functions, distribution spaces [See also 46T30]
- 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35]
- 46F10: Operations with distributions
- 46F12: Integral transforms in distribution spaces [See also 42-XX, 44-XX]
- 46F15: Hyperfunctions, analytic functionals [See also 32A25, 32A45, 32C35, 58J15]
- 46F20: Distributions and ultradistributions as boundary values of analytic functions [See also 30D40, 30E25, 32A40]
- 46F25: Distributions on infinite-dimensional spaces [See also 58C35]
- 46F30: Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
- 46F99: None of the above, but in this section

- 46Gxx: Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces) [See also 28-XX, 46Txx]
- 46G05: Derivatives [See also 46T20, 58C20, 58C25]
- 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]
- 46G12: Measures and integration on abstract linear spaces [See also 28C20, 46T12]
- 46G15: Functional analytic lifting theory [See also 28A51]
- 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]
- 46G25: (Spaces of) multilinear mappings, polynomials [See also 46E50, 46G20, 47H60]
- 46G99: None of the above, but in this section

- 46Hxx: Topological algebras, normed rings and algebras, Banach algebras {For group algebras, convolution algebras and measure algebras, see 43A10, 43A20}
- 46H05: General theory of topological algebras
- 46H10: Ideals and subalgebras
- 46H15: Representations of topological algebras
- 46H20: Structure, classification of topological algebras
- 46H25: Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
- 46H30: Functional calculus in topological algebras [See also 47A60]
- 46H35: Topological algebras of operators [See mainly 47Lxx]
- 46H40: Automatic continuity
- 46H70: Nonassociative topological algebras [See also 46K70, 46L70]
- 46H99: None of the above, but in this section

- 46Jxx: Commutative Banach algebras and commutative topological algebras [See also 46E25]
- 46J05: General theory of commutative topological algebras
- 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
- 46J15: Banach algebras of differentiable or analytic functions, $H^p$-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30]
- 46J20: Ideals, maximal ideals, boundaries
- 46J25: Representations of commutative topological algebras
- 46J30: Subalgebras
- 46J40: Structure, classification of commutative topological algebras
- 46J45: Radical Banach algebras
- 46J99: None of the above, but in this section

- 46Kxx: Topological (rings and) algebras with an involution [See also 16W10]
- 46K05: General theory of topological algebras with involution
- 46K10: Representations of topological algebras with involution
- 46K15: Hilbert algebras
- 46K50: Nonselfadjoint (sub)algebras in algebras with involution
- 46K70: Nonassociative topological algebras with an involution [See also 46H70, 46L70]
- 46K99: None of the above, but in this section

- 46Lxx: Selfadjoint operator algebras ($C^*$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx]
- 46L05: General theory of $C^*$-algebras
- 46L06: Tensor products of $C^*$-algebras
- 46L07: Operator spaces and completely bounded maps [See also 47L25]
- 46L08: $C^*$-modules
- 46L09: Free products of $C^*$-algebras
- 46L10: General theory of von Neumann algebras
- 46L30: States
- 46L35: Classifications of $C^*$-algebras
- 46L36: Classification of factors
- 46L37: Subfactors and their classification
- 46L40: Automorphisms
- 46L45: Decomposition theory for $C^*$-algebras
- 46L51: Noncommutative measure and integration
- 46L52: Noncommutative function spaces
- 46L53: Noncommutative probability and statistics
- 46L54: Free probability and free operator algebras
- 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
- 46L57: Derivations, dissipations and positive semigroups in $C^*$-algebras
- 46L60: Applications of selfadjoint operator algebras to physics [See also 46N50, 46N55, 47L90, 81T05, 82B10, 82C10]
- 46L65: Quantizations, deformations
- 46L70: Nonassociative selfadjoint operator algebras [See also 46H70, 46K70]
- 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
- 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]
- 46L87: Noncommutative differential geometry [See also 58B32, 58B34, 58J22]
- 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]
- 46L99: None of the above, but in this section

- 46Mxx: Methods of category theory in functional analysis [See also 18-XX]
- 46M05: Tensor products [See also 46A32, 46B28, 47A80]
- 46M07: Ultraproducts [See also 46B08, 46S20]
- 46M10: Projective and injective objects [See also 46A22]
- 46M15: Categories, functors {For $K$-theory, EXT, etc., see 19K33, 46L80, 46M18, 46M20}
- 46M18: Homological methods (exact sequences, right inverses, lifting, etc.)
- 46M20: Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.) [See also 14F05, 18Fxx, 19Kxx, 32Cxx, 32Lxx, 46L80, 46M15, 46M18, 55Rxx]
- 46M35: Abstract interpolation of topological vector spaces [See also 46B70]
- 46M40: Inductive and projective limits [See also 46A13]
- 46M99: None of the above, but in this section

- 46Nxx: Miscellaneous applications of functional analysis [See also 47Nxx]
- 46N10: Applications in optimization, convex analysis, mathematical programming, economics
- 46N20: Applications to differential and integral equations
- 46N30: Applications in probability theory and statistics
- 46N40: Applications in numerical analysis [See also 65Jxx]
- 46N50: Applications in quantum physics
- 46N55: Applications in statistical physics
- 46N60: Applications in biology and other sciences
- 46N99: None of the above, but in this section

- 46Sxx: Other (nonclassical) types of functional analysis [See also 47Sxx]
- 46S10: Functional analysis over fields other than ${\bf R}$ or ${\bf C}$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05]
- 46S20: Nonstandard functional analysis [See also 03H05]
- 46S30: Constructive functional analysis [See also 03F60]
- 46S40: Fuzzy functional analysis [See also 03E72]
- 46S50: Functional analysis in probabilistic metric linear spaces
- 46S60: Functional analysis on superspaces (supermanifolds) or graded spaces [See also 58A50 and 58C50]
- 46S99: None of the above, but in this section

- 46Txx: Nonlinear functional analysis [See also 47Hxx, 47Jxx, 58Cxx, 58Dxx]
- 46T05: Infinite-dimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx]
- 46T10: Manifolds of mappings
- 46T12: Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds [See also 28Cxx, 46G12, 60-XX]
- 46T20: Continuous and differentiable maps [See also 46G05]
- 46T25: Holomorphic maps [See also 46G20]
- 46T30: Distributions and generalized functions on nonlinear spaces [See also 46Fxx]
- 46T99: None of the above, but in this section

- 47-XX: Operator theory
- 47-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 47-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 47-02: Research exposition (monographs, survey articles)
- 47-03: Historical (must also be assigned at least one classification number from Section 01)
- 47-04: Explicit machine computation and programs (not the theory of computation or programming)
- 47-06: Proceedings, conferences, collections, etc.
- 47Axx: General theory of linear operators
- 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
- 47A06: Linear relations (multivalued linear operators)
- 47A07: Forms (bilinear, sesquilinear, multilinear)
- 47A10: Spectrum, resolvent
- 47A11: Local spectral properties
- 47A12: Numerical range, numerical radius
- 47A13: Several-variable operator theory (spectral, Fredholm, etc.)
- 47A15: Invariant subspaces [See also 47A46]
- 47A16: Cyclic vectors, hypercyclic and chaotic operators
- 47A20: Dilations, extensions, compressions
- 47A25: Spectral sets
- 47A30: Norms (inequalities, more than one norm, etc.)
- 47A35: Ergodic theory [See also 28Dxx, 37Axx]
- 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
- 47A45: Canonical models for contractions and nonselfadjoint operators
- 47A46: Chains (nests) of projections or of invariant subspaces, integrals along chains, etc.
- 47A48: Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
- 47A50: Equations and inequalities involving linear operators, with vector unknowns
- 47A52: Ill-posed problems, regularization [See also 35R25, 47J06, 65F22, 65J20, 65L08, 65M30, 65R30]
- 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]
- 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
- 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
- 47A57: Operator methods in interpolation, moment and extension problems [See also 30E05, 42A70, 42A82, 44A60]
- 47A58: Operator approximation theory
- 47A60: Functional calculus
- 47A62: Equations involving linear operators, with operator unknowns
- 47A63: Operator inequalities
- 47A64: Operator means, shorted operators, etc.
- 47A65: Structure theory
- 47A66: Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal operators
- 47A67: Representation theory
- 47A68: Factorization theory (including Wiener-Hopf and spectral factorizations)
- 47A70: (Generalized) eigenfunction expansions; rigged Hilbert spaces
- 47A75: Eigenvalue problems [See also 47J10, 49R05]
- 47A80: Tensor products of operators [See also 46M05]
- 47A99: None of the above, but in this section

- 47Bxx: Special classes of linear operators
- 47B06: Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
- 47B07: Operators defined by compactness properties
- 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]
- 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)
- 47B20: Subnormal operators, hyponormal operators, etc.
- 47B25: Symmetric and selfadjoint operators (unbounded)
- 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22]
- 47B33: Composition operators
- 47B34: Kernel operators
- 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
- 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations
- 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
- 47B38: Operators on function spaces (general)
- 47B39: Difference operators [See also 39A70]
- 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
- 47B44: Accretive operators, dissipative operators, etc.
- 47B47: Commutators, derivations, elementary operators, etc.
- 47B48: Operators on Banach algebras
- 47B49: Transformers, preservers (operators on spaces of operators)
- 47B50: Operators on spaces with an indefinite metric [See also 46C50]
- 47B60: Operators on ordered spaces
- 47B65: Positive operators and order-bounded operators
- 47B80: Random operators [See also 47H40, 60H25]
- 47B99: None of the above, but in this section

- 47Cxx: Individual linear operators as elements of algebraic systems
- 47C05: Operators in algebras
- 47C10: Operators in ${}^*$-algebras
- 47C15: Operators in $C^*$- or von Neumann algebras
- 47C99: None of the above, but in this section

- 47Dxx: Groups and semigroups of linear operators, their generalizations and applications
- 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}
- 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]
- 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}
- 47D08: Schrödinger and Feynman-Kac semigroups
- 47D09: Operator sine and cosine functions and higher-order Cauchy problems [See also 34G10]
- 47D60: $C$-semigroups, regularized semigroups
- 47D62: Integrated semigroups
- 47D99: None of the above, but in this section

- 47Exx: Ordinary differential operators [See also 34Bxx, 34Lxx]
- 47Fxx: Partial differential operators [See also 35Pxx, 58Jxx]
- 47Gxx: Integral, integro-differential, and pseudodifferential operators [See also 58Jxx]
- 47Hxx: Nonlinear operators and their properties {For global and geometric aspects, see 49J53, 58-XX, especially 58Cxx}
- 47H04: Set-valued operators [See also 28B20, 54C60, 58C06]
- 47H05: Monotone operators and generalizations
- 47H06: Accretive operators, dissipative operators, etc.
- 47H07: Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
- 47H08: Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
- 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
- 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]
- 47H11: Degree theory [See also 55M25, 58C30]
- 47H14: Perturbations of nonlinear operators [See also 47A55, 58J37, 70H09, 70K60, 81Q15]
- 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]
- 47H25: Nonlinear ergodic theorems [See also 28Dxx, 37Axx, 47A35]
- 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]
- 47H40: Random operators [See also 47B80, 60H25]
- 47H60: Multilinear and polynomial operators [See also 46G25]
- 47H99: None of the above, but in this section

- 47Jxx: Equations and inequalities involving nonlinear operators [See also 46Txx] {For global and geometric aspects, see 58-XX}
- 47J05: Equations involving nonlinear operators (general) [See also 47H10, 47J25]
- 47J06: Nonlinear ill-posed problems [See also 35R25, 47A52, 65F22, 65J20, 65L08, 65M30, 65R30]
- 47J07: Abstract inverse mapping and implicit function theorems [See also 46T20 and 58C15]
- 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05]
- 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09]
- 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40]
- 47J22: Variational and other types of inclusions [See also 34A60, 49J21, 49K21]
- 47J25: Iterative procedures [See also 65J15]
- 47J30: Variational methods [See also 58Exx]
- 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]
- 47J40: Equations with hysteresis operators [See also 34C55, 74N30]
- 47J99: None of the above, but in this section

- 47Lxx: Linear spaces and algebras of operators [See also 46Lxx]
- 47L05: Linear spaces of operators [See also 46A32 and 46B28]
- 47L07: Convex sets and cones of operators [See also 46A55]
- 47L10: Algebras of operators on Banach spaces and other topological linear spaces
- 47L15: Operator algebras with symbol structure
- 47L20: Operator ideals [See also 47B10]
- 47L22: Ideals of polynomials and of multilinear mappings
- 47L25: Operator spaces (= matricially normed spaces) [See also 46L07]
- 47L30: Abstract operator algebras on Hilbert spaces
- 47L35: Nest algebras, CSL algebras
- 47L40: Limit algebras, subalgebras of $C^*$-algebras
- 47L45: Dual algebras; weakly closed singly generated operator algebras
- 47L50: Dual spaces of operator algebras
- 47L55: Representations of (nonselfadjoint) operator algebras
- 47L60: Algebras of unbounded operators; partial algebras of operators
- 47L65: Crossed product algebras (analytic crossed products)
- 47L70: Nonassociative nonselfadjoint operator algebras
- 47L75: Other nonselfadjoint operator algebras
- 47L80: Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
- 47L90: Applications of operator algebras to physics
- 47L99: None of the above, but in this section

- 47Nxx: Miscellaneous applications of operator theory [See also 46Nxx]
- 47N10: Applications in optimization, convex analysis, mathematical programming, economics
- 47N20: Applications to differential and integral equations
- 47N30: Applications in probability theory and statistics
- 47N40: Applications in numerical analysis [See also 65Jxx]
- 47N50: Applications in the physical sciences
- 47N60: Applications in chemistry and life sciences
- 47N70: Applications in systems theory, circuits, and control theory
- 47N99: None of the above, but in this section

- 47Sxx: Other (nonclassical) types of operator theory [See also 46Sxx]
- 47S10: Operator theory over fields other than ${\bf R}$, ${\bf C}$ or the quaternions; non-Archimedean operator theory
- 47S20: Nonstandard operator theory [See also 03H05]
- 47S30: Constructive operator theory [See also 03F60]
- 47S40: Fuzzy operator theory [See also 03E72]
- 47S50: Operator theory in probabilistic metric linear spaces [See also 54E70]
- 47S99: None of the above, but in this section

- 49-XX: Calculus of variations and optimal control; optimization [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX]
- 49-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 49-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 49-02: Research exposition (monographs, survey articles)
- 49-03: Historical (must also be assigned at least one classification number from Section 01)
- 49-04: Explicit machine computation and programs (not the theory of computation or programming)
- 49-06: Proceedings, conferences, collections, etc.
- 49Jxx: Existence theories
- 49J05: Free problems in one independent variable
- 49J10: Free problems in two or more independent variables
- 49J15: Optimal control problems involving ordinary differential equations
- 49J20: Optimal control problems involving partial differential equations
- 49J21: Optimal control problems involving relations other than differential equations
- 49J27: Problems in abstract spaces [See also 90C48, 93C25]
- 49J30: Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
- 49J35: Minimax problems
- 49J40: Variational methods including variational inequalities [See also 47J20]
- 49J45: Methods involving semicontinuity and convergence; relaxation
- 49J50: Fréchet and Gateaux differentiability [See also 46G05, 58C20]
- 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56]
- 49J53: Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06]
- 49J55: Problems involving randomness [See also 93E20]
- 49J99: None of the above, but in this section

- 49Kxx: Optimality conditions
- 49K05: Free problems in one independent variable
- 49K10: Free problems in two or more independent variables
- 49K15: Problems involving ordinary differential equations
- 49K20: Problems involving partial differential equations
- 49K21: Problems involving relations other than differential equations
- 49K27: Problems in abstract spaces [See also 90C48, 93C25]
- 49K30: Optimal solutions belonging to restricted classes
- 49K35: Minimax problems
- 49K40: Sensitivity, stability, well-posedness [See also 90C31]
- 49K45: Problems involving randomness [See also 93E20]
- 49K99: None of the above, but in this section

- 49Lxx: Hamilton-Jacobi theories, including dynamic programming
- 49L20: Dynamic programming method
- 49L25: Viscosity solutions
- 49L99: None of the above, but in this section

