General General mathematics Mathematics for nonmathematicians (engineering, social sciences, etc.) Problem books Recreational mathematics [See also 97A20] Bibliographies External book reviews Dictionaries and other general reference works Formularies Philosophy of mathematics [See also 03A05] Methodology of mathematics, didactics [See also 97Cxx, 97Dxx] General applied mathematics {For physics, see 00A79 and Sections 70 through 86} Theory of mathematical modeling General methods of simulation Dimensional analysis Physics (use more specific entries from Sections 70 through 86 when possible) Miscellaneous topics Collections of abstracts of lectures Collections of articles of general interest Collections of articles of miscellaneous specific content Proceedings of conferences of general interest Proceedings of conferences of miscellaneous specific interest Festschriften Volumes of selected translations Miscellaneous volumes of translations Collections of reprinted articles [See also 01A75]    History and biography [See also the classification number -03 General histories, source books Ethnomathematics, general Paleolithic, Neolithic Indigenous cultures of the Americas Other indigenous cultures (non-European) Indigenous European cultures (pre-Greek, etc.) Egyptian Babylonian Greek, Roman China Japan Southeast Asia Islam (Medieval) India Medieval 15th and 16th centuries, Renaissance 17th century 18th century 19th century 20th century Twenty-first century Contemporary Future prospectives Biographies, obituaries, personalia, bibliographies Schools of mathematics Universities Other institutions and academies Collected or selected works; reprintings or translations of classics [See also 00B60] Sociology (and profession) of mathematics Historiography Bibliographic studies Miscellaneous topics    Mathematical logic and foundations Classical propositional logic Classical first-order logic Higher-order logic and type theory Subsystems of classical logic (including intuitionistic logic) Abstract deductive systems Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] Foundations of classical theories (including reverse mathematics) [See also 03F35] Mechanization of proofs and logical operations [See also 68T15] Combinatory logic and lambda-calculus [See also 68N18] Logic of knowledge and belief Temporal logic Modal logic {For knowledge and belief see 03B42; for temporal logic see 03B44; for provability logic see also 03F45} Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} Probability and inductive logic [See also 60A05] Many-valued logic Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05] Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.) Intermediate logics Other nonclassical logic Logic of natural languages [See also 68T50, 91F20] Logic in computer science [See also 68-XX] Other applications of logic None of the above, but in this section Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05] Basic properties of first-order languages and structures Quantifier elimination, model completeness and related topics Finite structures [See also 68Q15, 68Q19] Denumerable structures Ultraproducts and related constructions Model-theoretic forcing Other model constructions Categoricity and completeness of theories Interpolation, preservation, definability Classification theory, stability and related concepts Models with special properties (saturated, rigid, etc.) Properties of classes of models Set-theoretic model theory Effective and recursion-theoretic model theory [See also 03D45] Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] Models of arithmetic and set theory [See also 03Hxx] Model theory of ordered structures; o-minimality Models of other mathematical theories Other classical first-order model theory Logic on admissible sets Other infinitary logic Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48] Second- and higher-order model theory Nonclassical models (Boolean-valued, sheaf, etc.) Abstract model theory Applications of model theory [See also 03C60] None of the above, but in this section Thue and Post systems, etc. Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15] Turing machines and related notions [See also 68Q05] Complexity of computation [See also 68Q15, 68Q17] Recursive functions and relations, subrecursive hierarchies Recursively (computably) enumerable sets and degrees Other Turing degree structures Other degrees and reducibilities Undecidability and degrees of sets of sentences Word problems, etc. [See also 06B25, 08A50, 20F10, 68R15] Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55] Recursive equivalence types of sets and structures, isols Hierarchies Computability and recursion theory on ordinals, admissible sets, etc. Higher-type and set recursion theory Inductive definability Abstract and axiomatic computability and recursion theory Applications of computability and recursion theory None of the above, but in this section Proof theory, general Cut-elimination and normal-form theorems Structure of proofs Functionals in proof theory Recursive ordinals and ordinal notations Complexity of proofs Relative consistency and interpretations First-order arithmetic and fragments Second- and higher-order arithmetic and fragments [See also 03B30] Gödel numberings in proof theory Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25] Metamathematics of constructive systems Linear logic and other substructural logics [See also 03B47] Intuitionistic mathematics Constructive and recursive analysis [See also 03B30, 03D45, 26E40, 46S30, 47S30] Other constructive mathematics [See also 03D45] None of the above, but in this section Boolean algebras [See also 06Exx] Lattices and related structures [See also 06Bxx] Quantum logic [See also 06C15, 81P10] Cylindric and polyadic algebras; relation algebras Lukasiewicz and Post algebras [See also 06D25, 06D30] Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35] Categorical logic, topoi [See also 18B25, 18C05, 18C10] None of the above, but in this section Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] Other applications of nonstandard models (economics, physics, etc.) Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05] None of the above, but in this section    Combinatorics {For finite fields, see 11Txx} Block designs [See also 51E05, 62K10] Triple systems Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13] Orthogonal arrays, Latin squares, Room squares Matrices (incidence, Hadamard, etc.) Finite geometries [See also 51D20, 51Exx] Other designs, configurations [See also 51E30] Matroids, geometric lattices [See also 52B40, 90C27] Packing and covering [See also 11H31, 52C15, 52C17] Tessellation and tiling problems [See also 52C20, 52C22] Polyominoes None of the above, but in this section Trees Degree sequences Topological graph theory, imbedding [See also 57M15, 57M25] Distance in graphs Coloring of graphs and hypergraphs Perfect graphs Directed graphs (digraphs), tournaments Signed, gain and biased graphs Graphs and groups [See also 20F65] Enumeration of graphs and maps Extremal problems [See also 90C35] Paths and cycles [See also 90B10] Connectivity Eulerian and Hamiltonian graphs Graphs and matrices Generalized Ramsey theory Isomorphism problems (reconstruction conjecture, etc.) Graph representations (geometric and intersection representations, etc.) Hypergraphs Dominating sets, independent sets, cliques Factorization, matching, covering and packing Structural characterization of types of graphs Graph labelling (graceful graphs, bandwidth, etc.) Random graphs Graph minors Graph algorithms [See also 68R10, 68W05] Applications None of the above, but in this section Extremal set theory Ramsey theory Transversal (matching) theory Probabilistic methods None of the above, but in this section Symmetric functions Tableaux, representations of the symmetric group [See also 20C30] Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Group actions on designs, geometries and codes Group actions on posets and homology groups of posets [See also 06A11] Association schemes, strongly regular graphs Orthogonal polynomials [See also 33C45, 33C50, 33D45] None of the above, but in this section    Order, lattices, ordered algebraic structures [See also 18B35] Total order Partial order, general Combinatorics of partially ordered sets Algebraic aspects of posets [See also 05E25] Semilattices [See also 20M10; for topological semilattices see 22A26] Galois correspondences, closure operators None of the above, but in this section Structure theory Ideals, congruence relations Representation theory Varieties of lattices Complete lattices, completions Free lattices, projective lattices, word problems [See also 03D40, 08A50, 20F10] Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] None of the above, but in this section Modular lattices, Desarguesian lattices Semimodular lattices, geometric lattices Complemented lattices, orthocomplemented lattices and posets [See also 03G12, 81P10] Complemented modular lattices, continuous geometries None of the above, but in this section Structure and representation theory Complete distributivity Pseudocomplemented lattices Heyting algebras [See also 03G25] Frames, locales {For topological questions see 54-XX} Post algebras [See also 03G20] De Morgan algebras, Lukasiewicz algebras [See also 03G20] MV-algebras Lattices and duality Fuzzy lattices (soft algebras) and related topics None of the above, but in this section Structure theory Chain conditions, complete algebras Stone space and related constructions Ring-theoretic properties [See also 16E50, 16G30] Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45] Boolean functions [See also 94C10] None of the above, but in this section Ordered semigroups and monoids [See also 20Mxx] Quantales Noether lattices Ordered groups [See also 20F60] Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80} Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12] BCK-algebras, BCI-algebras [See also 03G25] None of the above, but in this section    General algebraic systems Relational systems, laws of composition Structure theory Subalgebras, congruence relations Automorphisms, endomorphisms Operations, polynomials, primal algebras Equational compactness Word problems [See also 03D40, 06B25, 20F10, 68R15] Partial algebras Unary algebras Finitary algebras Infinitary algebras Heterogeneous algebras Applications of universal algebra in computer science Fuzzy algebraic structures None of the above, but in this section Equational logic, Malcev (Maltsev) conditions Congruence modularity, congruence distributivity Lattices of varieties Free algebras Products, amalgamated products, and other kinds of limits and colimits [See also 18A30] Subdirect products and subdirect irreducibility Injectives, projectives None of the above, but in this section Categories of algebras [See also 18C05] Axiomatic model classes [See also 03Cxx, in particular 03C60] Quasivarieties None of the above, but in this section    Number theory Multiplicative structure; Euclidean algorithm; greatest common divisors Congruences; primitive roots; residue systems Power residues, reciprocity Arithmetic functions; related numbers; inversion formulas Primes Factorization; primality Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] Radix representation; digital problems {For metric results, see 11K16} Other representations None of the above, but in this section Density, gaps, topology Additive bases [See also 05B10] Arithmetic progressions [See also 11N13] Representation functions Recurrences {For applications to special functions, see 33-XX} Fibonacci and Lucas numbers and polynomials and generalizations Sequences (mod $m$) Farey sequences; the sequences ${1^k, 2^k, \cdots}$ Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] Bernoulli and Euler numbers and polynomials Bell and Stirling numbers Other combinatorial number theory Special sequences and polynomials Automata sequences None of the above, but in this section Polynomials [See also 13F20] Matrices, determinants [See also 15A36] None of the above, but in this section Linear equations Quadratic and bilinear equations Cubic and quartic equations Higher degree equations; Fermat's equation Counting solutions of Diophantine equations Multiplicative and norm form equations Thue-Mahler equations Exponential equations Rational numbers as sums of fractions Equations in many variables [See also 11P55] Diophantine inequalities [See also 11J25] Congruences in many variables Representation problems [See also 11P55] $p$-adic and power series fields None of the above, but in this section Quadratic forms over general fields Quadratic forms over local rings and fields Forms over real fields Quadratic forms over global rings and fields General binary quadratic forms General ternary and quaternary quadratic forms; forms of more than two variables Sums of squares and representations by other particular quadratic forms Bilinear and Hermitian forms Class numbers of quadratic and Hermitian forms Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) Classical groups [See also 14Lxx, 20Gxx] $K$-theory of quadratic and Hermitian forms Galois cohomology of linear algebraic groups [See also 20G10] Forms of degree higher than two Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24] Quadratic spaces; Clifford algebras [See also 15A63, 15A66] $p$-adic theory None of the above, but in this section Elliptic curves over global fields [See also 14H52] Elliptic curves over local fields [See also 14G20, 14H52] Drinfeld modules; higher-dimensional motives, etc. [See also 14L05] Abelian varieties of dimension $\gtr 1$ [See also 14Kxx] Complex multiplication and moduli of abelian varieties [See also 14K22] Elliptic and modular units [See also 11R27] Arithmetic aspects of modular and Shimura varieties [See also 14G35] Curves over finite and local fields [See also 14H25] Varieties over finite and local fields [See also 14G15, 14G20] Curves of arbitrary genus or genus $\ne 1$ over global fields [See also 14H25] Varieties over global fields [See also 14G25] $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] Geometric class field theory [See also 11R37, 14C35, 19F05] Heights [See also 14G40] Polylogarithms and relations with $K$-theory None of the above, but in this section Lattices and convex bodies [See also 11P21, 52C05, 52C07] Nonconvex bodies Lattice packing and covering [See also 05B40, 52C15, 52C17] Products of linear forms Minima of forms Quadratic forms (reduction theory, extreme forms, etc.) Automorphism groups of lattices Mean value and transfer theorems Relations with coding theory None of the above, but in this section Homogeneous approximation to one number Markov and Lagrange spectra and generalizations Simultaneous