- 49Mxx: Numerical methods [See also 90Cxx, 65Kxx]
- 49M05: Methods based on necessary conditions
- 49M15: Newton-type methods
- 49M20: Methods of relaxation type
- 49M25: Discrete approximations
- 49M27: Decomposition methods
- 49M29: Methods involving duality
- 49M30: Other methods
- 49M37: Methods of nonlinear programming type [See also 90C30, 65Kxx]
- 49M99: None of the above, but in this section

- 49Nxx: Miscellaneous topics
- 49N05: Linear optimal control problems [See also 93C05]
- 49N10: Linear-quadratic problems
- 49N15: Duality theory
- 49N20: Periodic optimization
- 49N25: Impulsive optimal control problems
- 49N30: Problems with incomplete information [See also 93C41]
- 49N35: Optimal feedback synthesis [See also 93B52]
- 49N45: Inverse problems
- 49N60: Regularity of solutions
- 49N70: Differential games
- 49N75: Pursuit and evasion games
- 49N90: Applications of optimal control and differential games [See also 90C90, 93C95]
- 49N99: None of the above, but in this section

- 49Qxx: Manifolds [See also 58Exx]
- 49Q05: Minimal surfaces [See also 53A10, 58E12]
- 49Q10: Optimization of shapes other than minimal surfaces [See also 90C90]
- 49Q12: Sensitivity analysis
- 49Q15: Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35]
- 49Q20: Variational problems in a geometric measure-theoretic setting
- 49Q99: None of the above, but in this section

- 49Rxx: Variational methods for eigenvalues of operators [See also 47A75]
- 49R05: Variational methods for eigenvalues of operators [See also 47A75] (should also be assigned at least one other classification number in Section 49)
- 49R99: None of the above, but in this section

- 49Sxx: Variational principles of physics
- 49S05: Variational principles of physics (should also be assigned at least one other classification number in section 49)
- 49S99: None of the above, but in this section

- 51-XX: Geometry {For algebraic geometry, see 14-XX}
- 51-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 51-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 51-02: Research exposition (monographs, survey articles)
- 51-03: Historical (must also be assigned at least one classification number from Section 01)
- 51-04: Explicit machine computation and programs (not the theory of computation or programming)
- 51-06: Proceedings, conferences, collections, etc.
- 51Axx: Linear incidence geometry
- 51A05: General theory and projective geometries
- 51A10: Homomorphism, automorphism and dualities
- 51A15: Structures with parallelism
- 51A20: Configuration theorems
- 51A25: Algebraization [See also 12Kxx, 20N05]
- 51A30: Desarguesian and Pappian geometries
- 51A35: Non-Desarguesian affine and projective planes
- 51A40: Translation planes and spreads
- 51A45: Incidence structures imbeddable into projective geometries
- 51A50: Polar geometry, symplectic spaces, orthogonal spaces
- 51A99: None of the above, but in this section

- 51Bxx: Nonlinear incidence geometry
- 51B05: General theory
- 51B10: Möbius geometries
- 51B15: Laguerre geometries
- 51B20: Minkowski geometries
- 51B25: Lie geometries
- 51B99: None of the above, but in this section

- 51Cxx: Ring geometry (Hjelmslev, Barbilian, etc.)
- 51C05: Ring geometry (Hjelmslev, Barbilian, etc.)
- 51C99: None of the above, but in this section

- 51Dxx: Geometric closure systems
- 51D05: Abstract (Maeda) geometries
- 51D10: Abstract geometries with exchange axiom
- 51D15: Abstract geometries with parallelism
- 51D20: Combinatorial geometries [See also 05B25, 05B35]
- 51D25: Lattices of subspaces [See also 05B35]
- 51D30: Continuous geometries and related topics [See also 06Cxx]
- 51D99: None of the above, but in this section

- 51Exx: Finite geometry and special incidence structures
- 51E05: General block designs [See also 05B05]
- 51E10: Steiner systems
- 51E12: Generalized quadrangles, generalized polygons
- 51E14: Finite partial geometries (general), nets, partial spreads
- 51E15: Affine and projective planes
- 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]
- 51E21: Blocking sets, ovals, $k$-arcs
- 51E22: Linear codes and caps in Galois spaces [See also 94B05]
- 51E23: Spreads and packing problems
- 51E24: Buildings and the geometry of diagrams
- 51E25: Other finite nonlinear geometries
- 51E26: Other finite linear geometries
- 51E30: Other finite incidence structures [See also 05B30]
- 51E99: None of the above, but in this section

- 51Fxx: Metric geometry
- 51F05: Absolute planes
- 51F10: Absolute spaces
- 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]
- 51F20: Congruence and orthogonality [See also 20H05]
- 51F25: Orthogonal and unitary groups [See also 20H05]
- 51F99: None of the above, but in this section

- 51Gxx: Ordered geometries (ordered incidence structures, etc.)
- 51G05: Ordered geometries (ordered incidence structures, etc.)
- 51G99: None of the above, but in this section

- 51Hxx: Topological geometry
- 51H05: General theory
- 51H10: Topological linear incidence structures
- 51H15: Topological nonlinear incidence structures
- 51H20: Topological geometries on manifolds [See also 57-XX]
- 51H25: Geometries with differentiable structure [See also 53Cxx, 53C70]
- 51H30: Geometries with algebraic manifold structure [See also 14-XX]
- 51H99: None of the above, but in this section

- 51Jxx: Incidence groups
- 51Kxx: Distance geometry
- 51K05: General theory
- 51K10: Synthetic differential geometry
- 51K99: None of the above, but in this section

- 51Lxx: Geometric order structures [See also 53C75]
- 51L05: Geometry of orders of nondifferentiable curves
- 51L10: Directly differentiable curves
- 51L15: $n$-vertex theorems via direct methods
- 51L20: Geometry of orders of surfaces
- 51L99: None of the above, but in this section

- 51Mxx: Real and complex geometry
- 51M04: Elementary problems in Euclidean geometries
- 51M05: Euclidean geometries (general) and generalizations
- 51M09: Elementary problems in hyperbolic and elliptic geometries
- 51M10: Hyperbolic and elliptic geometries (general) and generalizations
- 51M15: Geometric constructions
- 51M16: Inequalities and extremum problems {For convex problems, see 52A40}
- 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]
- 51M25: Length, area and volume [See also 26B15]
- 51M30: Line geometries and their generalizations [See also 53A25]
- 51M35: Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) [See also 14M15]
- 51M99: None of the above, but in this section

- 51Nxx: Analytic and descriptive geometry
- 51N05: Descriptive geometry [See also 65D17, 68U07]
- 51N10: Affine analytic geometry
- 51N15: Projective analytic geometry
- 51N20: Euclidean analytic geometry
- 51N25: Analytic geometry with other transformation groups
- 51N30: Geometry of classical groups [See also 20Gxx, 14L35]
- 51N35: Questions of classical algebraic geometry [See also 14Nxx]
- 51N99: None of the above, but in this section

- 51Pxx: Geometry and physics (should also be assigned at least one other classification number from Sections 70--86)
- 51P05: Geometry and physics (should also be assigned at least one other classification number from Sections 70--86)
- 51P99: None of the above, but in this section

- 52-XX: Convex and discrete geometry
- 52-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 52-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 52-02: Research exposition (monographs, survey articles)
- 52-03: Historical (must also be assigned at least one classification number from Section 01)
- 52-04: Explicit machine computation and programs (not the theory of computation or programming)
- 52-06: Proceedings, conferences, collections, etc.
- 52Axx: General convexity
- 52A01: Axiomatic and generalized convexity
- 52A05: Convex sets without dimension restrictions
- 52A07: Convex sets in topological vector spaces [See also 46A55]
- 52A10: Convex sets in $2$ dimensions (including convex curves) [See also 53A04]
- 52A15: Convex sets in $3$ dimensions (including convex surfaces) [See also 53A05, 53C45]
- 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
- 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
- 52A22: Random convex sets and integral geometry [See also 53C65, 60D05]
- 52A23: Asymptotic theory of convex bodies [See also 46B06]
- 52A27: Approximation by convex sets
- 52A30: Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
- 52A35: Helly-type theorems and geometric transversal theory
- 52A37: Other problems of combinatorial convexity
- 52A38: Length, area, volume [See also 26B15, 28A75, 49Q20]
- 52A39: Mixed volumes and related topics
- 52A40: Inequalities and extremum problems
- 52A41: Convex functions and convex programs [See also 26B25, 90C25]
- 52A55: Spherical and hyperbolic convexity
- 52A99: None of the above, but in this section

- 52Bxx: Polytopes and polyhedra
- 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]
- 52B10: Three-dimensional polytopes
- 52B11: $n$-dimensional polytopes
- 52B12: Special polytopes (linear programming, centrally symmetric, etc.)
- 52B15: Symmetry properties of polytopes
- 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
- 52B22: Shellability
- 52B35: Gale and other diagrams
- 52B40: Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) [See also 05B35, 52Cxx]
- 52B45: Dissections and valuations (Hilbert's third problem, etc.)
- 52B55: Computational aspects related to convexity {For computational geometry and algorithms, see 68Q25, 68U05; for numerical algorithms, see 65Yxx} [See also 68Uxx]
- 52B60: Isoperimetric problems for polytopes
- 52B70: Polyhedral manifolds
- 52B99: None of the above, but in this section

- 52Cxx: Discrete geometry
- 52C05: Lattices and convex bodies in $2$ dimensions [See also 11H06, 11H31, 11P21]
- 52C07: Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]
- 52C10: Erdős problems and related topics of discrete geometry [See also 11Hxx]
- 52C15: Packing and covering in $2$ dimensions [See also 05B40, 11H31]
- 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31]
- 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]
- 52C22: Tilings in $n$ dimensions [See also 05B45, 51M20]
- 52C23: Quasicrystals, aperiodic tilings
- 52C25: Rigidity and flexibility of structures [See also 70B15]
- 52C26: Circle packings and discrete conformal geometry
- 52C30: Planar arrangements of lines and pseudolines
- 52C35: Arrangements of points, flats, hyperplanes [See also 32S22]
- 52C40: Oriented matroids
- 52C45: Combinatorial complexity of geometric structures [See also 68U05]
- 52C99: None of the above, but in this section

- 53-XX: Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx}
- 53-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 53-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 53-02: Research exposition (monographs, survey articles)
- 53-03: Historical (must also be assigned at least one classification number from Section 01)
- 53-04: Explicit machine computation and programs (not the theory of computation or programming)
- 53-06: Proceedings, conferences, collections, etc.
- 53Axx: Classical differential geometry
- 53A04: Curves in Euclidean space
- 53A05: Surfaces in Euclidean space
- 53A07: Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
- 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
- 53A15: Affine differential geometry
- 53A17: Kinematics
- 53A20: Projective differential geometry
- 53A25: Differential line geometry
- 53A30: Conformal differential geometry
- 53A35: Non-Euclidean differential geometry
- 53A40: Other special differential geometries
- 53A45: Vector and tensor analysis
- 53A55: Differential invariants (local theory), geometric objects
- 53A60: Geometry of webs [See also 14C21, 20N05]
- 53A99: None of the above, but in this section

- 53Bxx: Local differential geometry
- 53B05: Linear and affine connections
- 53B10: Projective connections
- 53B15: Other connections
- 53B20: Local Riemannian geometry
- 53B21: Methods of Riemannian geometry
- 53B25: Local submanifolds [See also 53C40]
- 53B30: Lorentz metrics, indefinite metrics
- 53B35: Hermitian and Kählerian structures [See also 32Cxx]
- 53B40: Finsler spaces and generalizations (areal metrics)
- 53B50: Applications to physics
- 53B99: None of the above, but in this section

- 53Cxx: Global differential geometry [See also 51H25, 58-XX; for related bundle theory, see 55Rxx, 57Rxx]
- 53C05: Connections, general theory
- 53C07: Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) [See also 32Q20]
- 53C08: Gerbes, differential characters: differential geometric aspects
- 53C10: $G$-structures
- 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32]
- 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
- 53C17: Sub-Riemannian geometry
- 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
- 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]
- 53C22: Geodesics [See also 58E10]
- 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
- 53C24: Rigidity results
- 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
- 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
- 53C27: Spin and Spin${}^c$ geometry
- 53C28: Twistor methods [See also 32L25]
- 53C29: Issues of holonomy
- 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
- 53C35: Symmetric spaces [See also 32M15, 57T15]
- 53C38: Calibrations and calibrated geometries
- 53C40: Global submanifolds [See also 53B25]
- 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
- 53C43: Differential geometric aspects of harmonic maps [See also 58E20]
- 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
- 53C45: Global surface theory (convex surfaces à la A. D. Aleksandrov)
- 53C50: Lorentz manifolds, manifolds with indefinite metrics
- 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
- 53C56: Other complex differential geometry [See also 32Cxx]
- 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20]
- 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
- 53C70: Direct methods ($G$-spaces of Busemann, etc.)
- 53C75: Geometric orders, order geometry [See also 51Lxx]
- 53C80: Applications to physics
- 53C99: None of the above, but in this section

- 53Dxx: Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx]
- 53D05: Symplectic manifolds, general
- 53D10: Contact manifolds, general
- 53D12: Lagrangian submanifolds; Maslov index
- 53D15: Almost contact and almost symplectic manifolds
- 53D17: Poisson manifolds; Poisson groupoids and algebroids
- 53D18: Generalized geometries (à la Hitchin)
- 53D20: Momentum maps; symplectic reduction
- 53D22: Canonical transformations
- 53D25: Geodesic flows
- 53D30: Symplectic structures of moduli spaces
- 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
- 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category [See also 14J33]
- 53D40: Floer homology and cohomology, symplectic aspects
- 53D42: Symplectic field theory; contact homology
- 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
- 53D50: Geometric quantization
- 53D55: Deformation quantization, star products
- 53D99: None of the above, but in this section

- 53Zxx: Applications to physics
- 53Z05: Applications to physics
- 53Z99: None of the above, but in this section

- 54-XX: General topology {For the topology of manifolds of all dimensions, see 57Nxx}
- 54-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 54-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 54-02: Research exposition (monographs, survey articles)
- 54-03: Historical (must also be assigned at least one classification number from Section 01)
- 54-04: Explicit machine computation and programs (not the theory of computation or programming)
- 54-06: Proceedings, conferences, collections, etc.
- 54Axx: Generalities
- 54A05: Topological spaces and generalizations (closure spaces, etc.)
- 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
- 54A15: Syntopogeneous structures
- 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
- 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets) [See also 03Exx] {For ultrafilters, see 54D80}
- 54A35: Consistency and independence results [See also 03E35]
- 54A40: Fuzzy topology [See also 03E72]
- 54A99: None of the above, but in this section

- 54Bxx: Basic constructions
- 54Cxx: Maps and general types of spaces defined by maps
- 54C05: Continuous maps
- 54C08: Weak and generalized continuity
- 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
- 54C15: Retraction
- 54C20: Extension of maps
- 54C25: Embedding
- 54C30: Real-valued functions [See also 26-XX]
- 54C35: Function spaces [See also 46Exx, 58D15]
- 54C40: Algebraic properties of function spaces [See also 46J10]
- 54C45: $C$- and $C^*$-embedding
- 54C50: Special sets defined by functions [See also 26A21]
- 54C55: Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties) [See also 55M15]
- 54C56: Shape theory [See also 55P55, 57N25]
- 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]
- 54C65: Selections [See also 28B20]
- 54C70: Entropy
- 54C99: None of the above, but in this section

- 54Dxx: Fairly general properties
- 54D05: Connected and locally connected spaces (general aspects)
- 54D10: Lower separation axioms ($T_0$--$T_3$, etc.)
- 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
- 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.)
- 54D25: “$P$-minimal” and “$P$-closed” spaces
- 54D30: Compactness
- 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)
- 54D40: Remainders
- 54D45: Local compactness, $\sigma$-compactness
- 54D50: $k$-spaces
- 54D55: Sequential spaces
- 54D60: Realcompactness and realcompactification
- 54D65: Separability
- 54D70: Base properties
- 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)
- 54D99: None of the above, but in this section

- 54Exx: Spaces with richer structures
- 54E05: Proximity structures and generalizations
- 54E15: Uniform structures and generalizations
- 54E17: Nearness spaces
- 54E18: $p$-spaces, $M$-spaces, $\sigma$-spaces, etc.
- 54E20: Stratifiable spaces, cosmic spaces, etc.
- 54E25: Semimetric spaces
- 54E30: Moore spaces
- 54E35: Metric spaces, metrizability
- 54E40: Special maps on metric spaces
- 54E45: Compact (locally compact) metric spaces
- 54E50: Complete metric spaces
- 54E52: Baire category, Baire spaces
- 54E55: Bitopologies
- 54E70: Probabilistic metric spaces
- 54E99: None of the above, but in this section

- 54Fxx: Special properties
- 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces [See also 06B30, 06F30]
- 54F15: Continua and generalizations
- 54F35: Higher-dimensional local connectedness [See also 55Mxx, 55Nxx]
- 54F45: Dimension theory [See also 55M10]
- 54F50: Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03]
- 54F55: Unicoherence, multicoherence
- 54F65: Topological characterizations of particular spaces
- 54F99: None of the above, but in this section

- 54Gxx: Peculiar spaces
- 54G05: Extremally disconnected spaces, $F$-spaces, etc.
- 54G10: $P$-spaces
- 54G12: Scattered spaces
- 54G15: Pathological spaces
- 54G20: Counterexamples
- 54G99: None of the above, but in this section

- 54Hxx: Connections with other structures, applications
- 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05]
- 54H10: Topological representations of algebraic systems [See also 22-XX]
- 54H11: Topological groups [See also 22A05]
- 54H12: Topological lattices, etc. [See also 06B30, 06F30]
- 54H13: Topological fields, rings, etc. [See also 12Jxx] {For algebraic aspects, see 13Jxx, 16W80}
- 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]
- 54H20: Topological dynamics [See also 28Dxx, 37Bxx]
- 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]
- 54H99: None of the above, but in this section

- 54Jxx: Nonstandard topology [See also 03H05]
- 54J05: Nonstandard topology [See also 03H05]
- 54J99: None of the above, but in this section

- 55-XX: Algebraic topology
- 55-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 55-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 55-02: Research exposition (monographs, survey articles)
- 55-03: Historical (must also be assigned at least one classification number from Section 01)
- 55-04: Explicit machine computation and programs (not the theory of computation or programming)
- 55-06: Proceedings, conferences, collections, etc.
- 55Mxx: Classical topics {For the topology of Euclidean spaces and manifolds, see 57Nxx}
- 55M05: Duality
- 55M10: Dimension theory [See also 54F45]
- 55M15: Absolute neighborhood retracts [See also 54C55]
- 55M20: Fixed points and coincidences [See also 54H25]
- 55M25: Degree, winding number
- 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirel'man) category of a space
- 55M35: Finite groups of transformations (including Smith theory) [See also 57S17]
- 55M99: None of the above, but in this section

- 55Nxx: Homology and cohomology theories [See also 57Txx]
- 55N05: Čech types
- 55N07: Steenrod-Sitnikov homologies
- 55N10: Singular theory
- 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX}
- 55N20: Generalized (extraordinary) homology and cohomology theories
- 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90]
- 55N25: Homology with local coefficients, equivariant cohomology
- 55N30: Sheaf cohomology [See also 18F20, 32C35, 32L10]
- 55N32: Orbifold cohomology
- 55N33: Intersection homology and cohomology
- 55N34: Elliptic cohomology
- 55N35: Other homology theories
- 55N40: Axioms for homology theory and uniqueness theorems
- 55N45: Products and intersections
- 55N91: Equivariant homology and cohomology [See also 19L47]
- 55N99: None of the above, but in this section

- 55Pxx: Homotopy theory {For simple homotopy type, see 57Q10}
- 55P05: Homotopy extension properties, cofibrations
- 55P10: Homotopy equivalences
- 55P15: Classification of homotopy type
- 55P20: Eilenberg-Mac Lane spaces
- 55P25: Spanier-Whitehead duality
- 55P30: Eckmann-Hilton duality
- 55P35: Loop spaces
- 55P40: Suspensions
- 55P42: Stable homotopy theory, spectra
- 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
- 55P45: $H$-spaces and duals
- 55P47: Infinite loop spaces
- 55P48: Loop space machines, operads [See also 18D50]
- 55P50: String topology
- 55P55: Shape theory [See also 54C56, 55Q07]
- 55P57: Proper homotopy theory
- 55P60: Localization and completion
- 55P62: Rational homotopy theory
- 55P65: Homotopy functors
- 55P91: Equivariant homotopy theory [See also 19L47]
- 55P92: Relations between equivariant and nonequivariant homotopy theory
- 55P99: None of the above, but in this section

- 55Qxx: Homotopy groups
- 55Q05: Homotopy groups, general; sets of homotopy classes
- 55Q07: Shape groups
- 55Q10: Stable homotopy groups
- 55Q15: Whitehead products and generalizations
- 55Q20: Homotopy groups of wedges, joins, and simple spaces
- 55Q25: Hopf invariants
- 55Q35: Operations in homotopy groups
- 55Q40: Homotopy groups of spheres
- 55Q45: Stable homotopy of spheres
- 55Q50: $J$-morphism [See also 19L20]
- 55Q51: $v_n$-periodicity
- 55Q52: Homotopy groups of special spaces
- 55Q55: Cohomotopy groups
- 55Q70: Homotopy groups of special types [See also 55N05, 55N07]
- 55Q91: Equivariant homotopy groups [See also 19L47]
- 55Q99: None of the above, but in this section

- 55Rxx: Fiber spaces and bundles [See also 18F15, 32Lxx, 46M20, 57R20, 57R22, 57R25]
- 55R05: Fiber spaces
- 55R10: Fiber bundles
- 55R12: Transfer
- 55R15: Classification
- 55R20: Spectral sequences and homology of fiber spaces [See also 55Txx]
- 55R25: Sphere bundles and vector bundles
- 55R35: Classifying spaces of groups and $H$-spaces
- 55R37: Maps between classifying spaces
- 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20]
- 55R45: Homology and homotopy of $B{\rm O}$ and $B{\rm U}$; Bott periodicity
- 55R50: Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX}
- 55R55: Fiberings with singularities
- 55R60: Microbundles and block bundles [See also 57N55, 57Q50]
- 55R65: Generalizations of fiber spaces and bundles
- 55R70: Fibrewise topology
- 55R80: Discriminantal varieties, configuration spaces
- 55R91: Equivariant fiber spaces and bundles [See also 19L47]
- 55R99: None of the above, but in this section

- 55Sxx: Operations and obstructions
- 55S05: Primary cohomology operations
- 55S10: Steenrod algebra
- 55S12: Dyer-Lashof operations
- 55S15: Symmetric products, cyclic products
- 55S20: Secondary and higher cohomology operations
- 55S25: $K$-theory operations and generalized cohomology operations [See also 19D55, 19Lxx]
- 55S30: Massey products
- 55S35: Obstruction theory
- 55S36: Extension and compression of mappings
- 55S37: Classification of mappings
- 55S40: Sectioning fiber spaces and bundles
- 55S45: Postnikov systems, $k$-invariants
- 55S91: Equivariant operations and obstructions [See also 19L47]
- 55S99: None of the above, but in this section

- 55Txx: Spectral sequences [See also 18G40, 55R20]
- 55T05: General
- 55T10: Serre spectral sequences
- 55T15: Adams spectral sequences
- 55T20: Eilenberg-Moore spectral sequences [See also 57T35]
- 55T25: Generalized cohomology
- 55T99: None of the above, but in this section

- 55Uxx: Applied homological algebra and category theory [See also 18Gxx]
- 55U05: Abstract complexes
- 55U10: Simplicial sets and complexes
- 55U15: Chain complexes
- 55U20: Universal coefficient theorems, Bockstein operator
- 55U25: Homology of a product, Künneth formula
- 55U30: Duality
- 55U35: Abstract and axiomatic homotopy theory
- 55U40: Topological categories, foundations of homotopy theory
- 55U99: None of the above, but in this section

- 57-XX: Manifolds and cell complexes {For complex manifolds, see 32Qxx}
- 57-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 57-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 57-02: Research exposition (monographs, survey articles)
- 57-03: Historical (must also be assigned at least one classification number from Section 01)
- 57-04: Explicit machine computation and programs (not the theory of computation or programming)
- 57-06: Proceedings, conferences, collections, etc.
- 57Mxx: Low-dimensional topology
- 57M05: Fundamental group, presentations, free differential calculus
- 57M07: Topological methods in group theory
- 57M10: Covering spaces
- 57M12: Special coverings, e.g. branched
- 57M15: Relations with graph theory [See also 05Cxx]
- 57M20: Two-dimensional complexes
- 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
- 57M27: Invariants of knots and 3-manifolds
- 57M30: Wild knots and surfaces, etc., wild embeddings
- 57M35: Dehn's lemma, sphere theorem, loop theorem, asphericity
- 57M40: Characterizations of $E^3$ and $S^3$ (Poincaré conjecture) [See also 57N12]
- 57M50: Geometric structures on low-dimensional manifolds
- 57M60: Group actions in low dimensions
- 57M99: None of the above, but in this section

- 57Nxx: Topological manifolds
- 57N05: Topology of $E^2$, $2$-manifolds
- 57N10: Topology of general $3$-manifolds [See also 57Mxx]
- 57N12: Topology of $E^3$ and $S^3$ [See also 57M40]
- 57N13: Topology of $E^4$, $4$-manifolds [See also 14Jxx, 32Jxx]
- 57N15: Topology of $E^n$, $n$-manifolds ($4 \less n \less \infty$)
- 57N16: Geometric structures on manifolds [See also 57M50]
- 57N17: Topology of topological vector spaces
- 57N20: Topology of infinite-dimensional manifolds [See also 58Bxx]
- 57N25: Shapes [See also 54C56, 55P55, 55Q07]
- 57N30: Engulfing
- 57N35: Embeddings and immersions
- 57N37: Isotopy and pseudo-isotopy
- 57N40: Neighborhoods of submanifolds
- 57N45: Flatness and tameness
- 57N50: $S^{n-1}\subset E^n$, Schoenflies problem
- 57N55: Microbundles and block bundles [See also 55R60, 57Q50]
- 57N60: Cellularity
- 57N65: Algebraic topology of manifolds
- 57N70: Cobordism and concordance
- 57N75: General position and transversality
- 57N80: Stratifications
- 57N99: None of the above, but in this section

- 57Pxx: Generalized manifolds [See also 18F15]
- 57P05: Local properties of generalized manifolds
- 57P10: Poincaré duality spaces
- 57P99: None of the above, but in this section

- 57Qxx: PL-topology
- 57Q05: General topology of complexes
- 57Q10: Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]
- 57Q12: Wall finiteness obstruction for CW-complexes
- 57Q15: Triangulating manifolds
- 57Q20: Cobordism
- 57Q25: Comparison of PL-structures: classification, Hauptvermutung
- 57Q30: Engulfing
- 57Q35: Embeddings and immersions
- 57Q37: Isotopy
- 57Q40: Regular neighborhoods
- 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}
- 57Q50: Microbundles and block bundles [See also 55R60, 57N55]
- 57Q55: Approximations
- 57Q60: Cobordism and concordance
- 57Q65: General position and transversality
- 57Q91: Equivariant PL-topology
- 57Q99: None of the above, but in this section

- 57Rxx: Differential topology {For foundational questions of differentiable manifolds, see 58Axx; for infinite-dimensional manifolds, see 58Bxx}
- 57R05: Triangulating
- 57R10: Smoothing
- 57R12: Smooth approximations
- 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
- 57R17: Symplectic and contact topology
- 57R18: Topology and geometry of orbifolds
- 57R19: Algebraic topology on manifolds
- 57R20: Characteristic classes and numbers
- 57R22: Topology of vector bundles and fiber bundles [See also 55Rxx]
- 57R25: Vector fields, frame fields
- 57R27: Controllability of vector fields on $C^\infty$ and real-analytic manifolds [See also 49Qxx, 37C10, 93B05]
- 57R30: Foliations; geometric theory
- 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10]
- 57R35: Differentiable mappings
- 57R40: Embeddings
- 57R42: Immersions
- 57R45: Singularities of differentiable mappings
- 57R50: Diffeomorphisms
- 57R52: Isotopy
- 57R55: Differentiable structures
- 57R56: Topological quantum field theories
- 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
- 57R58: Floer homology
- 57R60: Homotopy spheres, Poincaré conjecture
- 57R65: Surgery and handlebodies
- 57R67: Surgery obstructions, Wall groups [See also 19J25]
- 57R70: Critical points and critical submanifolds
- 57R75: ${\rm O}$- and ${\rm SO}$-cobordism
- 57R77: Complex cobordism (${\rm U}$- and ${\rm SU}$-cobordism) [See also 55N22]
- 57R80: $h$- and $s$-cobordism
- 57R85: Equivariant cobordism
- 57R90: Other types of cobordism [See also 55N22]
- 57R91: Equivariant algebraic topology of manifolds
- 57R95: Realizing cycles by submanifolds
- 57R99: None of the above, but in this section

- 57Sxx: Topological transformation groups [See also 20F34, 22-XX, 37-XX, 54H15, 58D05]
- 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
- 57S10: Compact groups of homeomorphisms
- 57S15: Compact Lie groups of differentiable transformations
- 57S17: Finite transformation groups
- 57S20: Noncompact Lie groups of transformations
- 57S25: Groups acting on specific manifolds
- 57S30: Discontinuous groups of transformations
- 57S99: None of the above, but in this section

- 57Txx: Homology and homotopy of topological groups and related structures
- 57T05: Hopf algebras [See also 16T05]
- 57T10: Homology and cohomology of Lie groups
- 57T15: Homology and cohomology of homogeneous spaces of Lie groups
- 57T20: Homotopy groups of topological groups and homogeneous spaces
- 57T25: Homology and cohomology of $H$-spaces
- 57T30: Bar and cobar constructions [See also 18G55, 55Uxx]
- 57T35: Applications of Eilenberg-Moore spectral sequences [See also 55R20, 55T20]
- 57T99: None of the above, but in this section

- 58-XX: Global analysis, analysis on manifolds [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx] {For geometric integration theory, see 49Q15}
- 58-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 58-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 58-02: Research exposition (monographs, survey articles)
- 58-03: Historical (must also be assigned at least one classification number from Section 01)
- 58-04: Explicit machine computation and programs (not the theory of computation or programming)
- 58-06: Proceedings, conferences, collections, etc.
- 58Axx: General theory of differentiable manifolds [See also 32Cxx]
- 58A03: Topos-theoretic approach to differentiable manifolds
- 58A05: Differentiable manifolds, foundations
- 58A07: Real-analytic and Nash manifolds [See also 14P20, 32C07]
- 58A10: Differential forms
- 58A12: de Rham theory [See also 14Fxx]
- 58A14: Hodge theory [See also 14C30, 14Fxx, 32J25, 32S35]
- 58A15: Exterior differential systems (Cartan theory)
- 58A17: Pfaffian systems
- 58A20: Jets
- 58A25: Currents [See also 32C30, 53C65]
- 58A30: Vector distributions (subbundles of the tangent bundles)
- 58A32: Natural bundles
- 58A35: Stratified sets [See also 32S60]
- 58A40: Differential spaces
- 58A50: Supermanifolds and graded manifolds [See also 14A22, 32C11]
- 58A99: None of the above, but in this section

- 58Bxx: Infinite-dimensional manifolds
- 58B05: Homotopy and topological questions
- 58B10: Differentiability questions
- 58B12: Questions of holomorphy [See also 32-XX, 46G20]
- 58B15: Fredholm structures [See also 47A53]
- 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]
- 58B25: Group structures and generalizations on infinite-dimensional manifolds [See also 22E65, 58D05]
- 58B32: Geometry of quantum groups
- 58B34: Noncommutative geometry (à la Connes)
- 58B99: None of the above, but in this section

- 58Cxx: Calculus on manifolds; nonlinear operators [See also 46Txx, 47Hxx, 47Jxx]
- 58C05: Real-valued functions
- 58C06: Set valued and function-space valued mappings [See also 47H04, 54C60]
- 58C07: Continuity properties of mappings
- 58C10: Holomorphic maps [See also 32-XX]
- 58C15: Implicit function theorems; global Newton methods
- 58C20: Differentiation theory (Gateaux, Fréchet, etc.) [See also 26Exx, 46G05]
- 58C25: Differentiable maps
- 58C30: Fixed point theorems on manifolds [See also 47H10]
- 58C35: Integration on manifolds; measures on manifolds [See also 28Cxx]
- 58C40: Spectral theory; eigenvalue problems [See also 47J10, 58E07]
- 58C50: Analysis on supermanifolds or graded manifolds
- 58C99: None of the above, but in this section