homogeneous approximation, linear forms Approximation by numbers from a fixed field Inhomogeneous linear forms Diophantine inequalities [See also 11D75] Small fractional parts of polynomials and generalizations Approximation in non-Archimedean valuations Approximation to algebraic numbers Continued fractions and generalizations [See also 11A55, 11K50] Distribution modulo one [See also 11K06] Irrationality; linear independence over a field Transcendence (general theory) Measures of irrationality and of transcendence Metric theory Algebraic independence; Gelfond's method Linear forms in logarithms; Baker's method Transcendence theory of elliptic and abelian functions Transcendence theory of other special functions Transcendence theory of Drinfel'd and $t$-modules Results involving abelian varieties Analogues of methods in Nevanlinna theory (work of Vojta et al.) None of the above, but in this section Trigonometric and exponential sums, general Gauss and Kloosterman sums; generalizations Estimates on exponential sums Jacobsthal and Brewer sums; other complete character sums Weyl sums Sums over primes Sums over arbitrary intervals Estimates on character sums None of the above, but in this section $\zeta (s)$ and $L(s, \chi)$ Real zeros of $L(s, \chi)$; results on $L(1, \chi)$ Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses Hurwitz and Lerch zeta functions Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas Zeta and $L$-functions in characteristic $p$ 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} Tauberian theorems [See also 40E05] None of the above, but in this section Distribution of primes Primes in progressions [See also 11B25] Distribution of integers with specified multiplicative constraints Turán theory [See also 30Bxx] Primes represented by polynomials; other multiplicative structure of polynomial values Sieves Applications of sieve methods Asymptotic results on arithmetic functions Asymptotic results on counting functions for algebraic and topological structures Rate of growth of arithmetic functions Distribution functions associated with additive and positive multiplicative functions Other results on the distribution of values or the characterization of arithmetic functions Distribution of integers in special residue classes Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx] Generalized primes and integers None of the above, but in this section Waring's problem and variants Lattice points in specified regions Goldbach-type theorems; other additive questions involving primes Applications of the Hardy-Littlewood method [See also 11D85] Inverse problems of additive number theory Elementary theory of partitions [See also 05A17] Analytic theory of partitions Partitions; congruences and congruential restrictions None of the above, but in this section Algebraic numbers; rings of algebraic integers PV-numbers and generalizations; other special algebraic numbers Polynomials (irreducibility, etc.) Quadratic extensions Cubic and quartic extensions Cyclotomic extensions Other abelian and metabelian extensions Other number fields Iwasawa theory Units and factorization Class numbers, class groups, discriminants Galois theory Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] Galois cohomology [See also 12Gxx, 16H05, 19A31] Class field theory Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55] Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] Distribution of prime ideals [See also 11N05] Density theorems Other analytic theory [See also 11Nxx] Quaternion and other division algebras: arithmetic, zeta functions Other algebras and orders, and their zeta and $L$-functions [See also 11S45, 16H05, 16Kxx] Adèle rings and groups Arithmetic theory of algebraic function fields [See also 14-XX] Cyclotomic function fields (class groups, Bernoulli objects, etc.) Class groups and Picard groups of orders $K$-theory of global fields [See also 19Fxx] Totally real and totally positive fields [See also 12J15] None of the above, but in this section Polynomials Ramification and extension theory Galois theory Integral representations Galois cohomology [See also 12Gxx, 16H05] Class field theory; $p$-adic formal groups [See also 14L05] Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50] Zeta functions and $L$-functions [See also 11M41, 19F27] Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16H05, 16Kxx] $K$-theory of local fields [See also 19Fxx] Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) Other nonanalytic theory Prehomogeneous vector spaces None of the above, but in this section Decidability [See also 03B25] Ultraproducts [See also 03C20] Model theory [See also 03Cxx] Nonstandard arithmetic [See also 03H15] None of the above, but in this section Factorization Primality Algorithms; complexity [See also 68Q25] Analytic computations Algebraic number theory computations Computer solution of Diophantine equations Calculation of integer sequences Evaluation of constants Continued fraction calculations Values of arithmetic functions; tables None of the above, but in this section    Field theory and polynomials Polynomials: factorization Polynomials: location of zeros (algebraic theorems) {For the analytic theory, see 26C10, 30C15} Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx] None of the above, but in this section Polynomials (irreducibility, etc.) Special polynomials Equations Skew fields, division rings [See also 11R52, 11R54, 11S45, 16Kxx] Finite fields (field-theoretic aspects) Hilbertian fields; Hilbert's irreducibility theorem Field arithmetic None of the above, but in this section Algebraic extensions Separable extensions, Galois theory Inverse Galois theory Inseparable extensions Transcendental extensions None of the above, but in this section Galois cohomology [See also 14F22, 16H05, 16K50] Cohomological dimension None of the above, but in this section Differential algebra [See also 13Nxx] Difference algebra [See also 39Axx] Abstract differential equations [See also 34Mxx] $p$-adic differential equations [See also 11S80, 14G20] None of the above, but in this section Normed fields Valued fields Formally $p$-adic fields Ordered fields Topological semifields General valuation theory [See also 13A18] Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10] Krasner-Tate algebras [See mainly 32P05; see also 46S10, 47S10] None of the above, but in this section Near-fields [See also 16Y30] Semifields [See also 16Y60] None of the above, but in this section Decidability [See also 03B25] Ultraproducts [See also 03C20] Model theory [See also 03C60] Nonstandard arithmetic [See also 03H15] None of the above, but in this section    Commutative rings and algebras Graded rings [See also 16W50] Divisibility Radical theory Ideals; multiplicative ideal theory Valuations and their generalizations [See also 12J20] Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22] Actions of groups on commutative rings; invariant theory [See also 14L24] None of the above, but in this section Extension theory Galois theory Morphisms Integral dependence Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) Going up; going down; going between Polynomials over commutative rings [See also 11C08, 13F20, 13M10] Quotients and localization Completion [See also 13J10] Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] None of the above, but in this section Structure, classification theorems Projective and free modules and ideals [See also 19A13] Injective and flat modules and ideals Torsion modules and ideals Other special types Cohen-Macaulay modules [See also 13H10] Dimension theory, depth, related rings (catenary, etc.) Class groups [See also 11R29] Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] None of the above, but in this section Noetherian rings and modules Artinian rings and modules, finite-dimensional algebras Rings and modules of finite generation or presentation; number of generators None of the above, but in this section Dedekind, Prüfer and Krull rings and their generalizations Euclidean rings and generalizations Principal ideal rings Factorial rings, unique factorization domains [See also 14M05] Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] Formal power series rings [See also 13J05] Valuation rings [See also 13A18] Excellent rings Seminormal rings Rings with straightening laws, Hodge algebras Face and Stanley-Reisner rings; simplicial complexes [See also 55U10] None of the above, but in this section Regular local rings Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] Multiplicity theory and related topics [See also 14C17] None of the above, but in this section Power series rings [See also 13F25] Analytical algebras and rings [See also 32B05] Complete rings, completion [See also 13B35] Henselian rings [See also 13B40] Global topological rings Ordered rings [See also 06F25] Real algebra [See also 12D15, 14Pxx] None of the above, but in this section Structure Polynomials None of the above, but in this section Modules of differentials Rings of differential operators and their modules [See also 16S32, 32C38] Derivations None of the above, but in this section Polynomials, factorization [See also 12Y05] Polynomial ideals, Gröbner bases [See also 13F20] None of the above, but in this section    Algebraic geometry Relevant commutative algebra [See also 13-XX] Varieties and morphisms Schemes and morphisms Generalizations (algebraic spaces, stacks) Noncommutative algebraic geometry Elementary questions None of the above, but in this section Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] Deformations of singularities [See also 14D15, 32S30] Infinitesimal methods [See also 13D10] Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10] Local cohomology [See also 13D45, 32C36] Formal neighborhoods Local structure of morphisms: étale, flat, etc. [See also 13B40] None of the above, but in this section Parametrization (Chow and Hilbert schemes) Chow groups and rings Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] Divisors, linear systems, invertible sheaves Pencils, nets, webs [See also 53A60] Picard groups Algebraic cycles Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture Torelli problem [See also 32G20] Applications of methods of algebraic $K$-theory [See also 19Exx] Riemann-Roch theorems [See also 19E20, 19L10] None of the above, but in this section Structure of families (Picard-Lefschetz, monodromy, etc.) Fibrations, degenerations Variation of Hodge structures [See also 32G20] Arithmetic ground fields (finite, local, global) Formal methods; deformations [See also 13D10, 14B07, 32Gxx] Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Fine and coarse moduli spaces None of the above, but in this section Rational and birational maps Birational automorphisms, Cremona group and generalizations Rationality questions Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] Coverings [See also 14H30] Ramification problems [See also 11S15] Embeddings Minimal model program (Mori theory, extremal rays) None of the above, but in this section Vector bundles, sheaves, related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] Differentials and other special sheaves [See also 13Nxx, 32C38] Vanishing theorems [See also 32L20] Étale and other Grothendieck topologies and cohomologies Brauer groups of schemes [See also 12G05, 16K50] Classical real and complex cohomology $p$-adic cohomology, crystalline cohomology Homotopy theory; fundamental groups [See also 14H30] de Rham cohomology [See also 14C30, 32C35, 32L10] Motivic cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Topological properties None of the above, but in this section Rational points Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture) Finite ground fields Local ground fields Rigid analytic geometry Global ground fields Other nonalgebraically closed ground fields Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Modular and Shimura varieties [See also 11F41, 11F46, 11G18] Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50] Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40] None of the above, but in this section Algebraic functions; function fields [See also 11R58] Families, moduli (algebraic) Families, moduli (analytic) [See also 30F10, 32Gxx] Singularities, local rings [See also 13Hxx, 14B05] Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx] Coverings, fundamental group [See also 14E20, 14F35] Automorphisms Jacobians, Prym varieties [See also 32G20] Theta functions; Schottky problem [See also 14K25, 32G20] Special curves and curves of low genus Plane and space curves Special divisors (gonality, Brill-Noether theory) Elliptic curves [See also 11G05, 11G07, 14Kxx] Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx] Vector bundles on curves and their moduli [See also 14D20, 14F05] Relationships with integrable systems Relationships with physics None of the above, but in this section Families, moduli, classification: algebraic theory Moduli, classification: analytic theory; relations with modular forms [See also 32G13] Singularities [See also 14B05, 14E15] Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] Special surfaces {For Hilbert modular surfaces, see 14G35} Rational and ruled surfaces Elliptic surfaces $K3$ surfaces and Enriques surfaces Surfaces of general type $3$-folds Calabi-Yau manifolds, mirror symmetry $4$-folds $n$-folds ($n>4$) Fano varieties Automorphisms of surfaces and higher-dimensional varieties Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] Hypersurfaces Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) Relationships with physics None of the above, but in this section Isogeny Algebraic theory Algebraic moduli, classification [See also 11G15] Subvarieties Arithmetic ground fields [See also 11Dxx, 11Fxx, 11Gxx, 14Gxx] Analytic theory; abelian integrals and differentials Complex multiplication [See also 11G15] Theta functions [See also 14H42] Picard schemes, higher Jacobians [See also 14H40, 32G20] None of the above, but in this section Formal groups, $p$-divisible groups [See also 55N22] Group varieties Group schemes Affine algebraic groups, hyperalgebra constructions [See also 17B45, 18D35] Geometric invariant theory [See also 13A50] Group actions on varieties or schemes (quotients) [See also 13A50, 14L24] Classical groups (geometric aspects) [See also 20Gxx, 51N30] Other algebraic groups (geometric aspects) None of the above, but in this section Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F45, 13H10] Linkage [See also 13C40] Low codimension problems Complete intersections [See also 13C40] Determinantal varieties [See also 13C40] Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] Rational and unirational varieties Toric varieties, Newton polyhedra [See also 52B20] Supervarieties [See also 32C11, 58A50] None of the above, but in this section Projective techniques [See also 51N35] Enumerative problems (combinatorial problems) Classical problems, Schubert calculus Configurations of linear subspaces Varieties of low degree Adjunction problems Gromov-Witten invariants, quantum cohomology [See also 53D45] None of the above, but in this section Real algebraic sets [See also 12Dxx] Semialgebraic sets and related spaces Real analytic and semianalytic sets [See also 32B20, 32C05] Nash functions and manifolds [See also 32C07, 58A07] Topology of real algebraic varieties None of the above, but in this section Curves Surfaces, hypersurfaces Higher-dimensional varieties Effectivity None of the above, but in this section Classification of affine varieties Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Jacobian problem Group actions on affine varieties [See also 13A50, 14L30] Affine fibrations [See also 14D06] None of the above, but in this section    Linear and multilinear algebra; matrix theory    Associative rings and algebras {For the commutative case, see 13-XX} Category-theoretic methods and results (except as in 16D90) [See also 18-XX] Applications of logic [See also 03Cxx] None of the above, but in this section General module theory Bimodules Ideals Infinite-dimensional simple rings (except as in 16Kxx) Free, projective, and flat modules and ideals [See also 19A13] Injective modules, self-injective rings [See also 16L60] Simple and semisimple modules, primitive rings and ideals Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation Other classes of modules and ideals [See also 16G50] Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality None of the above, but in this section Representations of Artinian rings Representations of quivers and partially ordered sets Representations of orders, lattices, algebras over commutative rings [See also 16H05] Cohen-Macaulay modules Representation type (finite, tame, wild, etc.) Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers None of the above, but in this section Finite-dimensional {For crossed products, see 16S35} Infinite-dimensional and general Brauer groups [See also 12G05, 14F22] None of the above, but in this section Noncommutative local and semilocal rings, perfect rings Quasi-Frobenius rings [See also 16D50] None of the above, but in this section Jacobson radical, quasimultiplication Nil and nilpotent radicals, sets, ideals, rings Prime and semiprime rings [See also 16D60, 16U10] General radicals and rings {For radicals in module categories, see 16S90} None of the above, but in this section Finite rings and finite-dimensional algebras {For semisimple, see 16K20; for commutative, see 11Txx, 13Mxx} Artinian rings and modules Noetherian rings and modules Localization and Noetherian rings [See also 16U20] Chain conditions on annihilators and summands: Goldie-type conditions [See also 16U20], Krull dimension Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence Growth rate, Gelfand-Kirillov dimension None of the above, but in this section $T$-ideals, identities, varieties of rings and algebras Semiprime p.i. rings, rings embeddable in matrices over commutative rings Trace rings and invariant theory Identities other than those of matrices over commutative rings Other kinds of identities (generalized polynomial, rational, involution) None of the above, but in this section Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) Centralizing and normalizing extensions Universal enveloping algebras of Lie algebras [See mainly 17B35] Rings of differential operators [See also 13N10, 32C38] Group rings [See also 20C05, 20C07], Laurent polynomial rings Twisted and skew group rings, crossed products Ordinary and skew polynomial rings and semigroup rings [See also 20M25] Quadratic and Koszul algebras Rings arising from non-commutative algebraic geometry Smash products of general Hopf actions [See also 16W30] Endomorphism rings; matrix rings [See also 15-XX] Rings of functions, subdirect products, sheaves of rings Extensions of rings by ideals Deformations of rings [See also 13D10, 14D15] Maximal ring of quotients, torsion theories, radicals on module categories [See also 13D30, 18E40] {For radicals of rings, see 16Nxx} None of the above, but in this section Integral domains Ore rings, multiplicative sets, Ore localization Divisibility, noncommutative UFDs Units, groups of units Center, normalizer (invariant elements) Generalizations of commutativity None of the above, but in this section Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] Automorphisms and endomorphisms Actions of groups and semigroups; invariant theory Derivations, actions of Lie algebras Coalgebras, bialgebras, Hopf algebras [See also 16S40, 57T05]; rings, modules, etc. on which these act Ring-theoretic aspects of quantum groups [See also 17B37, 20G42, 81R50] Graded rings and modules ``Super'' (or ``skew'') structure [See also 17A70, 17Bxx, 17C70] {For exterior algebras, see 15A75; for Clifford algebras, see 11E88, 15A66} Valuations, completions, formal power series and related constructions [See also 13Jxx] Filtered rings; filtrational and graded techniques Topological and ordered rings and modules [See also 06F25, 13Jxx] None of the above, but in this section Near-rings [See also 12K05] Semirings [See also 12K10] None of the above, but in this section    Nonassociative rings and algebras General theory Power-associative rings Noncommutative Jordan algebras Flexible algebras Algebras satisfying other identities Leibniz algebras Division algebras Automorphisms, derivations, other operators Ternary compositions Other $n$-ary compositions $(n \ge 3)$ Quadratic algebras (but not quadratic Jordan algebras) Free algebras Structure theory Radical theory Superalgebras Composition algebras Valued algebras None of the above, but in this section Identities, free Lie (super)algebras Structure theory Representations, algebraic theory (weights) Representations, analytic theory Simple, semisimple, reductive (super)algebras (roots) Exceptional (super)algebras Solvable, nilpotent (super)algebras Universal enveloping (super)algebras [See also 16S30] Quantum groups (quantized enveloping algebras) and related deformations [See also 16W35, 20G42, 81R50, 82B23] Automorphisms, derivations, other operators Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx] Modular Lie (super)algebras Homological methods in Lie (super)algebras Cohomology of Lie (super)algebras Lie (super)algebras associated with other structures (associative, Jordan, etc.) [See also 16W10, 17C40, 17C50] Lie bialgebras Poisson algebras Infinite-dimensional Lie (super)algebras [See also 22E65] Lie algebras of vector fields and related (super) algebras Kac-Moody (super)algebras (structure and representation theory) Virasoro and related algebras Vertex operators; vertex operator algebras and related structures Graded Lie (super)algebras Color Lie (super)algebras Applications to integrable systems Applications to physics None of the above, but in this section Identities and free Jordan structures Structure theory Radicals Simple, semisimple algebras Idempotents, Peirce decompositions Associated groups, automorphisms Associated manifolds Associated geometries Exceptional Jordan structures Jordan structures associated with other structures [See also 16W10] Finite-dimensional structures Division algebras Jordan structures on Banach spaces and algebras [See also 46H70, 46L70] Super structures Applications to physics None of the above, but in this section Alternative rings Malcev (Maltsev) rings and algebras Right alternative rings $(\gamma, \delta)$-rings, including $(1,-1)$-rings Lie-admissible algebras Genetic algebras None of the above, but in this section    Category theory; homological algebra {For commutative rings Definitions, generalizations Graphs, diagram schemes, precategories [See especially 20L05] Foundations, relations to logic and deductive systems [See also 03-XX] Epimorphisms, monomorphisms, special classes of morphisms, null morphisms Special properties of functors (faithful, full, etc.) Natural morphisms, dinatural morphisms Functor categories, comma categories Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) Factorization of morphisms, substructures, quotient structures, congruences, amalgams Categories admitting limits (complete categories), functors preserving limits, completions Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) None of the above, but in this section Equational categories [See also 03C05, 08C05] Theories (e.g. algebraic theories), structure, and semantics [See also 03G30] Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples [See also 18Gxx] Algebras and Kleisli categories associated with monads Sketches and generalizations Accessible and locally presentable categories Categorical semantics of formal languages [See also 68Q55, 68Q65] None of the above, but in this section Double categories, $2$-categories, bicategories and generalizations Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] Closed categories (closed monoidal and Cartesian closed categories, etc.) Enriched categories (over closed or monoidal categories) Strong functors, strong adjunctions Fibered categories Structured objects in a category (group objects, etc.) Operads [See also 55P48] None of the above, but in this section Preadditive, additive categories Exact categories, abelian categories Grothendieck categories Embedding theorems [See also 18B15] Derived functors and satellites Derived categories, triangulated categories Localization of categories Torsion theories, radicals [See also 13D30, 16S90] None of the above, but in this section Projectives and injectives [See also 13C10, 13C11, 16D40, 16D50] Resolutions; derived functors [See also 13D02, 16E05, 18E25] Ext and Tor, generalizations, Künneth formula [See also 55U25] Homological dimension [See also 13D05, 16E10] Relative homological algebra, projective classes Simplicial sets, simplicial objects (in a category) [See also 55U10] Chain complexes [See also 18E30, 55U15] Spectral sequences, hypercohomology [See also 55Txx] Nonabelian homological algebra Homotopical algebra Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22] None of the above, but in this section    $K$-theory [See also 16E20, 18F25] Stability for projective modules [See also 13C10] Efficient generation Frobenius induction, Burnside and representation rings $K_0$ of group rings and orders $K_0$ of other rings None of the above, but in this section Stable range conditions Stability for linear groups $K_1$ of group rings and orders [See also 57Q10] Congruence subgroup problems [See also 20H05] None of the above, but in this section Central extensions and Schur multipliers Symbols, presentations and stability of $K_2$ $K_2$ and the Brauer group Excision for $K_2$ None of the above, but in this section $Q$- and plus-constructions Algebraic $K$-theory of spaces Symmetric monoidal categories [See also 18D10] Karoubi-Villamayor-Gersten $K$-theory Negative $K$-theory, NK and Nil Higher symbols, Milnor $K$-theory Computations of higher $K$-theory of rings [See also 13D15, 16E20] $K$-theory and homology; cyclic homology and cohomology [See also 18G60] None of the above, but in this section $K$-theory of schemes [See also 14C35] Algebraic cycles and motivic cohomology [See also 14C25, 14C35] Relations with cohomology theories [See also 14Fxx] None of the above, but in this section Generalized class field theory [See also 11G45] Symbols and arithmetic [See also 11R37] Étale cohomology, higher regulators, zeta and $L$-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10] None of the above, but in this section Stability for quadratic modules Witt groups of rings [See also 11E81] $L$-theory of group rings [See also 11E81] Hermitian $K$-theory, relations with $K$-theory of rings None of the above, but in this section Finiteness and other obstructions in $K_0$ Whitehead (and related) torsion Surgery obstructions [See also 57R67] Obstructions to group actions None of the above, but in this section $K_0$ as an ordered group, traces EXT and $K$-homology [See also 55N22] Kasparov theory ($KK$-theory) [See also 58J22] Index theory [See also 58J20, 58J22] None of the above, but in this section Riemann-Roch theorems, Chern characters $J$-homomorphism, Adams operations [See also 55Q50] Connective $K$-theory, cobordism [See also 55N22] Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] Computations, geometric applications None of the above, but in this section    Group theory and generalizations Axiomatics and elementary properties Metamathematical considerations {For word problems, see 20F10} Applications of logic to group theory None of the above, but in this section General theory for finite groups General theory for infinite groups Characterization theorems Primitive groups Multiply transitive finite groups Multiply transitive infinite groups Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX] Infinite automorphism groups [See also 12F10] Symmetric groups Subgroups of symmetric groups Computational methods None of the above, but in this section Group rings of finite groups and their modules [See also 16S34] Group rings of infinite groups and their modules [See also 16S34] Hecke algebras and their representations Integral representations of finite groups $p$-adic representations of finite groups Integral representations of infinite groups Ordinary representations and characters Modular representations and characters Projective representations and multipliers Representations of finite symmetric groups Representations of infinite symmetric groups Representations of finite groups of Lie type Representations of sporadic groups Applications of group representations to physics Computational methods None of the above, but in this section Classification of simple and nonsolvable groups Simple groups: alternating groups and groups of Lie type [See also 20Gxx] Simple groups: sporadic groups Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] Nilpotent groups, $p$-groups Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure Special subgroups (Frattini, Fitting, etc.) Series and lattices of subgroups Subnormal subgroups Products of subgroups Automorphisms Arithmetic and combinatorial problems None of the above, but in this section Free nonabelian groups Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations Subgroup theorems; subgroup growth Groups acting on trees [See also 20F65] Quasivarieties and varieties of groups Chains and lattices of subgroups, subnormal subgroups [See also 20F22] Limits, profinite groups Extensions, wreath products, and other compositions [See also 20J05] Local properties Residual properties and generalizations Maximal