- 58Dxx: Spaces and manifolds of mappings (including nonlinear versions of 46Exx) [See also 46Txx, 53Cxx]
- 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05]
- 58D07: Groups and semigroups of nonlinear operators [See also 17B65, 47H20]
- 58D10: Spaces of imbeddings and immersions
- 58D15: Manifolds of mappings [See also 46T10, 54C35]
- 58D17: Manifolds of metrics (esp. Riemannian)
- 58D19: Group actions and symmetry properties
- 58D20: Measures (Gaussian, cylindrical, etc.) on manifolds of maps [See also 28Cxx, 46T12]
- 58D25: Equations in function spaces; evolution equations [See also 34Gxx, 35K90, 35L90, 35R15, 37Lxx, 47Jxx]
- 58D27: Moduli problems for differential geometric structures
- 58D29: Moduli problems for topological structures
- 58D30: Applications (in quantum mechanics (Feynman path integrals), relativity, fluid dynamics, etc.)
- 58D99: None of the above, but in this section

- 58Exx: Variational problems in infinite-dimensional spaces
- 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
- 58E07: Abstract bifurcation theory
- 58E09: Group-invariant bifurcation theory
- 58E10: Applications to the theory of geodesics (problems in one independent variable)
- 58E11: Critical metrics
- 58E12: Applications to minimal surfaces (problems in two independent variables) [See also 49Q05]
- 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
- 58E17: Pareto optimality, etc., applications to economics [See also 90C29]
- 58E20: Harmonic maps [See also 53C43], etc.
- 58E25: Applications to control theory [See also 49-XX, 93-XX]
- 58E30: Variational principles
- 58E35: Variational inequalities (global problems)
- 58E40: Group actions
- 58E50: Applications
- 58E99: None of the above, but in this section

- 58Hxx: Pseudogroups, differentiable groupoids and general structures on manifolds
- 58Jxx: Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx]
- 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]
- 58J10: Differential complexes [See also 35Nxx]; elliptic complexes
- 58J15: Relations with hyperfunctions
- 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]
- 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20]
- 58J26: Elliptic genera
- 58J28: Eta-invariants, Chern-Simons invariants
- 58J30: Spectral flows
- 58J32: Boundary value problems on manifolds
- 58J35: Heat and other parabolic equation methods
- 58J37: Perturbations; asymptotics
- 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]
- 58J42: Noncommutative global analysis, noncommutative residues
- 58J45: Hyperbolic equations [See also 35Lxx]
- 58J47: Propagation of singularities; initial value problems
- 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
- 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
- 58J52: Determinants and determinant bundles, analytic torsion
- 58J53: Isospectrality
- 58J55: Bifurcation [See also 35B32]
- 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.)
- 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
- 58J70: Invariance and symmetry properties [See also 35A30]
- 58J72: Correspondences and other transformation methods (e.g. Lie-Bäcklund) [See also 35A22]
- 58J90: Applications
- 58J99: None of the above, but in this section

- 58Kxx: Theory of singularities and catastrophe theory [See also 32Sxx, 37-XX]
- 58K05: Critical points of functions and mappings
- 58K10: Monodromy
- 58K15: Topological properties of mappings
- 58K20: Algebraic and analytic properties of mappings
- 58K25: Stability
- 58K30: Global theory
- 58K35: Catastrophe theory
- 58K40: Classification; finite determinacy of map germs
- 58K45: Singularities of vector fields, topological aspects
- 58K50: Normal forms
- 58K55: Asymptotic behavior
- 58K60: Deformation of singularities
- 58K65: Topological invariants
- 58K70: Symmetries, equivariance
- 58K99: None of the above, but in this section

- 58Zxx: Applications to physics
- 58Z05: Applications to physics
- 58Z99: None of the above, but in this section

- 60-XX: Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX}
- 60-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 60-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 60-02: Research exposition (monographs, survey articles)
- 60-03: Historical (must also be assigned at least one classification number from Section 01)
- 60-04: Explicit machine computation and programs (not the theory of computation or programming)
- 60-06: Proceedings, conferences, collections, etc.
- 60-08: Computational methods (not classified at a more specific level) [See also 65C50]
- 60Axx: Foundations of probability theory
- 60Bxx: Probability theory on algebraic and topological structures
- 60B05: Probability measures on topological spaces
- 60B10: Convergence of probability measures
- 60B11: Probability theory on linear topological spaces [See also 28C20]
- 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
- 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
- 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
- 60B99: None of the above, but in this section

- 60Cxx: Combinatorial probability
- 60C05: Combinatorial probability
- 60C99: None of the above, but in this section

- 60Dxx: Geometric probability and stochastic geometry [See also 52A22, 53C65]
- 60Exx: Distribution theory [See also 62Exx, 62Hxx]
- 60E05: Distributions: general theory
- 60E07: Infinitely divisible distributions; stable distributions
- 60E10: Characteristic functions; other transforms
- 60E15: Inequalities; stochastic orderings
- 60E99: None of the above, but in this section

- 60Fxx: Limit theorems [See also 28Dxx, 60B12]
- 60F05: Central limit and other weak theorems
- 60F10: Large deviations
- 60F15: Strong theorems
- 60F17: Functional limit theorems; invariance principles
- 60F20: Zero-one laws
- 60F25: $L^p$-limit theorems
- 60F99: None of the above, but in this section

- 60Gxx: Stochastic processes
- 60G05: Foundations of stochastic processes
- 60G07: General theory of processes
- 60G09: Exchangeability
- 60G10: Stationary processes
- 60G12: General second-order processes
- 60G15: Gaussian processes
- 60G17: Sample path properties
- 60G18: Self-similar processes
- 60G20: Generalized stochastic processes
- 60G22: Fractional processes, including fractional Brownian motion
- 60G25: Prediction theory [See also 62M20]
- 60G30: Continuity and singularity of induced measures
- 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]
- 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
- 60G42: Martingales with discrete parameter
- 60G44: Martingales with continuous parameter
- 60G46: Martingales and classical analysis
- 60G48: Generalizations of martingales
- 60G50: Sums of independent random variables; random walks
- 60G51: Processes with independent increments; L\'evy processes
- 60G52: Stable processes
- 60G55: Point processes
- 60G57: Random measures
- 60G60: Random fields
- 60G70: Extreme value theory; extremal processes
- 60G99: None of the above, but in this section

- 60Hxx: Stochastic analysis [See also 58J65]
- 60H05: Stochastic integrals
- 60H07: Stochastic calculus of variations and the Malliavin calculus
- 60H10: Stochastic ordinary differential equations [See also 34F05]
- 60H15: Stochastic partial differential equations [See also 35R60]
- 60H20: Stochastic integral equations
- 60H25: Random operators and equations [See also 47B80]
- 60H30: Applications of stochastic analysis (to PDE, etc.)
- 60H35: Computational methods for stochastic equations [See also 65C30]
- 60H40: White noise theory
- 60H99: None of the above, but in this section

- 60Jxx: Markov processes
- 60J05: Discrete-time Markov processes on general state spaces
- 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
- 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]
- 60J22: Computational methods in Markov chains [See also 65C40]
- 60J25: Continuous-time Markov processes on general state spaces
- 60J27: Continuous-time Markov processes on discrete state spaces
- 60J28: Applications of continuous-time Markov processes on discrete state spaces
- 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
- 60J40: Right processes
- 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
- 60J50: Boundary theory
- 60J55: Local time and additive functionals
- 60J57: Multiplicative functionals
- 60J60: Diffusion processes [See also 58J65]
- 60J65: Brownian motion [See also 58J65]
- 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
- 60J68: Superprocesses
- 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
- 60J75: Jump processes
- 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
- 60J85: Applications of branching processes [See also 92Dxx]
- 60J99: None of the above, but in this section

- 60Kxx: Special processes
- 60K05: Renewal theory
- 60K10: Applications (reliability, demand theory, etc.)
- 60K15: Markov renewal processes, semi-Markov processes
- 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]
- 60K25: Queueing theory [See also 68M20, 90B22]
- 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]
- 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
- 60K37: Processes in random environments
- 60K40: Other physical applications of random processes
- 60K99: None of the above, but in this section

- 62-XX: Statistics
- 62-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 62-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 62-02: Research exposition (monographs, survey articles)
- 62-03: Historical (must also be assigned at least one classification number from Section 01)
- 62-04: Explicit machine computation and programs (not the theory of computation or programming)
- 62-06: Proceedings, conferences, collections, etc.
- 62-07: Data analysis
- 62-09: Graphical methods
- 62Axx: Foundational and philosophical topics
- 62A01: Foundations and philosophical topics
- 62A86: Fuzzy analysis in statistics
- 62A99: None of the above, but in this section

- 62Bxx: Sufficiency and information
- 62B05: Sufficient statistics and fields
- 62B10: Information-theoretic topics [See also 94A17]
- 62B15: Theory of statistical experiments
- 62B86: Fuzziness, sufficiency, and information
- 62B99: None of the above, but in this section

- 62Cxx: Decision theory [See also 90B50, 91B06; for game theory, see 91A35]
- 62C05: General considerations
- 62C07: Complete class results
- 62C10: Bayesian problems; characterization of Bayes procedures
- 62C12: Empirical decision procedures; empirical Bayes procedures
- 62C15: Admissibility
- 62C20: Minimax procedures
- 62C25: Compound decision problems
- 62C86: Decision theory and fuzziness
- 62C99: None of the above, but in this section

- 62Dxx: Sampling theory, sample surveys
- 62D05: Sampling theory, sample surveys
- 62D99: None of the above, but in this section

- 62Exx: Distribution theory [See also 60Exx]
- 62E10: Characterization and structure theory
- 62E15: Exact distribution theory
- 62E17: Approximations to distributions (nonasymptotic)
- 62E20: Asymptotic distribution theory
- 62E86: Fuzziness in connection with the topics on distributions in this section
- 62E99: None of the above, but in this section

- 62Fxx: Parametric inference
- 62F03: Hypothesis testing
- 62F05: Asymptotic properties of tests
- 62F07: Ranking and selection
- 62F10: Point estimation
- 62F12: Asymptotic properties of estimators
- 62F15: Bayesian inference
- 62F25: Tolerance and confidence regions
- 62F30: Inference under constraints
- 62F35: Robustness and adaptive procedures
- 62F40: Bootstrap, jackknife and other resampling methods
- 62F86: Parametric inference and fuzziness
- 62F99: None of the above, but in this section

- 62Gxx: Nonparametric inference
- 62G05: Estimation
- 62G07: Density estimation
- 62G08: Nonparametric regression
- 62G09: Resampling methods
- 62G10: Hypothesis testing
- 62G15: Tolerance and confidence regions
- 62G20: Asymptotic properties
- 62G30: Order statistics; empirical distribution functions
- 62G32: Statistics of extreme values; tail inference
- 62G35: Robustness
- 62G86: Nonparametric inference and fuzziness
- 62G99: None of the above, but in this section

- 62Hxx: Multivariate analysis [See also 60Exx]
- 62H05: Characterization and structure theory
- 62H10: Distribution of statistics
- 62H11: Directional data; spatial statistics
- 62H12: Estimation
- 62H15: Hypothesis testing
- 62H17: Contingency tables
- 62H20: Measures of association (correlation, canonical correlation, etc.)
- 62H25: Factor analysis and principal components; correspondence analysis
- 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]
- 62H35: Image analysis
- 62H86: Multivariate analysis and fuzziness
- 62H99: None of the above, but in this section

- 62Jxx: Linear inference, regression
- 62J02: General nonlinear regression
- 62J05: Linear regression
- 62J07: Ridge regression; shrinkage estimators
- 62J10: Analysis of variance and covariance
- 62J12: Generalized linear models
- 62J15: Paired and multiple comparisons
- 62J20: Diagnostics
- 62J86: Fuzziness, and linear inference and regression
- 62J99: None of the above, but in this section

- 62Kxx: Design of experiments [See also 05Bxx]
- 62K05: Optimal designs
- 62K10: Block designs
- 62K15: Factorial designs
- 62K20: Response surface designs
- 62K25: Robust parameter designs
- 62K86: Fuzziness and design of experiments
- 62K99: None of the above, but in this section

- 62Lxx: Sequential methods
- 62Mxx: Inference from stochastic processes
- 62M02: Markov processes: hypothesis testing
- 62M05: Markov processes: estimation
- 62M07: Non-Markovian processes: hypothesis testing
- 62M09: Non-Markovian processes: estimation
- 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
- 62M15: Spectral analysis
- 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]
- 62M30: Spatial processes
- 62M40: Random fields; image analysis
- 62M45: Neural nets and related approaches
- 62M86: Inference from stochastic processes and fuzziness
- 62M99: None of the above, but in this section

- 62Nxx: Survival analysis and censored data
- 62N01: Censored data models
- 62N02: Estimation
- 62N03: Testing
- 62N05: Reliability and life testing [See also 90B25]
- 62N86: Fuzziness, and survival analysis and censored data
- 62N99: None of the above, but in this section

- 62Pxx: Applications [See also 90-XX, 91-XX, 92-XX]
- 62P05: Applications to actuarial sciences and financial mathematics
- 62P10: Applications to biology and medical sciences
- 62P12: Applications to environmental and related topics
- 62P15: Applications to psychology
- 62P20: Applications to economics [See also 91Bxx]
- 62P25: Applications to social sciences
- 62P30: Applications in engineering and industry
- 62P35: Applications to physics
- 62P99: None of the above, but in this section

- 62Qxx: Statistical tables
- 62Q05: Statistical tables
- 62Q99: None of the above, but in this section

- 65-XX: Numerical analysis
- 65-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 65-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 65-02: Research exposition (monographs, survey articles)
- 65-03: Historical (must also be assigned at least one classification number from Section 01)
- 65-04: Explicit machine computation and programs (not the theory of computation or programming)
- 65-05: Experimental papers
- 65-06: Proceedings, conferences, collections, etc.
- 65Axx: Tables
- 65A05: Tables
- 65A99: None of the above, but in this section

- 65Bxx: Acceleration of convergence
- 65B05: Extrapolation to the limit, deferred corrections
- 65B10: Summation of series
- 65B15: Euler-Maclaurin formula
- 65B99: None of the above, but in this section

- 65Cxx: Probabilistic methods, simulation and stochastic differential equations {For theoretical aspects, see 68U20 and 60H35}
- 65C05: Monte Carlo methods
- 65C10: Random number generation
- 65C20: Models, numerical methods [See also 68U20]
- 65C30: Stochastic differential and integral equations
- 65C35: Stochastic particle methods [See also 82C80]
- 65C40: Computational Markov chains
- 65C50: Other computational problems in probability
- 65C60: Computational problems in statistics
- 65C99: None of the above, but in this section

- 65Dxx: Numerical approximation and computational geometry (primarily algorithms) {For theory, see 41-XX and 68Uxx}
- 65D05: Interpolation
- 65D07: Splines
- 65D10: Smoothing, curve fitting
- 65D15: Algorithms for functional approximation
- 65D17: Computer aided design (modeling of curves and surfaces) [See also 68U07]
- 65D18: Computer graphics, image analysis, and computational geometry [See also 51N05, 68U05]
- 65D19: Computational issues in computer and robotic vision
- 65D20: Computation of special functions, construction of tables [See also 33F05]
- 65D25: Numerical differentiation
- 65D30: Numerical integration
- 65D32: Quadrature and cubature formulas
- 65D99: None of the above, but in this section

- 65Exx: Numerical methods in complex analysis (potential theory, etc.) {For numerical methods in conformal mapping, see also 30C30}
- 65E05: Numerical methods in complex analysis (potential theory, etc.) {For numerical methods in conformal mapping, see also 30C30}
- 65E99: None of the above, but in this section

- 65Fxx: Numerical linear algebra
- 65F05: Direct methods for linear systems and matrix inversion
- 65F08: Preconditioners for iterative methods
- 65F10: Iterative methods for linear systems [See also 65N22]
- 65F15: Eigenvalues, eigenvectors
- 65F18: Inverse eigenvalue problems
- 65F20: Overdetermined systems, pseudoinverses
- 65F22: Ill-posedness, regularization
- 65F25: Orthogonalization
- 65F30: Other matrix algorithms
- 65F35: Matrix norms, conditioning, scaling [See also 15A12, 15A60]
- 65F40: Determinants
- 65F50: Sparse matrices
- 65F60: Matrix exponential and similar matrix functions
- 65F99: None of the above, but in this section