subgroups Simple groups [See also 20D05] General structure theorems General theorems concerning automorphisms of groups Groups with a $BN$-pair; buildings [See also 51E24] Conjugacy classes None of the above, but in this section Generators, relations, and presentations Cancellation theory; application of van Kampen diagrams [See also 57M05] Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 68Q70] Commutator calculus Derived series, central series, and generalizations Solvable groups, supersolvable groups [See also 20D10] Formations of groups, Fitting classes [See also 20D10] Nilpotent groups [See also 20D15] Generalizations of solvable and nilpotent groups Other classes of groups defined by subgroup chains FC-groups and their generalizations Automorphism groups of groups [See also 20E36] Representations of groups as automorphism groups of algebraic systems Fundamental groups and their automorphisms [See also 57M05, 57Sxx] Braid groups; Artin groups Other groups related to topology or analysis Associated Lie structures Engel conditions Periodic groups; locally finite groups Reflection and Coxeter groups [See also 22E40, 51F15] Ordered groups [See mainly 06F15] Geometric group theory [See also 05C25, 20E08, 57Mxx] Hyperbolic groups and nonpositively curved groups Asymptotic properties of groups None of the above, but in this section Representation theory Cohomology theory Linear algebraic groups over arbitrary fields Linear algebraic groups over the reals, the complexes, the quaternions Linear algebraic groups over local fields and their integers Linear algebraic groups over global fields and their integers Linear algebraic groups over adèles and other rings and schemes Linear algebraic groups over finite fields Quantum groups (quantized function algebras) and their representations [See also 16W35, 17B37, 81R50] Applications to physics None of the above, but in this section Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20] Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] Other matrix groups over fields Other matrix groups over rings Other matrix groups over finite fields None of the above, but in this section Finite abelian groups Torsion groups, primary groups and generalized primary groups Torsion-free groups, finite rank Torsion-free groups, infinite rank Mixed groups Direct sums, direct products, etc. Subgroups Automorphisms, homomorphisms, endomorphisms, etc. Extensions Homological and categorical methods Topological methods [See also 22A05, 22B05] None of the above, but in this section Free semigroups, generators and relations, word problems Varieties of semigroups General structure theory Radical theory Ideal theory Commutative semigroups Mappings of semigroups Regular semigroups Inverse semigroups Orthodox semigroups Semigroups of transformations, etc. [See also 47D03, 47H20, 54H15] Semigroup rings, multiplicative semigroups of rings [See also 16S36, 16Y60] Representation of semigroups; actions of semigroups on sets Semigroups in automata theory, linguistics, etc. [See also 03D05, 68Q70, 68T50] Connections of semigroups with homological algebra and category theory None of the above, but in this section Sets with a single binary operation (groupoids) Loops, quasigroups [See also 05Bxx] Ternary systems (heaps, semiheaps, heapoids, etc.) $n$-ary systems $(n\ge 3)$ Hypergroups Fuzzy groups [See also 03E72] None of the above, but in this section    Topological groups, Lie groups {For transformation groups, see Structure of general topological groups Analysis on general topological groups Structure of topological semigroups Analysis on topological semigroups Topological groupoids (including differentiable and Lie groupoids) [See also 58H05] Representations of general topological groups and semigroups Topological semilattices, lattices and applications [See also 06B30, 06B35, 06F30] Other topological algebraic systems and their representations None of the above, but in this section General properties and structure of LCA groups Structure of group algebras of LCA groups None of the above, but in this section General properties and structure of locally compact groups Unitary representations of locally compact groups Other representations of locally compact groups Group algebras of locally compact groups Representations of group algebras $C^*$-algebras and $W$*-algebras in relation to group representations [See also 46Lxx] Induced representations Duality theorems Ergodic theory on groups [See also 28Dxx] Automorphism groups of locally compact groups None of the above, but in this section General theory of group and pseudogroup actions {For topological properties of spaces with an action, see 57S20} Measurable group actions [See also 22D40, 28Dxx, 37Axx] Homogeneous spaces {For general actions on manifolds or preserving geometrical structures, see 57M60, 57Sxx; for discrete subgroups of Lie groups see especially 22E40} Groups as automorphisms of other structures    Real functions [See also 54C30] Continuity and differentiation questions Implicit function theorems, Jacobians, transformations with several variables Calculus of vector functions Integration: length, area, volume [See also 28A75, 51M25] Integral formulas (Stokes, Gauss, Green, etc.) Convexity, generalizations Absolutely continuous functions, functions of bounded variation Special properties of functions of several variables, Hölder conditions, etc. Representation and superposition of functions None of the above, but in this section Polynomials: analytic properties, etc. [See also 12Dxx, 12Exx] Polynomials: location of zeros [See also 12D10, 30C15, 65H05] Rational functions [See also 14Pxx] None of the above, but in this section Inequalities for trigonometric functions and polynomials Inequalities involving other types of functions Inequalities involving derivatives and differential and integral operators Inequalities for sums, series and integrals Other analytical inequalities None of the above, but in this section Real-analytic functions [See also 32B05, 32C05] $C^\infty$-functions, quasi-analytic functions [See also 58C25] Calculus of functions on infinite-dimensional spaces [See also 46G05, 58Cxx] Calculus of functions taking values in infinite-dimensional spaces [See also 46E40, 46G10, 58Cxx] Set-valued functions [See also 28B20, 54C60] {For nonsmooth analysis, see 49J52, 58Cxx, 90Cxx} Non-Archimedean analysis [See also 12J25] Nonstandard analysis [See also 03H05, 28E05, 54J05] Constructive real analysis [See also 03F60] Fuzzy real analysis [See also 03E72, 28E10] Means [See also 47A64] None of the above, but in this section    Measure and integration {For analysis on manifolds, see 58-XX} Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] Real- or complex-valued set functions Contents, measures, outer measures, capacities Abstract differentiation theory, differentiation of set functions [See also 26A24] Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence Integration with respect to measures and other set functions Spaces of measures, convergence of measures [See also 46E27, 60Bxx] Measures and integrals in product spaces Integration and disintegration of measures Lifting theory [See also 46G15] Measures on Boolean rings, measure algebras [See also 54H10] Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] Hausdorff and packing measures Fractals [See also 37Fxx] None of the above, but in this section Vector-valued set functions, measures and integrals [See also 46G10] Group- or semigroup-valued set functions, measures and integrals Set functions, measures and integrals with values in ordered spaces Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14] None of the above, but in this section Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures Set functions and measures on topological groups, Haar measures, invariant measures [See also 22Axx, 43A05] Set functions and measures on topological spaces (regularity of measures, etc.) Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] None of the above, but in this section Nonstandard measure theory [See also 03H05, 26E35] Fuzzy measure theory [See also 03E72, 26E50, 94D05] Other connections with logic and set theory None of the above, but in this section    Functions of a complex variable {For analysis on manifolds, see Monogenic properties of complex functions (including polygenic and areolar monogenic functions) Inequalities in the complex domain None of the above, but in this section Power series (including lacunary series) Random power series Boundary behavior of power series, over-convergence Analytic continuation Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX] Completeness problems, closure of a system of functions Continued fractions [See also 11A55, 40A15] None of the above, but in this section Polynomials 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} Conformal mappings of special domains Covering theorems in conformal mapping theory Numerical methods in conformal mapping theory [See also 65E05] General theory of conformal mappings Kernel functions and applications Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) Coefficient problems for univalent and multivalent functions General theory of univalent and multivalent functions Quasiconformal mappings in the plane Quasiconformal mappings in $R^n$, other generalizations Extremal problems for conformal and quasiconformal mappings, variational methods Extremal problems for conformal and quasiconformal mappings, other methods Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination Capacity and harmonic measure in the complex plane [See also 31A15] None of the above, but in this section Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] Representations of entire functions by series and integrals Special classes of entire functions and growth estimates Entire functions, general theory Meromorphic functions, general theory Distribution of values, Nevanlinna theory Cluster sets, prime ends, boundary behavior Bloch functions, normal functions, normal families Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part ${H}^p$-classes Quasi-analytic and other classes of functions None of the above, but in this section Compact Riemann surfaces and uniformization [See also 14H15, 32G15] Harmonic functions on Riemann surfaces Classification theory of Riemann surfaces Ideal boundary theory Differentials on Riemann surfaces Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx] Kleinian groups [See also 20H10] Conformal metrics (hyperbolic, Poincaré, distance functions) Klein surfaces Teichmüller theory [See also 32G15] None of the above, but in this section Non-Archimedean function theory [See also 12J25]; nonstandard function theory [See also 03H05] Finely holomorphic functions and topological function theory Generalizations of Bers or Vekua type (pseudoanalytic, $p$-analytic, etc.) Discrete analytic functions Other generalizations of analytic functions (including abstract-valued functions) Functions of hypercomplex variables and generalized variables None of the above, but in this section    Potential theory {For probabilistic potential theory, see 60J45} Harmonic, subharmonic, superharmonic functions Integral representations, integral operators, integral equations methods Potentials and capacity, harmonic measure, extremal length [See also 30C85] Boundary behavior (theorems of Fatou type, etc.) Boundary value and inverse problems Biharmonic, polyharmonic functions and equations, Poisson's equation Connections with differential equations None of the above, but in this section Harmonic, subharmonic, superharmonic functions Integral representations, integral operators, integral equations methods Potentials and capacities, extremal length Boundary value and inverse problems Boundary behavior Biharmonic and polyharmonic equations and functions Connections with differential equations None of the above, but in this section Harmonic, subharmonic, superharmonic functions Pluriharmonic and plurisubharmonic functions [See also 32U05] Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14] Potentials and capacities Discrete potential theory and numerical methods Dirichlet spaces Martin boundary theory [See also 60J50] Fine potential theory Other generalizations (nonlinear potential theory, etc.) None of the above, but in this section    Several complex variables and analytic spaces {For infinite- Power series, series of functions Special domains (Reinhardt, Hartogs, circular, tube) Holomorphic functions Multifunctions Entire functions Special families of functions Bloch functions, normal functions Normal families of functions, mappings Meromorphic functions Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30} Integral representations; canonical kernels (Szegó, Bergman, etc.) Integral representations, constructed kernels (e.g. Cauchy, Fantappiè-type kernels) Local theory of residues [See also 32C30] 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} ${H}^p$-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15] Bergman spaces Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] Algebras of holomorphic functions [See also 30H05, 46J10, 46J15] Boundary behavior of holomorphic functions Hyperfunctions [See also 46F15] Harmonic analysis of several complex variables [See mainly 43-XX] Singular integrals Zero sets of holomorphic functions Banach algebra techniques [See mainly 46Jxx] Functional analysis techniques [See mainly 46Exx] None of the above, but in this section Analytic algebras and generalizations, preparation theorems Germs of analytic sets, local parametrization Analytic subsets of affine space Semi-analytic sets and subanalytic sets [See also 14P15] Triangulation and related questions None of the above, but in this section Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15] Deformations of special (e.g. CR) structures Deformations of fiber bundles Deformations of submanifolds and subspaces Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15] Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30] Moduli and deformations for ordinary differential equations (e.g. Khnizhnik-Zamolodchikov equation) [See also 34Mxx] Applications to physics None of the above, but in this section Holomorphic mappings, (holomorphic) embeddings and related questions Meromorphic mappings Boundary uniqueness of mappings Picard-type theorems and generalizations {For function-theoretic properties, see 32A22} Value distribution theory in higher dimensions {For function-theoretic properties, see 32A22} Proper mappings, finiteness theorems Boundary regularity of mappings Iteration problems None of the above, but in this section Compactification of analytic spaces Algebraic dependence theorems Compact surfaces Compact $3$-folds Compact $n$-folds Transcendental methods of algebraic geometry [See also 14C30] Compact Kähler manifolds: generalizations, classification Applications to physics None of the above, but in this section Banach analytic spaces [See also 58Bxx] Formal and graded complex spaces [See also 58C50] Differentiable functions on analytic spaces, differentiable spaces [See also 58C25] None of the above, but in this section Holomorphic bundles and generalizations Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] Bundle convexity [See also 32F10] Vanishing theorems Twistor theory, double fibrations [See also 53C28] Applications to physics None of the above, but in this section Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10] Homogeneous complex manifolds [See also 14M17, 57T15] Almost homogeneous manifolds and spaces [See also 14M17] Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15] Automorphism groups of ${\bf C}^n$ and affine manifolds Complex vector fields None of the above, but in this section General theory of automorphic functions of several complex variables Automorphic forms Automorphic functions in symmetric domains None of the above, but in this section Local dynamics around a fixed point Bifurcation theory Iterations of polynomial automorphism Dynamics of holomorphic and meromorphic maps Hyperbolic maps Classification of Fatou components Invariant currents and measures, ergodic theory Holomorphic vector fields Holomorphic foliations and laminations Computer graphs None of the above, but in this section Local singularities [See also 14J17] Invariants of analytic local rings Equisingularity (topological and analytic) [See also 14E15] Global theory of singularities; cohomological properties [See also 14E15] Relations with arrangements of hyperplanes [See also 52C35] Surface and hypersurface singularities [See also 14J17] Deformations of singularities; vanishing cycles [See also 14B07] Mixed Hodge theory of singular varieties [See also 14C30, 14D07] Monodromy; relations with differential equations and $D$-modules Modifications; resolution of singularities [See also 14E15] Topological aspects: Lefschetz theorems, topological classification, invariants Milnor fibration; relations with knot theory [See also 57M25, 57Q45] Stratifications; constructible sheaves; intersection cohomology [See also 58Kxx] Singularities of holomorphic vector fields and foliations Other operations on singularities None of the above, but in this section Domains of holomorphy Strongly pseudoconvex domains Worm domains Finite type domains Geometric and analytic invariants on weakly pseudoconvex boundaries Exhaustion functions Peak functions None of the above, but in this section Plurisubharmonic functions and generalizations [See also 31C10] Plurisubharmonic exhaustion functions General pluripotential theory Capacity theory and generalizations Lelong numbers Removable sets Pluricomplex Green functions Currents None of the above, but in this section CR structures, CR operators, and generalizations CR functions CR manifolds as boundaries of domains Analysis on CR manifolds Extension of functions and other analytic objects from CR manifolds Embeddings of CR manifolds Finite type conditions on CR manifolds Real submanifolds in complex manifolds None of the above, but in this section    Special functions (33-XX deals with the properties of functions Exponential and trigonometric functions Gamma, beta and polygamma functions Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) Higher logarithm functions None of the above, but in this section Classical hypergeometric functions, $_2F_1$ Bessel and Airy functions, cylinder functions, $_0F_1$ Confluent hypergeometric functions, Whittaker functions, $_1F_1$ Generalized hypergeometric series, $_pF_q$ Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] Other special orthogonal polynomials and functions Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable Orthogonal polynomials and functions associated with root systems Spherical harmonics Hypergeometric integrals and functions defined by them ($E$, $G$ and ${H}$ functions) Appell, Horn and Lauricella functions Hypergeometric functions associated with root systems Other hypergeometric functions and integrals in several variables Elliptic integrals as hypergeometric functions Connections with groups and algebras, and related topics Applications None of the above, but in this section $q$-gamma functions, $q$-beta functions and integrals Basic hypergeometric functions in one variable, ${}_r\phi_s$ Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Basic hypergeometric integrals and functions defined by them Bibasic functions and multiple bases Basic hypergeometric functions associated with root systems Other basic hypergeometric functions and integrals in several variables Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics Applications None of the above, but in this section Elliptic functions and integrals Lamé, Mathieu, and spheroidal wave functions Mittag-Leffler functions and generalizations Other wave functions Painlevé-type functions Other functions defined by series and integrals Other functions coming from differential, difference and integral equations Special functions in characteristic $p$ (gamma functions, etc.) None of the above, but in this section Numerical approximation [See also 65D20] Symbolic computation (Gosper and Zeilberger algorithms, etc.) [See also 68W30] None of the above, but in this section    Ordinary differential equations Explicit solutions and reductions Implicit equations, differential-algebraic equations [See also 65L80] Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions Analytical theory: series, transformations, transforms, operational calculus, etc. [See also 44-XX] Geometric methods in differential equations Linear equations and systems, general Nonlinear equations and systems, general Differential equations of infinite order Discontinuous equations Differential equations with impulses Differential inequalities [See also 26D20] Theoretical approximation of solutions {For numerical analysis, see 65Lxx} Inverse problems Differential inclusions [See also 49J24, 49K24] None of the above, but in this section Linear boundary value problems Linear boundary value problems with nonlinear dependence on the spectral parameter Multi-parameter boundary value problems Boundary value problems with an indefinite weight Multipoint boundary value problems Nonlinear boundary value problems Singular nonlinear boundary value problems Positive solutions of nonlinear boundary value problems Weyl theory and its generalizations Sturm-Liouville theory [See also 34Lxx] Green functions Special equations (Mathieu, Hill, Bessel, etc.) Boundary value problems with impulses Boundary value problems on infinite intervals Boundary value problems on graphs and networks Applications None of the above, but in this section Asymptotic expansions Perturbations, asymptotics Multiple scale methods Singular perturbations, general theory Methods of nonstandard analysis Singular perturbations, turning point theory, WKB methods None of the above, but in this section General spectral theory Eigenfunction expansions, completeness of eigenfunctions Estimation of eigenvalues, upper and lower bounds Numerical approximation of eigenvalues and of other parts of the spectrum Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions Scattering theory Nonlinear ordinary differential operators Particular operators (Dirac, one-dimensional Schrödinger, etc.) None of the above, but in this section Entire and meromorphic solutions Oscillation, growth of solutions Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) Nonanalytic aspects Formal solutions, transform techniques Asymptotics, summation methods Singularities, monodromy, local behavior of solutions, normal forms Resurgence phenomena Stokes phenomena and connection problems (linear and nonlinear) Differential equations on complex manifolds Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) Painlevé and other special equations; classification, hierarchies; isomonodromic deformations Singular perturbation problems in the complex domain (complex WKB, turning points, steepest descent) [See also 34E20] None of the above, but in this section    Partial differential equations PDE with discontinuous coefficients or data Partial functional-differential or differential-difference equations, with or without deviating arguments Impulsive partial differential equations Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 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] Improperly posed problems for PDE Inverse problems (undetermined coefficients, etc.) for PDE Free boundary problems for PDE Partial differential inequalities Partial differential equations of infinite order Partial differential equations with randomness [See also 60H15] PDE with multivalued right-hand sides None of the above, but in this section    Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, Measure-preserving transformations One-parameter continuous families of measure-preserving transformations General groups of measure-preserving transformations [See mainly 22Fxx] Homogeneous flows [See also 22Fxx] Orbit equivalence, cocycles, ergodic equivalence relations Ergodicity, mixing, rates of mixing Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} Entropy and other invariants, isomorphism, classification Nonsingular (and infinite-measure preserving) transformations Relations with number theory and harmonic analysis [See also 11Kxx] Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] Relations with the theory of $C^*$-algebras [See mainly 46L55] Dynamical systems in statistical mechanics [See also 82Cxx] None of the above, but in this section Transformations and group actions with special properties (minimality, distality, proximality, etc.) Symbolic dynamics [See also 37Cxx, 37Dxx] Cellular automata Notions of recurrence Lyapunov functions and stability; attractors, repellers Index theory, Morse-Conley indices Gradient-like and recurrent behavior; isolated (locally-maximal) invariant sets Topological entropy Continua theory in dynamics Multi-dimensional shifts of finite type, tiling dynamics Nonautonomous dynamical systems None of the above, but in this section Smooth mappings and diffeomorphisms Vector fields, flows, ordinary differential equations Topological and differentiable equivalence, conjugacy, invariants, moduli, classification Generic properties, structural stability Fixed points, periodic points, fixed-point index theory Periodic orbits of vector fields and flows Homoclinic and heteroclinic orbits Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems Orbit growth Smooth ergodic theory, invariant measures [See also 37Dxx] Dimension theory of dynamical systems Approximate trajectories (pseudotrajectories, shadowing, etc.) Periodic and quasiperiodic flows and diffeomorphisms Nonautonomous smooth dynamical systems [See also 37B55] Monotone flows Attractors and repellers, topological structure Stability theory Symmetries, equivariant dynamical systems Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] None of the above, but in this section Hyperbolic orbits and sets Invariant manifold theory Morse-Smale systems Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) Partially hyperbolic systems and dominated splittings Thermodynamic formalism, variational principles, equilibrium states Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) Strange attractors, chaotic dynamics Hyperbolic systems with singularities (billiards, etc.) None of the above, but in this section Maps of the interval (piecewise continuous, continuous, smooth) Maps of the circle Combinatorial dynamics (types of periodic orbits) Universality, renormalization [See also 37F25] Maps of trees and graphs Homeomorphisms and diffeomorphisms of planes and surfaces Flows on surfaces Twist maps Rotation numbers and vectors None of the above, but in this section Relations and correspondences Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] Expanding maps; hyperbolicity; structural stability Combinatorics and topology Renormalization Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems Conformal densities and Hausdorff dimension Geometric limits Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations Small divisors, rotation domains and linearization; Fatou and Julia sets Holomorphic foliations and vector fields [See also 32M25, 32S65, 34Mxx] None of the above, but in this section Normal forms Bifurcations of singular points Bifurcations of limit cycles and periodic orbits Hyperbolic singular points with homoclinic trajectories Bifurcations connected with nontransversal intersection Infinite nonwandering sets arising in bifurcations Attractors and their bifurcations Symmetries, equivariant bifurcation theory None of the above, but in this section Foundations, general theory of cocycles, algebraic ergodic theory [See also 37Axx] Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] Bifurcation theory [See also 37Gxx] None of the above, but in this section General theory, relations with symplectic geometry and topology Symplectic mappings, fixed points Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20] Bifurcation problems Stability problems Obstructions to integrability (nonintegrability criteria) Completely integrable systems, topological structure of phase space, integration methods Perturbations, normal forms, small divisors, KAM theory, Arnold diffusion Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods Action-minimizing orbits and measures Contact systems [See also 53D10] Nonholonomic dynamical systems [See also 70F25] None of the above, but in this section General theory, nonlinear semigroups, evolution equations Normal forms, center manifold theory, bifurcation theory Stability problems Symmetries Inertial manifolds and other invariant attracting sets Attractors and their dimensions, Lyapunov exponents Invariant measures Hyperbolicity; Lyapunov functions Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] Lattice dynamics [See also 37K60] Special approximation methods (nonlinear Galerkin, etc.) None of the above, but in this section Simulation Time series analysis Symplectic integrators Computational methods for bifurcation problems Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy) None of the above, but in this section Dynamical systems in classical and celestial mechanics [See mainly 70Fxx, 70Hxx, 70Kxx] Dynamical systems in fluid mechanics, oceanography and meteorology [See mainly 76-XX, especially 76D05, 76F20, 86A05, 86A10] Dynamical systems in solid mechanics [See mainly 74Hxx] Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) Dynamical systems in