- 65Gxx: Error analysis and interval analysis
- 65G20: Algorithms with automatic result verification
- 65G30: Interval and finite arithmetic
- 65G40: General methods in interval analysis
- 65G50: Roundoff error
- 65G99: None of the above, but in this section

- 65Hxx: Nonlinear algebraic or transcendental equations
- 65Jxx: Numerical analysis in abstract spaces
- 65Kxx: Mathematical programming, optimization and variational techniques
- 65Lxx: Ordinary differential equations
- 65L03: Functional-differential equations
- 65L04: Stiff equations
- 65L05: Initial value problems
- 65L06: Multistep, Runge-Kutta and extrapolation methods
- 65L07: Numerical investigation of stability of solutions
- 65L08: Improperly posed problems
- 65L09: Inverse problems
- 65L10: Boundary value problems
- 65L11: Singularly perturbed problems
- 65L12: Finite difference methods
- 65L15: Eigenvalue problems
- 65L20: Stability and convergence of numerical methods
- 65L50: Mesh generation and refinement
- 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods
- 65L70: Error bounds
- 65L80: Methods for differential-algebraic equations
- 65L99: None of the above, but in this section

- 65Mxx: Partial differential equations, initial value and time-dependent initial-boundary value problems
- 65M06: Finite difference methods
- 65M08: Finite volume methods
- 65M12: Stability and convergence of numerical methods
- 65M15: Error bounds
- 65M20: Method of lines
- 65M22: Solution of discretized equations [See also 65Fxx, 65Hxx]
- 65M25: Method of characteristics
- 65M30: Improperly posed problems
- 65M32: Inverse problems
- 65M38: Boundary element methods
- 65M50: Mesh generation and refinement
- 65M55: Multigrid methods; domain decomposition
- 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
- 65M70: Spectral, collocation and related methods
- 65M75: Probabilistic methods, particle methods, etc.
- 65M80: Fundamental solutions, Green's function methods, etc.
- 65M85: Fictitious domain methods
- 65M99: None of the above, but in this section

- 65Nxx: Partial differential equations, boundary value problems
- 65N06: Finite difference methods
- 65N08: Finite volume methods
- 65N12: Stability and convergence of numerical methods
- 65N15: Error bounds
- 65N20: Ill-posed problems
- 65N21: Inverse problems
- 65N22: Solution of discretized equations [See also 65Fxx, 65Hxx]
- 65N25: Eigenvalue problems
- 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
- 65N35: Spectral, collocation and related methods
- 65N38: Boundary element methods
- 65N40: Method of lines
- 65N45: Method of contraction of the boundary
- 65N50: Mesh generation and refinement
- 65N55: Multigrid methods; domain decomposition
- 65N75: Probabilistic methods, particle methods, etc.
- 65N80: Fundamental solutions, Green's function methods, etc.
- 65N85: Fictitious domain methods
- 65N99: None of the above, but in this section

- 65Pxx: Numerical problems in dynamical systems [See also 37Mxx]
- 65P10: Hamiltonian systems including symplectic integrators
- 65P20: Numerical chaos
- 65P30: Bifurcation problems
- 65P40: Nonlinear stabilities
- 65P99: None of the above, but in this section

- 65Qxx: Difference and functional equations, recurrence relations
- 65Q10: Difference equations
- 65Q20: Functional equations
- 65Q30: Recurrence relations
- 65Q99: None of the above, but in this section

- 65Rxx: Integral equations, integral transforms
- 65R10: Integral transforms
- 65R20: Integral equations
- 65R30: Improperly posed problems
- 65R32: Inverse problems
- 65R99: None of the above, but in this section

- 65Sxx: Graphical methods
- 65S05: Graphical methods
- 65S99: None of the above, but in this section

- 65Txx: Numerical methods in Fourier analysis
- 65T40: Trigonometric approximation and interpolation
- 65T50: Discrete and fast Fourier transforms
- 65T60: Wavelets
- 65T99: None of the above, but in this section

- 65Yxx: Computer aspects of numerical algorithms
- 65Zxx: Applications to physics
- 65Z05: Applications to physics
- 65Z99: None of the above, but in this section

- 68-XX: Computer science {For papers involving machine computations and programs in a specific mathematical area, see Section –04 in that area}
- 68-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 68-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 68-02: Research exposition (monographs, survey articles)
- 68-03: Historical (must also be assigned at least one classification number from Section 01)
- 68-04: Explicit machine computation and programs (not the theory of computation or programming)
- 68-06: Proceedings, conferences, collections, etc.
- 68Mxx: Computer system organization
- 68M01: General
- 68M07: Mathematical problems of computer architecture
- 68M10: Network design and communication [See also 68R10, 90B18]
- 68M11: Internet topics [See also 68U35]
- 68M12: Network protocols
- 68M14: Distributed systems
- 68M15: Reliability, testing and fault tolerance [See also 94C12]
- 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]
- 68M99: None of the above, but in this section

- 68Nxx: Software
- 68N01: General
- 68N15: Programming languages
- 68N17: Logic programming
- 68N18: Functional programming and lambda calculus [See also 03B40]
- 68N19: Other programming techniques (object-oriented, sequential, concurrent, automatic, etc.)
- 68N20: Compilers and interpreters
- 68N25: Operating systems
- 68N30: Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
- 68N99: None of the above, but in this section

- 68Pxx: Theory of data
- 68P01: General
- 68P05: Data structures
- 68P10: Searching and sorting
- 68P15: Database theory
- 68P20: Information storage and retrieval
- 68P25: Data encryption [See also 94A60, 81P94]
- 68P30: Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]
- 68P99: None of the above, but in this section

- 68Qxx: Theory of computing
- 68Q01: General
- 68Q05: Models of computation (Turing machines, etc.) [See also 03D10, 68Q12, 81P68]
- 68Q10: Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) [See also 68Q85]
- 68Q12: Quantum algorithms and complexity [See also 68Q05, 81P68]
- 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19]
- 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15]
- 68Q19: Descriptive complexity and finite models [See also 03C13]
- 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
- 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32]
- 68Q32: Computational learning theory [See also 68T05]
- 68Q42: Grammars and rewriting systems
- 68Q45: Formal languages and automata [See also 03D05, 68Q70, 94A45]
- 68Q55: Semantics [See also 03B70, 06B35, 18C50]
- 68Q60: Specification and verification (program logics, model checking, etc.) [See also 03B70]
- 68Q65: Abstract data types; algebraic specification [See also 18C50]
- 68Q70: Algebraic theory of languages and automata [See also 18B20, 20M35]
- 68Q80: Cellular automata [See also 37B15]
- 68Q85: Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
- 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]
- 68Q99: None of the above, but in this section

- 68Rxx: Discrete mathematics in relation to computer science
- 68Txx: Artificial intelligence
- 68T01: General
- 68T05: Learning and adaptive systems [See also 68Q32, 91E40]
- 68T10: Pattern recognition, speech recognition {For cluster analysis, see 62H30}
- 68T15: Theorem proving (deduction, resolution, etc.) [See also 03B35]
- 68T20: Problem solving (heuristics, search strategies, etc.)
- 68T27: Logic in artificial intelligence
- 68T30: Knowledge representation
- 68T35: Languages and software systems (knowledge-based systems, expert systems, etc.)
- 68T37: Reasoning under uncertainty
- 68T40: Robotics [See also 93C85]
- 68T42: Agent technology
- 68T45: Machine vision and scene understanding
- 68T50: Natural language processing [See also 03B65]
- 68T99: None of the above, but in this section

- 68Uxx: Computing methodologies and applications
- 68U01: General
- 68U05: Computer graphics; computational geometry [See also 65D18]
- 68U07: Computer-aided design [See also 65D17]
- 68U10: Image processing
- 68U15: Text processing; mathematical typography
- 68U20: Simulation [See also 65Cxx]
- 68U35: Information systems (hypertext navigation, interfaces, decision support, etc.) [See also 68M11]
- 68U99: None of the above, but in this section

- 68Wxx: Algorithms {For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 05C85, 68Rxx}
- 68W01: General
- 68W05: Nonnumerical algorithms
- 68W10: Parallel algorithms
- 68W15: Distributed algorithms
- 68W20: Randomized algorithms
- 68W25: Approximation algorithms
- 68W27: Online algorithms
- 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
- 68W32: Algorithms on strings
- 68W35: VLSI algorithms
- 68W40: Analysis of algorithms [See also 68Q25]
- 68W99: None of the above, but in this section

- 70-XX: Mechanics of particles and systems {For relativistic mechanics, see 83A05 and 83C10; for statistical mechanics, see 82-XX}
- 70-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 70-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 70-02: Research exposition (monographs, survey articles)
- 70-03: Historical (must also be assigned at least one classification number from Section 01)
- 70-04: Explicit machine computation and programs (not the theory of computation or programming)
- 70-05: Experimental work
- 70-06: Proceedings, conferences, collections, etc.
- 70-08: Computational methods
- 70Axx: Axiomatics, foundations
- 70A05: Axiomatics, foundations
- 70A99: None of the above, but in this section

- 70Bxx: Kinematics [See also 53A17]
- 70Cxx: Statics
- 70C20: Statics
- 70C99: None of the above, but in this section

- 70Exx: Dynamics of a rigid body and of multibody systems
- 70E05: Motion of the gyroscope
- 70E15: Free motion of a rigid body [See also 70M20]
- 70E17: Motion of a rigid body with a fixed point
- 70E18: Motion of a rigid body in contact with a solid surface [See also 70F25]
- 70E20: Perturbation methods for rigid body dynamics
- 70E40: Integrable cases of motion
- 70E45: Higher-dimensional generalizations
- 70E50: Stability problems
- 70E55: Dynamics of multibody systems
- 70E60: Robot dynamics and control [See also 68T40, 70Q05, 93C85]
- 70E99: None of the above, but in this section

- 70Fxx: Dynamics of a system of particles, including celestial mechanics
- 70F05: Two-body problems
- 70F07: Three-body problems
- 70F10: $n$-body problems
- 70F15: Celestial mechanics
- 70F16: Collisions in celestial mechanics, regularization
- 70F17: Inverse problems
- 70F20: Holonomic systems
- 70F25: Nonholonomic systems
- 70F35: Collision of rigid or pseudo-rigid bodies
- 70F40: Problems with friction
- 70F45: Infinite particle systems
- 70F99: None of the above, but in this section

- 70Gxx: General models, approaches, and methods [See also 37-XX]
- 70G10: Generalized coordinates; event, impulse-energy, configuration, state, or phase space
- 70G40: Topological and differential-topological methods
- 70G45: Differential-geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) [See also 53Cxx, 53Dxx, 58Axx]
- 70G55: Algebraic geometry methods
- 70G60: Dynamical systems methods
- 70G65: Symmetries, Lie-group and Lie-algebra methods
- 70G70: Functional-analytic methods
- 70G75: Variational methods
- 70G99: None of the above, but in this section

- 70Hxx: Hamiltonian and Lagrangian mechanics [See also 37Jxx]
- 70H03: Lagrange's equations
- 70H05: Hamilton's equations
- 70H06: Completely integrable systems and methods of integration
- 70H07: Nonintegrable systems
- 70H08: Nearly integrable Hamiltonian systems, KAM theory
- 70H09: Perturbation theories
- 70H11: Adiabatic invariants
- 70H12: Periodic and almost periodic solutions
- 70H14: Stability problems
- 70H15: Canonical and symplectic transformations
- 70H20: Hamilton-Jacobi equations
- 70H25: Hamilton's principle
- 70H30: Other variational principles
- 70H33: Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
- 70H40: Relativistic dynamics
- 70H45: Constrained dynamics, Dirac's theory of constraints [See also 70F20, 70F25, 70Gxx]
- 70H50: Higher-order theories
- 70H99: None of the above, but in this section

- 70Jxx: Linear vibration theory
- 70J10: Modal analysis
- 70J25: Stability
- 70J30: Free motions
- 70J35: Forced motions
- 70J40: Parametric resonances
- 70J50: Systems arising from the discretization of structural vibration problems
- 70J99: None of the above, but in this section

- 70Kxx: Nonlinear dynamics [See also 34Cxx, 37-XX]
- 70K05: Phase plane analysis, limit cycles
- 70K20: Stability
- 70K25: Free motions
- 70K28: Parametric resonances
- 70K30: Nonlinear resonances
- 70K40: Forced motions
- 70K42: Equilibria and periodic trajectories
- 70K43: Quasi-periodic motions and invariant tori
- 70K44: Homoclinic and heteroclinic trajectories
- 70K45: Normal forms
- 70K50: Bifurcations and instability
- 70K55: Transition to stochasticity (chaotic behavior) [See also 37D45]
- 70K60: General perturbation schemes
- 70K65: Averaging of perturbations
- 70K70: Systems with slow and fast motions
- 70K75: Nonlinear modes
- 70K99: None of the above, but in this section

- 70Lxx: Random vibrations [See also 74H50]
- 70L05: Random vibrations [See also 74H50]
- 70L99: None of the above, but in this section

- 70Mxx: Orbital mechanics
- 70M20: Orbital mechanics
- 70M99: None of the above, but in this section

- 70Pxx: Variable mass, rockets
- 70P05: Variable mass, rockets
- 70P99: None of the above, but in this section

- 70Qxx: Control of mechanical systems [See also 60Gxx, 60Jxx]
- 70Q05: Control of mechanical systems
- 70Q99: None of the above, but in this section

- 70Sxx: Classical field theories [See also 37Kxx, 37Lxx, 78-XX, 81Txx, 83-XX]
- 70S05: Lagrangian formalism and Hamiltonian formalism
- 70S10: Symmetries and conservation laws
- 70S15: Yang-Mills and other gauge theories
- 70S20: More general nonquantum field theories
- 70S99: None of the above, but in this section

- 74-XX: Mechanics of deformable solids
- 74-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 74-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 74-02: Research exposition (monographs, survey articles)
- 74-03: Historical (must also be assigned at least one classification number from Section 01)
- 74-04: Explicit machine computation and programs (not the theory of computation or programming)
- 74-05: Experimental work
- 74-06: Proceedings, conferences, collections, etc.
- 74Axx: Generalities, axiomatics, foundations of continuum mechanics of solids
- 74A05: Kinematics of deformation
- 74A10: Stress
- 74A15: Thermodynamics
- 74A20: Theory of constitutive functions
- 74A25: Molecular, statistical, and kinetic theories
- 74A30: Nonsimple materials
- 74A35: Polar materials
- 74A40: Random materials and composite materials
- 74A45: Theories of fracture and damage
- 74A50: Structured surfaces and interfaces, coexistent phases
- 74A55: Theories of friction (tribology)
- 74A60: Micromechanical theories
- 74A65: Reactive materials
- 74A99: None of the above, but in this section

- 74Bxx: Elastic materials
- 74B05: Classical linear elasticity
- 74B10: Linear elasticity with initial stresses
- 74B15: Equations linearized about a deformed state (small deformations superposed on large)
- 74B20: Nonlinear elasticity
- 74B99: None of the above, but in this section

- 74Cxx: Plastic materials, materials of stress-rate and internal-variable type
- 74C05: Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials)
- 74C10: Small-strain, rate-dependent theories (including theories of viscoplasticity)
- 74C15: Large-strain, rate-independent theories (including nonlinear plasticity)
- 74C20: Large-strain, rate-dependent theories
- 74C99: None of the above, but in this section

- 74Dxx: Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
- 74D05: Linear constitutive equations
- 74D10: Nonlinear constitutive equations
- 74D99: None of the above, but in this section

- 74Exx: Material properties given special treatment
- 74E05: Inhomogeneity
- 74E10: Anisotropy
- 74E15: Crystalline structure
- 74E20: Granularity
- 74E25: Texture
- 74E30: Composite and mixture properties
- 74E35: Random structure
- 74E40: Chemical structure
- 74E99: None of the above, but in this section

- 74Fxx: Coupling of solid mechanics with other effects
- 74F05: Thermal effects
- 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
- 74F15: Electromagnetic effects
- 74F20: Mixture effects
- 74F25: Chemical and reactive effects
- 74F99: None of the above, but in this section

- 74Gxx: Equilibrium (steady-state) problems
- 74G05: Explicit solutions
- 74G10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.)
- 74G15: Numerical approximation of solutions
- 74G20: Local existence of solutions (near a given solution)
- 74G25: Global existence of solutions
- 74G30: Uniqueness of solutions
- 74G35: Multiplicity of solutions
- 74G40: Regularity of solutions
- 74G45: Bounds for solutions
- 74G50: Saint-Venant's principle
- 74G55: Qualitative behavior of solutions
- 74G60: Bifurcation and buckling
- 74G65: Energy minimization
- 74G70: Stress concentrations, singularities
- 74G75: Inverse problems
- 74G99: None of the above, but in this section

- 74Hxx: Dynamical problems
- 74H05: Explicit solutions
- 74H10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.)
- 74H15: Numerical approximation of solutions
- 74H20: Existence of solutions
- 74H25: Uniqueness of solutions
- 74H30: Regularity of solutions
- 74H35: Singularities, blowup, stress concentrations
- 74H40: Long-time behavior of solutions
- 74H45: Vibrations
- 74H50: Random vibrations
- 74H55: Stability
- 74H60: Dynamical bifurcation
- 74H65: Chaotic behavior
- 74H99: None of the above, but in this section

- 74Jxx: Waves
- 74J05: Linear waves
- 74J10: Bulk waves
- 74J15: Surface waves
- 74J20: Wave scattering
- 74J25: Inverse problems
- 74J30: Nonlinear waves
- 74J35: Solitary waves
- 74J40: Shocks and related discontinuities
- 74J99: None of the above, but in this section

- 74Kxx: Thin bodies, structures
- 74K05: Strings
- 74K10: Rods (beams, columns, shafts, arches, rings, etc.)
- 74K15: Membranes
- 74K20: Plates
- 74K25: Shells
- 74K30: Junctions
- 74K35: Thin films
- 74K99: None of the above, but in this section

- 74Lxx: Special subfields of solid mechanics
- 74Mxx: Special kinds of problems
- 74M05: Control, switches and devices (“smart materials”) [See also 93Cxx]
- 74M10: Friction
- 74M15: Contact
- 74M20: Impact
- 74M25: Micromechanics
- 74M99: None of the above, but in this section

- 74Nxx: Phase transformations in solids [See also 74A50, 80Axx, 82B26, 82C26]
- 74N05: Crystals
- 74N10: Displacive transformations
- 74N15: Analysis of microstructure
- 74N20: Dynamics of phase boundaries
- 74N25: Transformations involving diffusion
- 74N30: Problems involving hysteresis
- 74N99: None of the above, but in this section

- 74Pxx: Optimization [See also 49Qxx]
- 74P05: Compliance or weight optimization
- 74P10: Optimization of other properties
- 74P15: Topological methods
- 74P20: Geometrical methods
- 74P99: None of the above, but in this section

- 74Qxx: Homogenization, determination of effective properties
- 74Q05: Homogenization in equilibrium problems
- 74Q10: Homogenization and oscillations in dynamical problems
- 74Q15: Effective constitutive equations
- 74Q20: Bounds on effective properties
- 74Q99: None of the above, but in this section

- 74Rxx: Fracture and damage
- 74R05: Brittle damage
- 74R10: Brittle fracture
- 74R15: High-velocity fracture
- 74R20: Anelastic fracture and damage
- 74R99: None of the above, but in this section

- 74Sxx: Numerical methods [See also 65-XX, 74G15, 74H15]
- 74S05: Finite element methods
- 74S10: Finite volume methods
- 74S15: Boundary element methods
- 74S20: Finite difference methods
- 74S25: Spectral and related methods
- 74S30: Other numerical methods
- 74S60: Stochastic methods
- 74S70: Complex variable methods
- 74S99: None of the above, but in this section

- 76-XX: Fluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX}
- 76-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 76-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 76-02: Research exposition (monographs, survey articles)
- 76-03: Historical (must also be assigned at least one classification number from Section 01)
- 76-04: Explicit machine computation and programs (not the theory of computation or programming)
- 76-05: Experimental work
- 76-06: Proceedings, conferences, collections, etc.
- 76Axx: Foundations, constitutive equations, rheology
- 76A02: Foundations of fluid mechanics
- 76A05: Non-Newtonian fluids
- 76A10: Viscoelastic fluids
- 76A15: Liquid crystals [See also 82D30]
- 76A20: Thin fluid films
- 76A25: Superfluids (classical aspects)
- 76A99: None of the above, but in this section

- 76Bxx: Incompressible inviscid fluids
- 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]
- 76B07: Free-surface potential flows
- 76B10: Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
- 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]
- 76B20: Ship waves
- 76B25: Solitary waves [See also 35C11]
- 76B45: Capillarity (surface tension) [See also 76D45]
- 76B47: Vortex flows
- 76B55: Internal waves
- 76B60: Atmospheric waves [See also 86A10]
- 76B65: Rossby waves [See also 86A05, 86A10]
- 76B70: Stratification effects in inviscid fluids
- 76B75: Flow control and optimization [See also 49Q10, 93C20, 93C95]
- 76B99: None of the above, but in this section

- 76Dxx: Incompressible viscous fluids
- 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]
- 76D05: Navier-Stokes equations [See also 35Q30]
- 76D06: Statistical solutions of Navier-Stokes and related equations [See also 60H30, 76M35]
- 76D07: Stokes and related (Oseen, etc.) flows
- 76D08: Lubrication theory
- 76D09: Viscous-inviscid interaction
- 76D10: Boundary-layer theory, separation and reattachment, higher-order effects
- 76D17: Viscous vortex flows
- 76D25: Wakes and jets
- 76D27: Other free-boundary flows; Hele-Shaw flows
- 76D33: Waves
- 76D45: Capillarity (surface tension) [See also 76B45]
- 76D50: Stratification effects in viscous fluids
- 76D55: Flow control and optimization [See also 49Q10, 93C20, 93C95]
- 76D99: None of the above, but in this section

- 76Exx: Hydrodynamic stability
- 76E05: Parallel shear flows
- 76E06: Convection
- 76E07: Rotation
- 76E09: Stability and instability of nonparallel flows
- 76E15: Absolute and convective instability and stability
- 76E17: Interfacial stability and instability
- 76E19: Compressibility effects
- 76E20: Stability and instability of geophysical and astrophysical flows
- 76E25: Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
- 76E30: Nonlinear effects
- 76E99: None of the above, but in this section

- 76Fxx: Turbulence [See also 37-XX, 60Gxx, 60Jxx]
- 76F02: Fundamentals
- 76F05: Isotropic turbulence; homogeneous turbulence
- 76F06: Transition to turbulence
- 76F10: Shear flows
- 76F20: Dynamical systems approach to turbulence [See also 37-XX]
- 76F25: Turbulent transport, mixing
- 76F30: Renormalization and other field-theoretical methods [See also 81T99]
- 76F35: Convective turbulence [See also 76E15, 76Rxx]
- 76F40: Turbulent boundary layers
- 76F45: Stratification effects
- 76F50: Compressibility effects
- 76F55: Statistical turbulence modeling [See also 76M35]
- 76F60: $k$-$\varepsilon$ modeling
- 76F65: Direct numerical and large eddy simulation of turbulence
- 76F70: Control of turbulent flows
- 76F99: None of the above, but in this section

- 76Gxx: General aerodynamics and subsonic flows
- 76G25: General aerodynamics and subsonic flows
- 76G99: None of the above, but in this section

- 76Hxx: Transonic flows
- 76H05: Transonic flows
- 76H99: None of the above, but in this section

- 76Jxx: Supersonic flows
- 76J20: Supersonic flows
- 76J99: None of the above, but in this section

- 76Kxx: Hypersonic flows
- 76K05: Hypersonic flows
- 76K99: None of the above, but in this section

- 76Lxx: Shock waves and blast waves [See also 35L67]
- 76L05: Shock waves and blast waves [See also 35L67]
- 76L99: None of the above, but in this section

- 76Mxx: Basic methods in fluid mechanics [See also 65-XX]
- 76M10: Finite element methods
- 76M12: Finite volume methods
- 76M15: Boundary element methods
- 76M20: Finite difference methods
- 76M22: Spectral methods
- 76M23: Vortex methods
- 76M25: Other numerical methods
- 76M27: Visualization algorithms
- 76M28: Particle methods and lattice-gas methods
- 76M30: Variational methods
- 76M35: Stochastic analysis
- 76M40: Complex-variables methods
- 76M45: Asymptotic methods, singular perturbations
- 76M50: Homogenization
- 76M55: Dimensional analysis and similarity
- 76M60: Symmetry analysis, Lie group and algebra methods
- 76M99: None of the above, but in this section

- 76Nxx: Compressible fluids and gas dynamics, general
- 76Pxx: Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]
- 76Qxx: Hydro- and aero-acoustics
- 76Q05: Hydro- and aero-acoustics
- 76Q99: None of the above, but in this section

- 76Rxx: Diffusion and convection
- 76R05: Forced convection
- 76R10: Free convection
- 76R50: Diffusion [See also 60J60]
- 76R99: None of the above, but in this section

- 76Sxx: Flows in porous media; filtration; seepage
- 76S05: Flows in porous media; filtration; seepage
- 76S99: None of the above, but in this section

- 76Txx: Two-phase and multiphase flows
- 76Uxx: Rotating fluids
- 76U05: Rotating fluids
- 76U99: None of the above, but in this section

- 76Vxx: Reaction effects in flows [See also 80A32]
- 76V05: Reaction effects in flows [See also 80A32]
- 76V99: None of the above, but in this section

- 76Wxx: Magnetohydrodynamics and electrohydrodynamics
- 76W05: Magnetohydrodynamics and electrohydrodynamics
- 76W99: None of the above, but in this section

- 76Xxx: Ionized gas flow in electromagnetic fields; plasmic flow [See also 82D10]
- 76X05: Ionized gas flow in electromagnetic fields; plasmic flow [See also 82D10]
- 76X99: None of the above, but in this section

- 76Yxx: Quantum hydrodynamics and relativistic hydrodynamics [See also 82D50, 83C55, 85A30]
- 76Zxx: Biological fluid mechanics [See also 74F10, 74L15, 92Cxx]
- 76Z05: Physiological flows [See also 92C35]
- 76Z10: Biopropulsion in water and in air
- 76Z99: None of the above, but in this section

- 78-XX: Optics, electromagnetic theory {For quantum optics, see 81V80}
- 78-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 78-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 78-02: Research exposition (monographs, survey articles)
- 78-03: Historical (must also be assigned at least one classification number from Section 01)
- 78-04: Explicit machine computation and programs (not the theory of computation or programming)
- 78-05: Experimental work
- 78-06: Proceedings, conferences, collections, etc.
- 78Axx: General
- 78A02: Foundations
- 78A05: Geometric optics
- 78A10: Physical optics
- 78A15: Electron optics
- 78A20: Space charge waves
- 78A25: Electromagnetic theory, general
- 78A30: Electro- and magnetostatics
- 78A35: Motion of charged particles
- 78A37: Ion traps
- 78A40: Waves and radiation
- 78A45: Diffraction, scattering [See also 34E20 for WKB methods]
- 78A46: Inverse scattering problems
- 78A48: Composite media; random media
- 78A50: Antennas, wave-guides
- 78A55: Technical applications
- 78A57: Electrochemistry
- 78A60: Lasers, masers, optical bistability, nonlinear optics [See also 81V80]
- 78A70: Biological applications [See also 91D30, 92C30]
- 78A97: Mathematically heuristic optics and electromagnetic theory (must also be assigned at least one other classification number in this section)
- 78A99: Miscellaneous topics

- 78Mxx: Basic methods
- 78M05: Method of moments
- 78M10: Finite element methods
- 78M12: Finite volume methods, finite integration techniques
- 78M15: Boundary element methods
- 78M16: Multipole methods
- 78M20: Finite difference methods
- 78M22: Spectral methods
- 78M25: Other numerical methods
- 78M30: Variational methods
- 78M31: Monte Carlo methods
- 78M32: Neural and heuristic methods
- 78M34: Model reduction
- 78M35: Asymptotic analysis
- 78M40: Homogenization
- 78M50: Optimization
- 78M99: None of the above, but in this section

- 80-XX: Classical thermodynamics, heat transfer {For thermodynamics of solids, see 74A15}
- 80-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 80-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 80-02: Research exposition (monographs, survey articles)
- 80-03: Historical (must also be assigned at least one classification number from Section 01)
- 80-04: Explicit machine computation and programs (not the theory of computation or programming)
- 80-05: Experimental work
- 80-06: Proceedings, conferences, collections, etc.
- 80Axx: Thermodynamics and heat transfer
- 80A05: Foundations
- 80A10: Classical thermodynamics, including relativistic
- 80A17: Thermodynamics of continua [See also 74A15]
- 80A20: Heat and mass transfer, heat flow
- 80A22: Stefan problems, phase changes, etc. [See also 74Nxx]
- 80A23: Inverse problems
- 80A25: Combustion
- 80A30: Chemical kinetics [See also 76V05, 92C45, 92E20]
- 80A32: Chemically reacting flows [See also 92C45, 92E20]
- 80A50: Chemistry (general) [See mainly 92Exx]
- 80A99: None of the above, but in this section

- 80Mxx: Basic methods
- 80M10: Finite element methods
- 80M12: Finite volume methods
- 80M15: Boundary element methods
- 80M20: Finite difference methods
- 80M22: Spectral methods
- 80M25: Other numerical methods
- 80M30: Variational methods
- 80M31: Monte Carlo methods
- 80M35: Asymptotic analysis
- 80M40: Homogenization
- 80M50: Optimization
- 80M99: None of the above, but in this section

- 81-XX: Quantum theory
- 81-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 81-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 81-02: Research exposition (monographs, survey articles)
- 81-03: Historical (must also be assigned at least one classification number from Section 01)
- 81-04: Explicit machine computation and programs (not the theory of computation or programming)
- 81-05: Experimental papers
- 81-06: Proceedings, conferences, collections, etc.
- 81-08: Computational methods
- 81Pxx: Axiomatics, foundations, philosophy
- 81P05: General and philosophical
- 81P10: Logical foundations of quantum mechanics; quantum logic [See also 03G12, 06C15]
- 81P13: Contextuality
- 81P15: Quantum measurement theory
- 81P16: Quantum state spaces, operational and probabilistic concepts
- 81P20: Stochastic mechanics (including stochastic electrodynamics)
- 81P40: Quantum coherence, entanglement, quantum correlations
- 81P45: Quantum information, communication, networks [See also 94A15, 94A17]
- 81P50: Quantum state estimation, approximate cloning
- 81P68: Quantum computation [See also 68Q05, 68Q12]
- 81P70: Quantum coding (general)
- 81P94: Quantum cryptography [See also 94A60]
- 81P99: None of the above, but in this section

- 81Qxx: General mathematical topics and methods in quantum theory
- 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
- 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
- 81Q12: Non-selfadjoint operator theory in quantum theory
- 81Q15: Perturbation theories for operators and differential equations
- 81Q20: Semiclassical techniques, including WKB and Maslov methods
- 81Q30: Feynman integrals and graphs; applications of algebraic topology and algebraic geometry [See also 14D05, 32S40]
- 81Q35: Quantum mechanics on special spaces: manifolds, fractals, graphs, etc.
- 81Q37: Quantum dots, waveguides, ratchets, etc.
- 81Q40: Bethe-Salpeter and other integral equations
- 81Q50: Quantum chaos [See also 37Dxx]
- 81Q60: Supersymmetry and quantum mechanics
- 81Q65: Alternative quantum mechanics
- 81Q70: Differential-geometric methods, including holonomy, Berry and Hannay phases, etc.
- 81Q80: Special quantum systems, such as solvable systems
- 81Q93: Quantum control
- 81Q99: None of the above, but in this section

- 81Rxx: Groups and algebras in quantum theory
- 81R05: Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70]
- 81R10: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations [See also 17B65, 17B67, 22E65, 22E67, 22E70]
- 81R12: Relations with integrable systems [See also 17Bxx, 37J35]
- 81R15: Operator algebra methods [See also 46Lxx, 81T05]
- 81R20: Covariant wave equations
- 81R25: Spinor and twistor methods [See also 32L25]
- 81R30: Coherent states [See also 22E45]; squeezed states [See also 81V80]
- 81R40: Symmetry breaking
- 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]
- 81R60: Noncommutative geometry
- 81R99: None of the above, but in this section