biology [See mainly 92-XX, but also 91-XX] Dynamical systems in numerical analysis Dynamical systems in control Dynamical systems in optimization and economics None of the above, but in this section    Difference and functional equations General Difference equations, additive Stability and asymptotics of difference equations; oscillatory and periodic solutions, etc. Discrete version of topics in analysis Difference equations, scaling ($q$-differences) [See also 33Dxx] Multiplicative and other generalized difference equations, e.g. of Lyness type Difference operators [See also 47B39] None of the above, but in this section General Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX] Equations for real functions [See also 26A51, 26B25] Equations for complex functions [See also 30D05] Matrix and operator equations [See also 47Jxx] Equations for functions with more general domains and/or ranges Orthogonal additivity and other conditional equations Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] Systems of functional equations and inequalities Stability, separation, extension, and related topics [See also 46A22] None of the above, but in this section    Sequences, series, summability Convergence and divergence of series and sequences Convergence and divergence of integrals Convergence and divergence of continued fractions [See also 30B70] Convergence and divergence of infinite products Approximation to limiting values (summation of series, etc.) {For the Euler-Maclaurin summation formula, see 65B15} Convergence and divergence of series and sequences of functions None of the above, but in this section Matrix methods Integral methods Function-theoretic methods (including power series methods and semicontinuous methods) None of the above, but in this section General theorems Structure of summability fields Tauberian constants and oscillation limits Convergence factors and summability factors Summability and bounded fields of methods Inclusion and equivalence theorems None of the above, but in this section Tauberian theorems, general Growth estimates Lacunary inversion theorems Tauberian constants None of the above, but in this section Cesàro, Euler, Nörlund and Hausdorff methods Abel, Borel and power series methods None of the above, but in this section    Approximations and expansions {For all approximation theory in the    Fourier analysis    Abstract harmonic analysis {For other analysis on topological and    Integral transforms, operational calculus {For fractional    Integral equations    Functional analysis {For manifolds modeled on topological linear General theory of locally convex spaces Locally convex Fréchet spaces and (DF)-spaces Barrelled spaces, bornological spaces Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) Bornologies and related structures; Mackey convergence, etc. Other ``topological'' linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${\bf R}$, etc.) Duality theory Theorems of Hahn-Banach type; extension and lifting of functionals and operators [See also 46M10] Reflexivity and semi-reflexivity [See also 46B10] Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness) Spaces of linear operators; topological tensor products; approximation properties [See also 46B28, 46M05, 47L05, 47L20] Summability and bases [See also 46B15] Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42] Sequence spaces (including Köthe sequence spaces) [See also 46B45] Compactness in topological linear spaces; angelic spaces, etc. Convex sets in topological linear spaces; Choquet theory [See also 52A07] Graded Fréchet spaces and tame operators Topological invariants ((DN), ($\Omega$), etc.) Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.) Modular spaces None of the above, but in this section Isomorphic theory (including renorming) of Banach spaces Isometric theory of Banach spaces Local theory of Banach spaces Ultraproduct techniques in Banach space theory [See also 46M07] Probabilistic methods in Banach space theory [See also 60Bxx] Duality and reflexivity [See also 46A25] Summability and bases [See also 46A35] Geometry and structure of normed linear spaces Radon-Nikodym, Krein-Milman and related properties [See also 46G10] Classical Banach spaces in the general theory Nonseparable Banach spaces Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] Ordered normed spaces [See also 46A40, 46B42] Banach lattices [See also 46A40, 46B40] Banach sequence spaces [See also 46A45] Compactness in Banach (or normed) spaces Interpolation between normed linear spaces [See also 46M35] None of the above, but in this section Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.) [See also 46B70, 46M35] Characterizations of Hilbert spaces Spaces with indefinite inner product (Krein spaces, Pontryagin spaces, etc.) [See also 47B50] Generalizations of inner products (semi-inner products, partial inner products, etc.) None of the above, but in this section Lattices of continuous, differentiable or analytic functions Topological linear spaces of continuous, differentiable or analytic functions Banach spaces of continuous, differentiable or analytic functions Hilbert spaces of continuous, differentiable or analytic functions Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15} Spaces of measures [See also 28A33, 46Gxx] Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems Sobolev (and similar kinds of) spaces of functions of discrete variables Spaces of vector- and operator-valued functions Spaces of differentiable or holomorphic functions on infinite-dimensional spaces [See also 46G20, 46G25, 47H60] None of the above, but in this section General theory of topological algebras Ideals and subalgebras Representations of topological algebras Structure, classification of topological algebras Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) Functional calculus in topological algebras [See also 47A60] Topological algebras of operators [See mainly 47Lxx] Automatic continuity Nonassociative topological algebras [See also 46K70, 46L70] None of the above, but in this section General theory of commutative topological algebras Banach algebras of continuous functions, function algebras [See also 46E25] Banach algebras of differentiable or analytic functions, ${H}^p$-spaces [See also 30D55, 30H05, 32A35, 32A37, 32A38, 42B30] Ideals, maximal ideals, boundaries Representations of commutative topological algebras Subalgebras Structure, classification of commutative topological algebras Radical Banach algebras None of the above, but in this section General theory of topological algebras with involution Representations of topological algebras with involution Hilbert algebras Nonselfadjoint (sub)algebras in algebras with involution Nonassociative topological algebras with an involution [See also 46H70, 46L70] None of the above, but in this section Tensor products [See also 46A32, 46B28, 47A80] Ultraproducts [See also 46B08, 46S20] Projective and injective objects [See also 46A22] Categories, functors {For $K$-theory, EXT, etc., see 19K33, 46L80, 46M18, 46M20} Homological methods (exact sequences, right inverses, lifting, etc.) Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.) [See also 14F05, 18Fxx, 19Kxx, 32Cxx, 32Lxx, 46L80, 46M15, 46M18, 55Rxx] Abstract interpolation of topological vector spaces [See also 46B70] Inductive and projective limits [See also 46A13] None of the above, but in this section Applications in optimization, convex analysis, mathematical programming, economics Applications to differential and integral equations Applications in probability theory and statistics Applications in numerical analysis [See also 65Jxx] Applications in quantum physics Applications in statistical physics Applications in biology and other sciences None of the above, but in this section Functional analysis over fields other than R or C or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05] Nonstandard functional analysis [See also 03H05] Constructive functional analysis [See also 03F60] Fuzzy functional analysis [See also 03E72] Functional analysis in probabilistic metric linear spaces Functional analysis on superspaces (supermanifolds) or graded spaces [See also 58A50 and 58C50] None of the above, but in this section    Operator theory Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators Operators defined by compactness properties Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] Hermitian and normal operators (spectral measures, functional calculus, etc.) Subnormal operators, hyponormal operators, etc. Symmetric and selfadjoint operators (unbounded) Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22] Composition operators Kernel operators Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] Jacobi (tridiagonal) operators (matrices) and generalizations Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) Operators on function spaces (general) Difference operators [See also 39A70] Spectral operators, decomposable operators, well-bounded operators, etc. Accretive operators, dissipative operators, etc. Commutators, derivations, elementary operators, etc. Operators on Banach algebras Transformers (= operators on spaces of operators) Operators on spaces with an indefinite metric [See also 46C50] Operators on ordered spaces Positive operators and order-bounded operators Random operators [See also 60H25] None of the above, but in this section Operators in algebras Operators in $^*$-algebras Operators in $C^*$- or von Neumann algebras None of the above, but in this section Integral operators [See also 45P05] Integro-differential operators [See also 34K30, 35R10, 45J05, 45K05] Pseudodifferential operators [See also 35Sxx, 58Jxx] None of the above, but in this section Equations involving nonlinear operators (general) Nonlinear ill-posed problems Abstract inverse mapping and implicit function theorems [See also 46T20 and 58C15] Nonlinear eigenvalue problems Abstract bifurcation theory [See also 58E07, 58E09] Variational and other types of inequalities involving nonlinear operators (general) Methods for solving nonlinear operator equations (general) Variational methods [See also 58Exx] Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 58D25] Equations with hysteresis operators None of the above, but in this section Applications in optimization, convex analysis, mathematical programming, economics Applications to differential and integral equations Applications in probability theory and statistics Applications in numerical analysis [See also 65Jxx] Applications in quantum physics Applications in statistical physics Applications in biology and other sciences Applications in systems theory, circuits, etc. None of the above, but in this section Operator theory over fields other than R, C or the quaternions; non-Archimedean operator theory Nonstandard operator theory [See also 03H05] Constructive operator theory [See also 03F60] Fuzzy operator theory [See also 03E72] Operator theory in probabilistic metric linear spaces None of the above, but in this section    Calculus of variations and optimal control; optimization Free problems in one independent variable Free problems in two or more independent variables Optimal control problems involving ordinary differential equations Optimal control problems involving partial differential equations Optimal control problems involving integral equations Optimal control problems involving differential inclusions [See also 34A60] Optimal control problems involving equations with retarded arguments [See also 34K35] Problems in abstract spaces [See also 90C48, 93C25] Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) Minimax problems Variational methods including variational inequalities [See also 47J20] Methods involving semicontinuity and convergence; relaxation Fréchet and Gateaux differentiability [See also 46G05, 58C20] Nonsmooth analysis [See also 46G05, 58C50] Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06] Problems involving randomness [See also 93E20] None of the above, but in this section Free problems in one independent variable Free problems in two or more independent variables Problems involving ordinary differential equations Problems involving partial differential equations Problems involving integral equations Problems involving differential inclusions [See also 34A60] Problems involving equations with retarded arguments [See also 34K35] Problems in abstract spaces [See also 90C48, 93C25] Optimal solutions belonging to restricted classes Minimax problems Sensitivity, stability, well-posedness [See also 90C31] Problems involving randomness [See also 93E20] None of the above, but in this section Dynamic programming method Viscosity solutions None of the above, but in this section Methods based on necessary conditions Methods of Newton-Raphson, Galerkin and Ritz types Methods of relaxation type Discrete approximations Decomposition methods Methods involving duality Other methods, not based on necessary conditions (penalty function, etc.) Methods of nonlinear programming type [See also 90C30, 65Kxx] None of the above, but in this section Linear optimal control problems [See also 93C05] Linear-quadratic problems Duality theory Periodic optimization Impulsive optimal control problems Problems with incomplete information [See also 93C41] Optimal feedback synthesis [See also 93B52] Inverse problems Regularity of solutions Differential games Pursuit and evasion games Applications of optimal control and differential games [See also 90C90, 93C95] None of the above, but in this section    Geometry {For algebraic geometry, see 14-XX} General theory and projective geometries Homomorphism, automorphism and dualities Structures with parallelism Configuration theorems Algebraization [See also 12Kxx, 20N05] Desarguesian and Pappian geometries Non-Desarguesian affine and projective planes Translation planes and spreads Incidence structures imbeddable into projective geometries Polar geometry, symplectic spaces, orthogonal spaces None of the above, but in this section General theory Möbius geometries Laguerre geometries Minkowski geometries Lie geometries None of the above, but in this section Abstract (Maeda) geometries Abstract geometries with exchange axiom Abstract geometries with parallelism Combinatorial geometries [See also 05B25, 05B35] Lattices of subspaces [See also 05B35] Continuous geometries and related topics [See also 06Cxx] None of the above, but in this section General block designs [See also 05B05] Steiner systems Generalized quadrangles, generalized polygons Finite partial geometries (general), nets, partial spreads Affine and projective planes Combinatorial structures in finite projective spaces [See also 05Bxx] Blocking sets, ovals, $k$-arcs Linear codes and caps in Galois spaces [See