- 81Sxx: General quantum mechanics and problems of quantization
- 81S05: Canonical quantization, commutation relations and statistics
- 81S10: Geometry and quantization, symplectic methods [See also 53D50]
- 81S20: Stochastic quantization
- 81S22: Open systems, reduced dynamics, master equations, decoherence [See also 82C31]
- 81S25: Quantum stochastic calculus
- 81S30: Phase-space methods including Wigner distributions, etc.
- 81S40: Path integrals [See also 58D30]
- 81S99: None of the above, but in this section

- 81Txx: Quantum field theory; related classical field theories [See also 70Sxx]
- 81T05: Axiomatic quantum field theory; operator algebras
- 81T08: Constructive quantum field theory
- 81T10: Model quantum field theories
- 81T13: Yang-Mills and other gauge theories [See also 53C07, 58E15]
- 81T15: Perturbative methods of renormalization
- 81T16: Nonperturbative methods of renormalization
- 81T17: Renormalization group methods
- 81T18: Feynman diagrams
- 81T20: Quantum field theory on curved space backgrounds
- 81T25: Quantum field theory on lattices
- 81T27: Continuum limits
- 81T28: Thermal quantum field theory [See also 82B30]
- 81T30: String and superstring theories; other extended objects (e.g., branes) [See also 83E30]
- 81T40: Two-dimensional field theories, conformal field theories, etc.
- 81T45: Topological field theories [See also 57R56, 58Dxx]
- 81T50: Anomalies
- 81T55: Casimir effect
- 81T60: Supersymmetric field theories
- 81T70: Quantization in field theory; cohomological methods [See also 58D29]
- 81T75: Noncommutative geometry methods [See also 46L85, 46L87, 58B34]
- 81T80: Simulation and numerical modeling
- 81T99: None of the above, but in this section

- 81Uxx: Scattering theory [See also 34A55, 34L25, 34L40, 35P25, 47A40]
- 81U05: $2$-body potential scattering theory [See also 34E20 for WKB methods]
- 81U10: $n$-body potential scattering theory
- 81U15: Exactly and quasi-solvable systems
- 81U20: $S$-matrix theory, etc.
- 81U30: Dispersion theory, dispersion relations
- 81U35: Inelastic and multichannel scattering
- 81U40: Inverse scattering problems
- 81U99: None of the above, but in this section

- 81Vxx: Applications to specific physical systems
- 81V05: Strong interaction, including quantum chromodynamics
- 81V10: Electromagnetic interaction; quantum electrodynamics
- 81V15: Weak interaction
- 81V17: Gravitational interaction [See also 83Cxx and 83Exx]
- 81V19: Other fundamental interactions
- 81V22: Unified theories
- 81V25: Other elementary particle theory
- 81V35: Nuclear physics
- 81V45: Atomic physics
- 81V55: Molecular physics [See also 92E10]
- 81V65: Quantum dots [See also 82D20]
- 81V70: Many-body theory; quantum Hall effect
- 81V80: Quantum optics
- 81V99: None of the above, but in this section

- 82-XX: Statistical mechanics, structure of matter
- 82-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 82-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 82-02: Research exposition (monographs, survey articles)
- 82-03: Historical (must also be assigned at least one classification number from Section 01)
- 82-04: Explicit machine computation and programs (not the theory of computation or programming)
- 82-05: Experimental papers
- 82-06: Proceedings, conferences, collections, etc.
- 82-08: Computational methods
- 82Bxx: Equilibrium statistical mechanics
- 82B03: Foundations
- 82B05: Classical equilibrium statistical mechanics (general)
- 82B10: Quantum equilibrium statistical mechanics (general)
- 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
- 82B21: Continuum models (systems of particles, etc.)
- 82B23: Exactly solvable models; Bethe ansatz
- 82B24: Interface problems; diffusion-limited aggregation
- 82B26: Phase transitions (general)
- 82B27: Critical phenomena
- 82B28: Renormalization group methods [See also 81T17]
- 82B30: Statistical thermodynamics [See also 80-XX]
- 82B31: Stochastic methods
- 82B35: Irreversible thermodynamics, including Onsager-Machlup theory [See also 92E20]
- 82B40: Kinetic theory of gases
- 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
- 82B43: Percolation [See also 60K35]
- 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
- 82B80: Numerical methods (Monte Carlo, series resummation, etc.) [See also 65-XX, 81T80]
- 82B99: None of the above, but in this section

- 82Cxx: Time-dependent statistical mechanics (dynamic and nonequilibrium)
- 82C03: Foundations
- 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)
- 82C10: Quantum dynamics and nonequilibrium statistical mechanics (general)
- 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
- 82C21: Dynamic continuum models (systems of particles, etc.)
- 82C22: Interacting particle systems [See also 60K35]
- 82C23: Exactly solvable dynamic models [See also 37K60]
- 82C24: Interface problems; diffusion-limited aggregation
- 82C26: Dynamic and nonequilibrium phase transitions (general)
- 82C27: Dynamic critical phenomena
- 82C28: Dynamic renormalization group methods [See also 81T17]
- 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10]
- 82C32: Neural nets [See also 68T05, 91E40, 92B20]
- 82C35: Irreversible thermodynamics, including Onsager-Machlup theory
- 82C40: Kinetic theory of gases
- 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
- 82C43: Time-dependent percolation [See also 60K35]
- 82C44: Dynamics of disordered systems (random Ising systems, etc.)
- 82C70: Transport processes
- 82C80: Numerical methods (Monte Carlo, series resummation, etc.)
- 82C99: None of the above, but in this section

- 82Dxx: Applications to specific types of physical systems
- 82D05: Gases
- 82D10: Plasmas
- 82D15: Liquids
- 82D20: Solids
- 82D25: Crystals {For crystallographic group theory, see 20H15}
- 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
- 82D35: Metals
- 82D37: Semiconductors
- 82D40: Magnetic materials
- 82D45: Ferroelectrics
- 82D50: Superfluids
- 82D55: Superconductors
- 82D60: Polymers
- 82D75: Nuclear reactor theory; neutron transport
- 82D77: Quantum wave guides, quantum wires [See also 78A50]
- 82D80: Nanostructures and nanoparticles
- 82D99: None of the above, but in this section

- 83-XX: Relativity and gravitational theory
- 83-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 83-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 83-02: Research exposition (monographs, survey articles)
- 83-03: Historical (must also be assigned at least one classification number from Section 01)
- 83-04: Explicit machine computation and programs (not the theory of computation or programming)
- 83-05: Experimental work
- 83-06: Proceedings, conferences, collections, etc.
- 83-08: Computational methods
- 83Axx: Special relativity
- 83A05: Special relativity
- 83A99: None of the above, but in this section

- 83Bxx: Observational and experimental questions
- 83B05: Observational and experimental questions
- 83B99: None of the above, but in this section

- 83Cxx: General relativity
- 83C05: Einstein's equations (general structure, canonical formalism, Cauchy problems)
- 83C10: Equations of motion
- 83C15: Exact solutions
- 83C20: Classes of solutions; algebraically special solutions, metrics with symmetries
- 83C22: Einstein-Maxwell equations
- 83C25: Approximation procedures, weak fields
- 83C27: Lattice gravity, Regge calculus and other discrete methods
- 83C30: Asymptotic procedures (radiation, news functions, $\scr H$-spaces, etc.)
- 83C35: Gravitational waves
- 83C40: Gravitational energy and conservation laws; groups of motions
- 83C45: Quantization of the gravitational field
- 83C47: Methods of quantum field theory [See also 81T20]
- 83C50: Electromagnetic fields
- 83C55: Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
- 83C57: Black holes
- 83C60: Spinor and twistor methods; Newman-Penrose formalism
- 83C65: Methods of noncommutative geometry [See also 58B34]
- 83C75: Space-time singularities, cosmic censorship, etc.
- 83C80: Analogues in lower dimensions
- 83C99: None of the above, but in this section

- 83Dxx: Relativistic gravitational theories other than Einstein's, including asymmetric field theories
- 83D05: Relativistic gravitational theories other than Einstein's, including asymmetric field theories
- 83D99: None of the above, but in this section

- 83Exx: Unified, higher-dimensional and super field theories
- 83E05: Geometrodynamics
- 83E15: Kaluza-Klein and other higher-dimensional theories
- 83E30: String and superstring theories [See also 81T30]
- 83E50: Supergravity
- 83E99: None of the above, but in this section

- 83Fxx: Cosmology
- 83F05: Cosmology
- 83F99: None of the above, but in this section

- 85-XX: Astronomy and astrophysics {For celestial mechanics, see 70F15}
- 85-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 85-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 85-02: Research exposition (monographs, survey articles)
- 85-03: Historical (must also be assigned at least one classification number from Section 01)
- 85-04: Explicit machine computation and programs (not the theory of computation or programming)
- 85-05: Experimental work
- 85-06: Proceedings, conferences, collections, etc.
- 85-08: Computational methods
- 85Axx: Astronomy and astrophysics {For celestial mechanics, see 70F15}
- 85A04: General
- 85A05: Galactic and stellar dynamics
- 85A15: Galactic and stellar structure
- 85A20: Planetary atmospheres
- 85A25: Radiative transfer
- 85A30: Hydrodynamic and hydromagnetic problems [See also 76Y05]
- 85A35: Statistical astronomy
- 85A40: Cosmology {For relativistic cosmology, see 83F05}
- 85A99: Miscellaneous topics

- 86-XX: Geophysics [See also 76U05, 76V05]
- 86-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 86-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 86-02: Research exposition (monographs, survey articles)
- 86-03: Historical (must also be assigned at least one classification number from Section 01)
- 86-04: Explicit machine computation and programs (not the theory of computation or programming)
- 86-05: Experimental work
- 86-06: Proceedings, conferences, collections, etc.
- 86-08: Computational methods
- 86Axx: Geophysics [See also 76U05, 76V05]
- 86A04: General
- 86A05: Hydrology, hydrography, oceanography [See also 76Bxx, 76E20, 76Q05, 76Rxx, 76U05]
- 86A10: Meteorology and atmospheric physics [See also 76Bxx, 76E20, 76N15, 76Q05, 76Rxx, 76U05]
- 86A15: Seismology
- 86A17: Global dynamics, earthquake problems
- 86A20: Potentials, prospecting
- 86A22: Inverse problems [See also 35R30]
- 86A25: Geo-electricity and geomagnetism [See also 76W05, 78A25]
- 86A30: Geodesy, mapping problems
- 86A32: Geostatistics
- 86A40: Glaciology
- 86A60: Geological problems
- 86A99: Miscellaneous topics

- 90-XX: Operations research, mathematical programming
- 90-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 90-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 90-02: Research exposition (monographs, survey articles)
- 90-03: Historical (must also be assigned at least one classification number from Section 01)
- 90-04: Explicit machine computation and programs (not the theory of computation or programming)
- 90-06: Proceedings, conferences, collections, etc.
- 90-08: Computational methods
- 90Bxx: Operations research and management science
- 90B05: Inventory, storage, reservoirs
- 90B06: Transportation, logistics
- 90B10: Network models, deterministic
- 90B15: Network models, stochastic
- 90B18: Communication networks [See also 68M10, 94A05]
- 90B20: Traffic problems
- 90B22: Queues and service [See also 60K25, 68M20]
- 90B25: Reliability, availability, maintenance, inspection [See also 60K10, 62N05]
- 90B30: Production models
- 90B35: Scheduling theory, deterministic [See also 68M20]
- 90B36: Scheduling theory, stochastic [See also 68M20]
- 90B40: Search theory
- 90B50: Management decision making, including multiple objectives [See also 90C29, 90C31, 91A35, 91B06]
- 90B60: Marketing, advertising [See also 91B60]
- 90B70: Theory of organizations, manpower planning [See also 91D35]
- 90B80: Discrete location and assignment [See also 90C10]
- 90B85: Continuous location
- 90B90: Case-oriented studies
- 90B99: None of the above, but in this section

- 90Cxx: Mathematical programming [See also 49Mxx, 65Kxx]
- 90C05: Linear programming
- 90C06: Large-scale problems
- 90C08: Special problems of linear programming (transportation, multi-index, etc.)
- 90C09: Boolean programming
- 90C10: Integer programming
- 90C11: Mixed integer programming
- 90C15: Stochastic programming
- 90C20: Quadratic programming
- 90C22: Semidefinite programming
- 90C25: Convex programming
- 90C26: Nonconvex programming, global optimization
- 90C27: Combinatorial optimization
- 90C29: Multi-objective and goal programming
- 90C30: Nonlinear programming
- 90C31: Sensitivity, stability, parametric optimization
- 90C32: Fractional programming
- 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions)
- 90C34: Semi-infinite programming
- 90C35: Programming involving graphs or networks [See also 90C27]
- 90C39: Dynamic programming [See also 49L20]
- 90C40: Markov and semi-Markov decision processes
- 90C46: Optimality conditions, duality [See also 49N15]
- 90C47: Minimax problems [See also 49K35]
- 90C48: Programming in abstract spaces
- 90C49: Extreme-point and pivoting methods
- 90C51: Interior-point methods
- 90C52: Methods of reduced gradient type
- 90C53: Methods of quasi-Newton type
- 90C55: Methods of successive quadratic programming type
- 90C56: Derivative-free methods and methods using generalized derivatives [See also 49J52]
- 90C57: Polyhedral combinatorics, branch-and-bound, branch-and-cut
- 90C59: Approximation methods and heuristics
- 90C60: Abstract computational complexity for mathematical programming problems [See also 68Q25]
- 90C70: Fuzzy programming
- 90C90: Applications of mathematical programming
- 90C99: None of the above, but in this section

- 91-XX: Game theory, economics, social and behavioral sciences
- 91-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 91-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 91-02: Research exposition (monographs, survey articles)
- 91-03: Historical (must also be assigned at least one classification number from section 01)
- 91-04: Explicit machine computation and programs (not the theory of computation or programming)
- 91-06: Proceedings, conferences, collections, etc.
- 91-08: Computational methods
- 91Axx: Game theory
- 91A05: 2-person games
- 91A06: $n$-person games, $n>2$
- 91A10: Noncooperative games
- 91A12: Cooperative games
- 91A13: Games with infinitely many players
- 91A15: Stochastic games
- 91A18: Games in extensive form
- 91A20: Multistage and repeated games
- 91A22: Evolutionary games
- 91A23: Differential games [See also 49N70]
- 91A24: Positional games (pursuit and evasion, etc.) [See also 49N75]
- 91A25: Dynamic games
- 91A26: Rationality, learning
- 91A28: Signaling, communication
- 91A30: Utility theory for games [See also 91B16]
- 91A35: Decision theory for games [See also 62Cxx, 91B06, 90B50]
- 91A40: Game-theoretic models
- 91A43: Games involving graphs [See also 05C57]
- 91A44: Games involving topology or set theory
- 91A46: Combinatorial games
- 91A50: Discrete-time games
- 91A55: Games of timing
- 91A60: Probabilistic games; gambling [See also 60G40]
- 91A65: Hierarchical games
- 91A70: Spaces of games
- 91A80: Applications of game theory
- 91A90: Experimental studies
- 91A99: None of the above, but in this section

- 91Bxx: Mathematical economics {For econometrics, see 62P20}
- 91B02: Fundamental topics (basic mathematics, methodology; applicable to economics in general)
- 91B06: Decision theory [See also 62Cxx, 90B50, 91A35]
- 91B08: Individual preferences
- 91B10: Group preferences
- 91B12: Voting theory
- 91B14: Social choice
- 91B15: Welfare economics
- 91B16: Utility theory
- 91B18: Public goods
- 91B24: Price theory and market structure
- 91B25: Asset pricing models
- 91B26: Market models (auctions, bargaining, bidding, selling, etc.)
- 91B30: Risk theory, insurance
- 91B32: Resource and cost allocation
- 91B38: Production theory, theory of the firm
- 91B40: Labor market, contracts
- 91B42: Consumer behavior, demand theory
- 91B44: Informational economics
- 91B50: General equilibrium theory
- 91B51: Dynamic stochastic general equilibrium theory
- 91B52: Special types of equilibria
- 91B54: Special types of economies
- 91B55: Economic dynamics
- 91B60: Trade models
- 91B62: Growth models
- 91B64: Macro-economic models (monetary models, models of taxation)
- 91B66: Multisectoral models
- 91B68: Matching models
- 91B69: Heterogeneous agent models
- 91B70: Stochastic models
- 91B72: Spatial models
- 91B74: Models of real-world systems
- 91B76: Environmental economics (natural resource models, harvesting, pollution, etc.)
- 91B80: Applications of statistical and quantum mechanics to economics (econophysics)
- 91B82: Statistical methods; economic indices and measures
- 91B84: Economic time series analysis [See also 62M10]
- 91B99: None of the above, but in this section