also 94B05] Spreads and packing problems Buildings and the geometry of diagrams Other finite nonlinear geometries Other finite linear geometries Other finite incidence structures [See also 05B30] None of the above, but in this section Absolute planes Absolute spaces Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55] Congruence and orthogonality [See also 20H05] Orthogonal and unitary groups [See also 20H05] None of the above, but in this section General theory Topological linear incidence structures Topological nonlinear incidence structures Topological geometries on manifolds [See also 57-XX] Geometries with differentiable structure [See also 53Cxx, 53C70] Geometries with algebraic manifold structure [See also 14-XX] None of the above, but in this section General theory Projective incidence groups Kinematic spaces Representation by near-fields and near-algebras [See also 12K05, 16Y30] None of the above, but in this section General theory Synthetic differential geometry None of the above, but in this section Geometry of orders of nondifferentiable curves Directly differentiable curves $n$-vertex theorems via direct methods Geometry of orders of surfaces None of the above, but in this section Elementary problems in Euclidean geometries Euclidean geometries (general) and generalizations Elementary problems in hyperbolic and elliptic geometries Hyperbolic and elliptic geometries (general) and generalizations Geometric constructions Inequalities and extremum problems {For convex problems, see 52A40} Polyhedra and polytopes; regular figures, division of spaces [See also 51F15] Length, area and volume [See also 26B15] Line geometries and their generalizations [See also 53A25] Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) [See also 14M15] None of the above, but in this section Descriptive geometry [See also 65D17, 68U07] Affine analytic geometry Projective analytic geometry Euclidean analytic geometry Analytic geometry with other transformation groups Geometry of classical groups [See also 20Gxx, 14L35] Questions of classical algebraic geometry [See also 14Nxx] None of the above, but in this section    Convex and discrete geometry Axiomatic and generalized convexity Convex sets without dimension restrictions Convex sets in topological vector spaces [See also 46A55] Convex sets in $2$ dimensions (including convex curves) [See also 53A04] Convex sets in $3$ dimensions (including convex surfaces) [See also 53A05, 53C45] Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45] Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] Random convex sets and integral geometry [See also 53C65, 60D05] Approximation by convex sets Variants of convex sets (star-shaped, ($m, n$)-convex, etc.) Helly-type theorems and geometric transversal theory Other problems of combinatorial convexity Length, area, volume [See also 26B15, 28A75, 49Q20] Mixed volumes and related topics Inequalities and extremum problems Convex functions and convex programs [See also 26B25, 90C25] Spherical and hyperbolic convexity None of the above, but in this section Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] Three-dimensional polytopes $n$-dimensional polytopes Special polytopes (linear programming, centrally symmetric, etc.) Symmetry properties of polytopes Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] Shellability Gale and other diagrams Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) [See also 05B35, 52Cxx] Dissections and valuations (Hilbert's third problem, etc.) Computational aspects related to convexity {For computational geometry and algorithms, see 68Q25, 68U05; for numerical algorithms, see 65Yxx} [See also 68Uxx] Isoperimetric problems for polytopes Polyhedral manifolds None of the above, but in this section Lattices and convex bodies in $2$ dimensions [See also 11H06, 11H31, 11P21] Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21] Erdös problems and related topics of discrete geometry [See also 11Hxx] Packing and covering in $2$ dimensions [See also 05B40, 11H31] Packing and covering in $n$ dimensions [See also 05B40, 11H31] Tilings in $2$ dimensions [See also 05B45, 51M20] Tilings in $n$ dimensions [See also 05B45, 51M20] Quasicrystals, aperiodic tilings Rigidity and flexibility of structures [See also 70B15] Circle packings and discrete conformal geometry Planar arrangements of lines and pseudolines Arrangements of points, flats, hyperplanes [See also 32S22] Oriented matroids Combinatorial complexity of geometric structures [See also 68U05] None of the above, but in this section    Differential geometry {For differential topology, see 57Rxx. For Curves in Euclidean space Surfaces in Euclidean space Higher-dimensional and -codimensional surfaces in Euclidean $n$-space Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] Affine differential geometry Kinematics Projective differential geometry Differential line geometry Conformal differential geometry Non-Euclidean differential geometry Other special differential geometries Vector and tensor analysis Differential invariants (local theory), geometric objects Geometry of webs [See also 14C21, 20N05] None of the above, but in this section Linear and affine connections Projective connections Other connections Local Riemannian geometry Methods of Riemannian geometry Local submanifolds [See also 53C40] Lorentz metrics, indefinite metrics Hermitian and Kählerian structures [See also 32Cxx] Finsler spaces and generalizations (areal metrics) Applications to physics None of the above, but in this section Symplectic manifolds, general Contact manifolds, general Lagrangian submanifolds; Maslov index Almost contact and almost symplectic manifolds Poisson manifolds Momentum maps; symplectic reduction Canonical transformations Geodesic flows Symplectic structures of moduli spaces Global theory of symplectic and contact manifolds [See also 57Rxx] Floer homology and cohomology, symplectic aspects Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] Geometric quantization Deformation quantization, star products None of the above, but in this section    General topology {For the topology of manifolds of all dimensions, Connected and locally connected spaces (general aspects) Lower separation axioms ($T_0$--$T_3$, etc.) Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) Noncompact covering properties (paracompact, Lindelöf, etc.) ``$P$-minimal'' and ``$P$-closed'' spaces Compactness Extensions of spaces (compactifications, supercompactifications, completions, etc.) Remainders Local compactness, $\sigma$-compactness $k$-spaces Sequential spaces Realcompactness and realcompactification Separability Base properties Special constructions of spaces (spaces of ultrafilters, etc.) None of the above, but in this section Proximity structures and generalizations Uniform structures and generalizations Nearness spaces $p$-spaces, $M$-spaces, $\sigma$-spaces, etc. Stratifiable spaces, cosmic spaces, etc. Semimetric spaces Moore spaces Metric spaces, metrizability Special maps on metric spaces Compact (locally compact) metric spaces Complete metric spaces Baire category, Baire spaces Bitopologies Probabilistic metric spaces None of the above, but in this section Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces [See also 06B30, 06F30] Continua and generalizations Higher-dimensional local connectedness [See also 55Mxx, 55Nxx] Dimension theory [See also 55M10] Spaces of dimension $\leq 1$; curves, dendrites [See also 26A03] Unicoherence, multicoherence Topological characterizations of particular spaces None of the above, but in this section Extremally disconnected spaces, $F$-spaces, etc. $P$-spaces Scattered spaces Pathological spaces Counterexamples None of the above, but in this section    Algebraic topology Duality Dimension theory [See also 54F45] Absolute neighborhood retracts [See also 54C55] Fixed points and coincidences [See also 54H25] Degree, winding number Ljusternik-Schnirelman (Lyusternik-Shnirelman) category of a space Finite groups of transformations (including Smith theory) [See also 57S17] None of the above, but in this section Homotopy extension properties, cofibrations Homotopy equivalences Classification of homotopy type Eilenberg-Mac Lane spaces Spanier-Whitehead duality Eckmann-Hilton duality Loop spaces Suspensions Stable homotopy theory, spectra Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) ${H}$-spaces and duals Infinite loop spaces Loop space machines, operads [See also 18D50] Shape theory [See also 54C56, 55Q07] Proper homotopy theory Localization and completion Rational homotopy theory Homotopy functors Equivariant homotopy theory [See also 19L47] Relations between equivariant and nonequivariant homotopy theory None of the above, but in this section Homotopy groups, general; sets of homotopy classes Shape groups Stable homotopy groups Whitehead products and generalizations Homotopy groups of wedges, joins, and simple spaces Hopf invariants Operations in homotopy groups Homotopy groups of spheres Stable homotopy of spheres $J$-morphism [See also 19L20] $v_n$-periodicity Homotopy groups of special spaces Cohomotopy groups Homotopy groups of special types [See also 55N05, 55N07] Equivariant homotopy groups [See also 19L47] None of the above, but in this section Fiber spaces Fiber bundles Transfer Classification Spectral sequences and homology of fiber spaces [See also 55Txx] Sphere bundles and vector bundles Classifying spaces of groups and ${H}$-spaces Maps between classifying spaces Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] Homology and homotopy of $B{\rm O}$ and $B{\rm U}$; Bott periodicity Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX} Fiberings with singularities Microbundles and block bundles [See also 57N55, 57Q50] Generalizations of fiber spaces and bundles Fibrewise topology Discriminantal varieties, configuration spaces Equivariant fiber spaces and bundles [See also 19L47] None of the above, but in this section Primary cohomology operations Steenrod algebra Dyer-Lashof operations Symmetric products, cyclic products Secondary and higher cohomology operations $K$-theory operations and generalized cohomology operations [See also 19D55, 19Lxx] Massey products Obstruction theory Extension and compression of mappings Classification of mappings Sectioning fiber spaces and bundles Postnikov systems, $k$-invariants Equivariant operations and obstructions [See also 19L47] None of the above, but in this section General Serre spectral sequences Adams spectral sequences Eilenberg-Moore spectral sequences [See also 57T35] Generalized cohomology None of the above, but in this section    Manifolds and cell complexes {For complex manifolds, see 32Qxx} Fundamental group, presentations, free differential calculus Topological methods in group theory Covering spaces Special coverings, e.g. branched Relations with graph theory [See also 05Cxx] Two-dimensional complexes Knots and links in $S^3$ {For higher dimensions, see 57Q45} Invariants of knots and 3-manifolds Wild knots and surfaces, etc., wild embeddings Dehn's lemma, sphere theorem, loop theorem, asphericity Characterizations of $E^3$ and $S^3$ (Poincaré conjecture) [See also 57N12] Geometric structures on low-dimensional manifolds Group actions in low dimensions None of the above, but in this section Local properties of generalized manifolds Poincaré duality spaces None of the above, but in this section General topology of complexes Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28] Wall finiteness obstruction for CW-complexes Triangulating manifolds Cobordism Comparison of PL-structures: classification, Hauptvermutung Engulfing Embeddings and immersions Isotopy Regular neighborhoods Knots and links (in high dimensions) {For the low-dimensional case, see 57M25} Microbundles and block bundles [See also 55R60, 57N55] Approximations Cobordism and concordance General position and transversality Equivariant PL-topology None of the above, but in this section    Global analysis, analysis on manifolds [See also 32Cxx, 32Fxx, Pseudogroups and differentiable groupoids [See also 22A22, 22E65] Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.) [See also 57R32] Deformations of structures [See also 32Gxx, 58J10] None of the above, but in this section Critical points of functions and mappings Monodromy Topological properties of mappings Algebraic and analytic properties of mappings Stability Global theory Catastrophe theory Classification; finite determinacy of map germs Singularities of vector fields, topological aspects Normal forms Asymptotic behavior Deformation of singularities Topological invariants Symmetries, equivariance None of the above, but in this section    Probability theory and stochastic processes {For additional Axioms; other general questions Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} None of the above, but in this section Probability measures on topological spaces Convergence of probability measures Probability theory on linear topological spaces [See also 28C20] Limit theorems for vector-valued random variables (infinite-dimensional case) Probability measures on groups, Fourier transforms, factorization None of the above, but in this section Distributions: general theory Infinitely divisible distributions; stable distributions Characteristic functions; other transforms Inequalities; stochastic orderings None of the above, but in this section Central limit and other weak theorems Large deviations Strong theorems Functional limit theorems; invariance principles Zero-one laws $L^p$-limit theorems None of the above, but in this section Foundations of stochastic processes General theory of processes Exchangeability Stationary processes General second-order processes Gaussian processes Sample path properties Self-similar processes Generalized stochastic processes Prediction theory [See also 62M20] Continuity and singularity of induced measures Applications (signal detection, filtering, etc.) [See also 62M20, 93E10, 93E11, 94Axx] Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] Martingales with discrete parameter Martingales with continuous parameter Martingales and classical analysis Generalizations of martingales Sums of independent random variables; random walks Processes with independent increments Stable processes Point processes Random measures Random fields Extreme value theory; extremal processes None of the above, but in this section    Statistics Sufficient statistics and fields Information-theoretic topics [See also 94A17] Theory of statistical experiments None of the above, but in this section General considerations Complete class results Bayesian problems; characterization of Bayes procedures Empirical decision procedures; empirical Bayes procedures Admissibility Minimax procedures Compound decision problems None of the above, but in this section Characterization and structure theory Exact distribution theory Approximations to distributions (nonasymptotic) Asymptotic distribution theory None of the above, but in this section Hypothesis testing Asymptotic properties of tests Ranking and selection Point estimation Asymptotic properties of estimators Bayesian inference Tolerance and confidence regions Inference under constraints Robustness and adaptive procedures Bootstrap, jackknife and other resampling methods None of the above, but in this section Estimation Density estimation Nonparametric regression Resampling methods Hypothesis testing Tolerance and confidence regions Asymptotic properties Order statistics; empirical distribution functions Statistics of extreme values; tail inference Robustness None of the above, but in this section Characterization and structure theory Distribution of statistics Directional data; spatial statistics Estimation Hypothesis testing Contingency tables Measures of association (correlation, canonical correlation, etc.) Factor analysis and principal components; correspondence analysis Classification and discrimination; cluster analysis [See also 68T10] Image analysis None of the above, but in this section General nonlinear regression Linear regression Ridge regression; shrinkage estimators Analysis of variance and covariance Generalized linear models Paired and multiple comparisons Diagnostics None of the above, but in this section Optimal designs Block designs Factorial designs Response surface designs Robust parameter designs None of the above, but in this section Sequential design Sequential analysis Sequential estimation Optimal stopping [See also 60G40, 91A60] Stochastic approximation None of the above, but in this section Markov processes: hypothesis testing Markov processes: estimation Non-Markovian processes: hypothesis testing Non-Markovian processes: estimation Time series, auto-correlation, regression, etc. [See also 91B84] Spectral analysis Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] Spatial processes Random fields; image analysis Neural nets and related approaches None of the above, but in this section Censored data models Estimation Testing Reliability and life testing [See also 90B25] None of the above, but in this section Applications to actuarial sciences and financial mathematics Applications to biology and medical sciences Applications to environmental and related topics Applications to psychology Applications to economics [See also 91Bxx] Applications to social sciences Applications in engineering and industry Applications to physics None of the above, but in this section    Numerical analysis Extrapolation to the limit, deferred corrections Summation of series Euler-Maclaurin formula None of the above, but in this section Monte Carlo methods Random number generation Models, numerical methods [See also 68U20] Stochastic differential and integral equations Stochastic particle methods [See also 82C80] Computational Markov chains Other computational problems in probability Computational problems in statistics None of the above, but in this section Direct methods for linear systems and matrix inversion Iterative methods for linear systems [See also 65N22] Eigenvalues, eigenvectors Inverse eigenvalue problems Overdetermined systems, pseudoinverses Ill-posedness, regularization Orthogonalization Other matrix algorithms Matrix norms, conditioning, scaling [See also 15A12, 15A60] Determinants Sparse matrices None of the above, but in this section Algorithms with automatic result verification Interval and finite arithmetic General methods in interval analysis Roundoff error None of the above, but in this section Single equations Systems of equations Eigenvalues, eigenvectors [See also 47Hxx, 47Jxx, 58C40, 58E07, 90C30] Global methods, including homotopy approaches [See also 58C30, 90C30] None of the above, but in this section General theory Equations with linear operators (do not use 65Fxx) Equations with nonlinear operators (do not use 65Hxx) Improperly posed problems; regularization Inverse problems None of the above, but in this section Initial value problems Multistep, Runge-Kutta and extrapolation methods Numerical investigation of stability of solutions Improperly posed problems Inverse problems Boundary value problems Finite difference methods Eigenvalue problems Stability and convergence of numerical methods Mesh generation and refinement Finite elements, Rayleigh-Ritz, Galerkin and collocation methods Error bounds Methods for differential-algebraic equations None of the above, but in this section Finite difference methods Stability and convergence of numerical methods Error bounds Method of lines Method of characteristics Improperly posed problems Inverse problems Mesh generation and refinement Multigrid methods; domain decomposition Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Spectral, collocation and related methods None of the above, but in this section Finite difference methods Stability and convergence of numerical methods Error bounds Inverse problems Solution of discretized equations [See also 65Fxx, 65Hxx] Eigenvalue problems Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Spectral, collocation and related methods Boundary element methods Method of lines Method of contraction of the boundary Mesh generation and refinement Multigrid methods; domain decomposition None of the above, but in this section Hamiltonian systems including symplectic integrators Numerical chaos Bifurcation problems Nonlinear stabilities None of the above, but in this section Trigonometric approximation and interpolation Discrete and fast Fourier transforms Wavelets None of the above, but in this section Parallel computation Algorithms for specific classes of architectures Packaged methods Complexity and performance of numerical algorithms [See also 68Q25] None of the above, but in this section    Computer science {For papers involving machine computations and General Mathematical problems of computer architecture Network design and communication [See also 68R10, 90B18] Network protocols Distributed systems Reliability, testing and fault tolerance [See also 94C12] Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] None of the above, but in this section General Programming languages Logic programming Functional programming and lambda calculus [See also 03B40] Other programming techniques (object-oriented, sequential, concurrent, automatic, etc.) Compilers and interpreters Operating systems Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.) None of the above, but in this section General Data structures Searching and sorting Database theory Information storage and retrieval Data encryption [See also 94A60, 81P68] Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx] None of the above, but in this section General Models of computation (Turing machines, etc.) [See also 03D10, 81P68] Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.) [See also 68Q85] Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19] Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15] Descriptive complexity and finite models [See also 03C13] Analysis of algorithms and problem complexity [See also 68W40] Algorithmic information theory (Kolmogorov complexity, etc.) Computational learning theory [See also 68T05] Grammars and rewriting systems Formal languages and automata [See also 03D05, 68Q70, 94A45] Semantics [See also 03B70, 06B35, 18C50] Specification and verification (program logics, model checking, etc.) [See also 03B70] Abstract data types; algebraic specification [See also 18C50] Algebraic theory of languages and automata [See also 18B20, 20M35] Cellular automata [See also 37B15] Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) None of the above, but in this section General Combinatorics Graph theory [See also 05Cxx, 90B10, 90B35, 90C35] Combinatorics on words None of the above, but in this section General Learning and adaptive systems [See also 68Q32, 91E40] Pattern recognition, speech recognition {For cluster analysis, see 62H30} Theorem proving (deduction, resolution, etc.) [See also 03B35] Problem solving (heuristics, search strategies, etc.) Logic in artificial intelligence Knowledge representation Languages and software systems (knowledge-based systems, expert systems, etc.) Reasoning under uncertainty Robotics [See also 93C85] Machine vision and scene understanding Natural language processing [See also 03B65] None of the above, but in this section General Computer graphics; computational geometry [See also 65D18] Computer-aided design [See also 65D17] Image processing Text processing; mathematical typography Simulation [See also 65Cxx] Information systems (hypertext navigation, interfaces, decision support, etc.) None of the above, but in this section General Nonnumerical algorithms Parallel algorithms Distributed algorithms Randomized algorithms Approximation algorithms Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10] VLSI algorithms Analysis of algorithms [See also 68Q25] None of the above, but in this section    Mechanics of particles and systems {For relativistic mechanics, Kinematics of a particle Kinematics of a rigid body Mechanisms, robots [See also 68T40, 70Q05, 93C85] None of the above, but in this section Motion of the gyroscope Free motion of a rigid body [See also 70M20] Motion of a rigid body with a fixed point Motion of a rigid body in contact with a solid surface [See also 70F25] Perturbation methods for rigid body dynamics Integrable cases of motion Higher-dimensional generalizations Stability problems Dynamics of multibody systems Robot dynamics and control [See also 68T40, 70Q05, 93C85] None of the above, but in this section    Mechanics of deformable solids Explicit solutions Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) Numerical approximation of solutions Existence of solutions Uniqueness of solutions Regularity of solutions Singularities, blowup, stress concentrations Long-time behavior of solutions Vibrations Random vibrations Stability Dynamical bifurcation Chaotic behavior None of the above, but in this section Linear waves Bulk waves Surface waves Wave scattering Inverse problems Nonlinear waves Solitary waves Shocks and related discontinuities None of the above, but in this section Strings Rods (beams, columns, shafts, arches, rings, etc.) Membranes Plates Shells Junctions Thin films None of the above, but in this section Geophysical solid mechanics [See also 86-XX] Soil and rock mechanics Biomechanical solid mechanics [See also 92C10] None of the above, but in this section Crystals Displacive transformations Analysis of microstructure Dynamics of phase boundaries Transformations involving diffusion Problems involving hysteresis None of the above, but in this section    Fluid mechanics {For general continuum mechanics, see 74Axx, or    Optics, electromagnetic theory {For quantum optics, see 81V80}    Classical thermodynamics, heat transfer {For thermodynamics of Foundations Classical thermodynamics, including relativistic Thermodynamics of continua [See also 74A15] Heat and mass transfer, heat flow Stefan problems, phase changes, etc. [See also 74Nxx] Inverse problems Combustion Chemical kinetics [See also 76V05, 92C45, 92E20] Chemically reacting flows [See also 92C45, 92E20] Chemistry (general) [See mainly 92Exx] None of the above, but in this section    Quantum theory General and philosophical Logical foundations of quantum mechanics; quantum logic [See also 03G12, 06C15] Quantum measurement theory Stochastic mechanics (including stochastic electrodynamics) Quantum computation and quantum cryptography [See also 68Q05, 94A60] None of the above, but in this section Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations Selfadjoint operator theory in quantum theory, including spectral analysis Perturbation theories for operators and differential equations Semiclassical techniques including WKB and Maslov methods Feynman integrals and graphs; applications of algebraic topology and algebraic geometry [See also 14D05, 32S40] Bethe-Salpeter and other integral equations Quantum chaos [See also 37Dxx] Supersymmetric quantum mechanics Differential-geometric methods, including holonomy, Berry and Hannay phases, etc. None of the above, but in this section Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70] 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] Relations with integrable systems [See also 17Bxx, 37J35] Operator algebra methods [See also 46Lxx, 81T05] Covariant wave equations Spinor and twistor methods [See also 32L25] Coherent states [See also 22E45]; squeezed states [See also 81V80] Symmetry breaking Quantum groups and related algebraic methods [See also 16W35, 17B37] Noncommutative geometry None of the above, but in this section Commutation relations and statistics Geometry and quantization, symplectic methods [See also 53D50] Stochastic quantization Quantum stochastic calculus Phase space methods including Wigner distributions, etc. Path integrals [See also 58D30] None of the above, but in this section Axiomatic quantum field theory; operator algebras Constructive quantum field theory Model quantum field theories Yang-Mills and other gauge theories [See also 53C07, 58E15] Perturbative methods of renormalization Nonperturbative methods of renormalization Renormalization group methods Feynman diagrams Quantum field theory on curved space backgrounds Quantum field theory on lattices Continuum limits String and superstring theories; other extended objects (e.g., branes) [See also 83E30] Two-dimensional field theories, conformal field theories, etc. Topological field theories [See also 57R56, 58Dxx] Anomalies Supersymmetric field theories Quantization in field theory; cohomological methods [See also 58D29] Noncommutative geometry methods [See also 46L85, 46L87, 58B34] Simulation and numerical modeling None of the above, but in this section    Statistical mechanics, structure of matter Foundations Classical dynamic and nonequilibrium statistical mechanics (general) Quantum dynamics and nonequilibrium statistical mechanics (general) Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs Dynamic continuum models (systems of particles, etc.) Interacting particle systems [See also 60K35] Exactly solvable dynamic models [See also 37K60] Interface problems; diffusion-limited aggregation Dynamic and nonequilibrium phase transitions (general) Dynamic critical phenomena Dynamic renormalization group methods [See also 81T17] Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10] Neural nets [See also 68T05, 91E40, 92B20] Irreversible thermodynamics, including Onsager-Machlup theory Kinetic theory of gases