- 91Cxx: Social and behavioral sciences: general topics {For statistics, see 62-XX}
- 91C05: Measurement theory
- 91C15: One- and multidimensional scaling
- 91C20: Clustering [See also 62H30]
- 91C99: None of the above, but in this section

- 91Dxx: Mathematical sociology (including anthropology)
- 91Exx: Mathematical psychology
- 91E10: Cognitive psychology
- 91E30: Psychophysics and psychophysiology; perception
- 91E40: Memory and learning [See also 68T05]
- 91E45: Measurement and performance
- 91E99: None of the above, but in this section

- 91Fxx: Other social and behavioral sciences (mathematical treatment)
- 91Gxx: Mathematical finance
- 91G10: Portfolio theory
- 91G20: Derivative securities
- 91G30: Interest rates (stochastic models)
- 91G40: Credit risk
- 91G50: Corporate finance
- 91G60: Numerical methods (including Monte Carlo methods)
- 91G70: Statistical methods, econometrics
- 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
- 91G99: None of the above, but in this section

- 92-XX: Biology and other natural sciences
- 92-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 92-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 92-02: Research exposition (monographs, survey articles)
- 92-03: Historical (must also be assigned at least one classification number from Section 01)
- 92-04: Explicit machine computation and programs (not the theory of computation or programming)
- 92-06: Proceedings, conferences, collections, etc.
- 92-08: Computational methods
- 92Bxx: Mathematical biology in general
- 92B05: General biology and biomathematics
- 92B10: Taxonomy, cladistics, statistics
- 92B15: General biostatistics [See also 62P10]
- 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]
- 92B25: Biological rhythms and synchronization
- 92B99: None of the above, but in this section

- 92Cxx: Physiological, cellular and medical topics
- 92C05: Biophysics
- 92C10: Biomechanics [See also 74L15]
- 92C15: Developmental biology, pattern formation
- 92C17: Cell movement (chemotaxis, etc.)
- 92C20: Neural biology
- 92C30: Physiology (general)
- 92C35: Physiological flow [See also 76Z05]
- 92C37: Cell biology
- 92C40: Biochemistry, molecular biology
- 92C42: Systems biology, networks
- 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) [See also 80A30]
- 92C50: Medical applications (general)
- 92C55: Biomedical imaging and signal processing [See also 44A12, 65R10, 94A08, 94A12]
- 92C60: Medical epidemiology
- 92C80: Plant biology
- 92C99: None of the above, but in this section

- 92Dxx: Genetics and population dynamics
- 92D10: Genetics {For genetic algebras, see 17D92}
- 92D15: Problems related to evolution
- 92D20: Protein sequences, DNA sequences
- 92D25: Population dynamics (general)
- 92D30: Epidemiology
- 92D40: Ecology
- 92D50: Animal behavior
- 92D99: None of the above, but in this section

- 92Exx: Chemistry {For biochemistry, see 92C40}
- 92Fxx: Other natural sciences (should also be assigned at least one other classification number in this section)
- 92F05: Other natural sciences (should also be assigned at least one other classification number in section 92)
- 92F99: None of the above, but in this section

- 93-XX: Systems theory; control {For optimal control, see 49-XX}
- 93-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 93-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 93-02: Research exposition (monographs, survey articles)
- 93-03: Historical (must also be assigned at least one classification number from Section 01)
- 93-04: Explicit machine computation and programs (not the theory of computation or programming)
- 93-06: Proceedings, conferences, collections, etc.
- 93Axx: General
- 93A05: Axiomatic system theory
- 93A10: General systems
- 93A13: Hierarchical systems
- 93A14: Decentralized systems
- 93A15: Large scale systems
- 93A30: Mathematical modeling (models of systems, model-matching, etc.)
- 93A99: None of the above, but in this section

- 93Bxx: Controllability, observability, and system structure
- 93B03: Attainable sets
- 93B05: Controllability
- 93B07: Observability
- 93B10: Canonical structure
- 93B11: System structure simplification
- 93B12: Variable structure systems
- 93B15: Realizations from input-output data
- 93B17: Transformations
- 93B18: Linearizations
- 93B20: Minimal systems representations
- 93B25: Algebraic methods
- 93B27: Geometric methods
- 93B28: Operator-theoretic methods [See also 47A48, 47A57, 47B35, 47N70]
- 93B30: System identification
- 93B35: Sensitivity (robustness)
- 93B36: $H^\infty$-control
- 93B40: Computational methods
- 93B50: Synthesis problems
- 93B51: Design techniques (robust design, computer-aided design, etc.)
- 93B52: Feedback control
- 93B55: Pole and zero placement problems
- 93B60: Eigenvalue problems
- 93B99: None of the above, but in this section

- 93Cxx: Control systems
- 93C05: Linear systems
- 93C10: Nonlinear systems
- 93C15: Systems governed by ordinary differential equations [See also 34H05]
- 93C20: Systems governed by partial differential equations
- 93C23: Systems governed by functional-differential equations [See also 34K35]
- 93C25: Systems in abstract spaces
- 93C30: Systems governed by functional relations other than differential equations (such as hybrid and switching systems)
- 93C35: Multivariable systems
- 93C40: Adaptive control
- 93C41: Problems with incomplete information
- 93C42: Fuzzy control systems
- 93C55: Discrete-time systems
- 93C57: Sampled-data systems
- 93C62: Digital systems
- 93C65: Discrete event systems
- 93C70: Time-scale analysis and singular perturbations
- 93C73: Perturbations
- 93C80: Frequency-response methods
- 93C83: Control problems involving computers (process control, etc.)
- 93C85: Automated systems (robots, etc.) [See also 68T40, 70B15, 70Q05]
- 93C95: Applications
- 93C99: None of the above, but in this section

- 93Dxx: Stability
- 93D05: Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.)
- 93D09: Robust stability
- 93D10: Popov-type stability of feedback systems
- 93D15: Stabilization of systems by feedback
- 93D20: Asymptotic stability
- 93D21: Adaptive or robust stabilization
- 93D25: Input-output approaches
- 93D30: Scalar and vector Lyapunov functions
- 93D99: None of the above, but in this section

- 93Exx: Stochastic systems and control
- 93E03: Stochastic systems, general
- 93E10: Estimation and detection [See also 60G35]
- 93E11: Filtering [See also 60G35]
- 93E12: System identification
- 93E14: Data smoothing
- 93E15: Stochastic stability
- 93E20: Optimal stochastic control
- 93E24: Least squares and related methods
- 93E25: Other computational methods
- 93E35: Stochastic learning and adaptive control
- 93E99: None of the above, but in this section

- 94-XX: Information and communication, circuits
- 94-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 94-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 94-02: Research exposition (monographs, survey articles)
- 94-03: Historical (must also be assigned at least one classification number from Section 01)
- 94-04: Explicit machine computation and programs (not the theory of computation or programming)
- 94-06: Proceedings, conferences, collections, etc.
- 94Axx: Communication, information
- 94A05: Communication theory [See also 60G35, 90B18]
- 94A08: Image processing (compression, reconstruction, etc.) [See also 68U10]
- 94A11: Application of orthogonal and other special functions
- 94A12: Signal theory (characterization, reconstruction, filtering, etc.)
- 94A13: Detection theory
- 94A14: Modulation and demodulation
- 94A15: Information theory, general [See also 62B10, 81P45]
- 94A17: Measures of information, entropy
- 94A20: Sampling theory
- 94A24: Coding theorems (Shannon theory)
- 94A29: Source coding [See also 68P30]
- 94A34: Rate-distortion theory
- 94A40: Channel models (including quantum)
- 94A45: Prefix, length-variable, comma-free codes [See also 20M35, 68Q45]
- 94A50: Theory of questionnaires
- 94A55: Shift register sequences and sequences over finite alphabets
- 94A60: Cryptography [See also 11T71, 14G50, 68P25, 81P94]
- 94A62: Authentication and secret sharing [See also 81P94]
- 94A99: None of the above, but in this section

- 94Bxx: Theory of error-correcting codes and error-detecting codes
- 94B05: Linear codes, general
- 94B10: Convolutional codes
- 94B12: Combined modulation schemes (including trellis codes)
- 94B15: Cyclic codes
- 94B20: Burst-correcting codes
- 94B25: Combinatorial codes
- 94B27: Geometric methods (including applications of algebraic geometry) [See also 11T71, 14G50]
- 94B30: Majority codes
- 94B35: Decoding
- 94B40: Arithmetic codes [See also 11T71, 14G50]
- 94B50: Synchronization error-correcting codes
- 94B60: Other types of codes
- 94B65: Bounds on codes
- 94B70: Error probability
- 94B75: Applications of the theory of convex sets and geometry of numbers (covering radius, etc.) [See also 11H31, 11H71]
- 94B99: None of the above, but in this section

- 94Cxx: Circuits, networks
- 94C05: Analytic circuit theory
- 94C10: Switching theory, application of Boolean algebra; Boolean functions [See also 06E30]
- 94C12: Fault detection; testing
- 94C15: Applications of graph theory [See also 05Cxx, 68R10]
- 94C30: Applications of design theory [See also 05Bxx]
- 94C99: None of the above, but in this section

- 94Dxx: Fuzzy sets and logic (in connection with questions of Section 94) [See also 03B52, 03E72, 28E10]

- 97-XX: Mathematics education
- 97-00: General reference works (handbooks, dictionaries, bibliographies, etc.)
- 97-01: Instructional exposition (textbooks, tutorial papers, etc.)
- 97-02: Research exposition (monographs, survey articles)
- 97-03: Historical (must also be assigned at least one classification number from Section 01)
- 97-04: Explicit machine computation and programs (not the theory of computation or programming)
- 97-06: Proceedings, conferences, collections, etc.
- 97Axx: General, mathematics and education
- 97A10: Comprehensive works, reference books
- 97A20: Recreational mathematics, games [See also 00A08]
- 97A30: History of mathematics and mathematics education [See also 01-XX]
- 97A40: Mathematics and society
- 97A50: Bibliographies [See also 01-00]
- 97A70: Theses and postdoctoral theses
- 97A80: Popularization of mathematics
- 97A99: None of the above, but in this section

- 97Bxx: Educational policy and systems
- 97B10: Educational research and planning
- 97B20: General education
- 97B30: Vocational education
- 97B40: Higher education
- 97B50: Teacher education {For research aspects, see 97C70}
- 97B60: Adult and further education
- 97B70: Syllabuses, educational standards
- 97B99: None of the above, but in this section

- 97Cxx: Psychology of mathematics education, research in mathematics education
- 97C10: Comprehensive works
- 97C20: Affective behavior
- 97C30: Cognitive processes, learning theories
- 97C40: Intelligence and aptitudes
- 97C50: Language and verbal communities
- 97C60: Sociological aspects of learning
- 97C70: Teaching-learning processes
- 97C99: None of the above, but in this section

- 97Dxx: Education and instruction in mathematics
- 97D10: Comprehensive works, comparative studies
- 97D20: Philosophical and theoretical contributions (maths didactics)
- 97D30: Objectives and goals
- 97D40: Teaching methods and classroom techniques
- 97D50: Teaching problem solving and heuristic strategies {For research aspects, see 97Cxx}
- 97D60: Student assessment, achievement control and rating
- 97D70: Learning difficulties and student errors
- 97D80: Teaching units and draft lessons
- 97D99: None of the above, but in this section

- 97Exx: Foundations of mathematics
- 97E10: Comprehensive works
- 97E20: Philosophy and mathematics
- 97E30: Logic
- 97E40: Language of mathematics
- 97E50: Reasoning and proving in the mathematics classroom
- 97E60: Sets, relations, set theory
- 97E99: None of the above, but in this section

- 97Fxx: Arithmetic, number theory
- 97F10: Comprehensive works
- 97F20: Pre-numerical stage, concept of numbers
- 97F30: Natural numbers
- 97F40: Integers, rational numbers
- 97F50: Real numbers, complex numbers
- 97F60: Number theory
- 97F70: Measures and units
- 97F80: Ratio and proportion, percentages
- 97F90: Real life mathematics, practical arithmetic
- 97F99: None of the above, but in this section

- 97Gxx: Geometry
- 97G10: Comprehensive works
- 97G20: Informal geometry
- 97G30: Areas and volumes
- 97G40: Plane and solid geometry
- 97G50: Transformation geometry
- 97G60: Plane and spherical trigonometry
- 97G70: Analytic geometry. Vector algebra
- 97G80: Descriptive geometry
- 97G99: None of the above, but in this section

- 97Hxx: Algebra
- 97H10: Comprehensive works
- 97H20: Elementary algebra
- 97H30: Equations and inequalities
- 97H40: Groups, rings, fields
- 97H50: Ordered algebraic structures
- 97H60: Linear algebra
- 97H99: None of the above, but in this section

- 97Ixx: Analysis
- 97I10: Comprehensive works
- 97I20: Mappings and functions
- 97I30: Sequences and series
- 97I40: Differential calculus
- 97I50: Integral calculus
- 97I60: Functions of several variables
- 97I70: Functional equations
- 97I80: Complex analysis
- 97I99: None of the above, but in this section

- 97Kxx: Combinatorics, graph theory, probability theory, statistics
- 97K10: Comprehensive works
- 97K20: Combinatorics
- 97K30: Graph theory
- 97K40: Descriptive statistics
- 97K50: Probability theory
- 97K60: Distributions and stochastic processes
- 97K70: Foundations and methodology of statistics
- 97K80: Applied statistics
- 97K99: None of the above, but in this section

- 97Mxx: Mathematical modeling, applications of mathematics
- 97M10: Modeling and interdisciplinarity
- 97M20: Mathematics in vocational training and career education
- 97M30: Financial and insurance mathematics
- 97M40: Operations research, economics
- 97M50: Physics, astronomy, technology, engineering
- 97M60: Biology, chemistry, medicine
- 97M70: Behavioral and social sciences
- 97M80: Arts, music, language, architecture
- 97M99: None of the above, but in this section

- 97Nxx: Numerical mathematics
- 97N10: Comprehensive works
- 97N20: Rounding, estimation, theory of errors
- 97N30: Numerical algebra
- 97N40: Numerical analysis
- 97N50: Interpolation and approximation
- 97N60: Mathematical programming
- 97N70: Discrete mathematics
- 97N80: Mathematical software, computer programs
- 97N99: None of the above, but in this section

- 97Pxx: Computer science
- 97P10: Comprehensive works
- 97P20: Theory of computer science
- 97P30: System software
- 97P40: Programming languages
- 97P50: Programming techniques
- 97P60: Hardware
- 97P70: Computer science and society
- 97P99: None of the above, but in this section

- 97Qxx: Computer science education
- 97Q10: Comprehensive works
- 97Q20: Affective aspects in teaching computer science
- 97Q30: Cognitive processes
- 97Q40: Sociological aspects
- 97Q50: Objectives
- 97Q60: Teaching methods and classroom techniques
- 97Q70: Student assessment
- 97Q80: Teaching units
- 97Q99: None of the above, but in this section

- 97Rxx: Computer science applications
- 97R10: Comprehensive works, collections of programs
- 97R20: Applications in mathematics
- 97R30: Applications in sciences
- 97R40: Artificial intelligence
- 97R50: Data bases, information systems
- 97R60: Computer graphics
- 97R70: User programs, administrative applications
- 97R80: Recreational computing
- 97R99: None of the above, but in this section

- 97Uxx: Educational material and media, educational technology
- 97U10: Comprehensive works
- 97U20: Textbooks. Textbook research
- 97U30: Teachers' manuals and planning aids
- 97U40: Problem books. Competitions. Examinations
- 97U50: Computer assisted instruction; e-learning
- 97U60: Manipulative materials
- 97U70: Technological tools, calculators
- 97U80: Audiovisual media
- 97U99: None of the above, but in this section