=========== Higher level sections 00-XX General 01-XX History and biography 03-XX Mathematical logic and foundations 04-XX This section has been deleted {For set theory see 03Exx} 05-XX Combinatorics {For finite fields, see 11Txx} 06-XX Order, lattices, ordered algebraic structures [See also 18B35] 08-XX General algebraic systems 11-XX Number theory 12-XX Field theory and polynomials 13-XX Commutative rings and algebras /- {for the noncommutative case, see 16-XX} 14-XX Algebraic geometry 15-XX Linear and multilinear algebra; matrix theory {(finite and infinite)} 16-XX Associative rings and algebras {For the commutative case, see 13-XX} 17-XX Nonassociative rings and algebras 18-XX Category theory; abstract homological algebra /+ abstract 19-XX $K$-theory [See also 16E20, 18F25] 20-XX Group theory and generalizations 22-XX Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX} 26-XX Real functions [See also 54C30] 28-XX Measure and integration {For analysis on manifolds, see 58-XX} 30-XX Functions of a complex variable {For analysis on manifolds, see 58-XX} 31-XX Potential theory {For probabilistic potential theory, see 60J45} 32-XX Several complex variables and analytic spaces {For infinite-dimensional holomorphy see also 46G20, 58B12} 33-XX Special functions {33-XX deals with the properties of functions as functions. For orthogonal functions [See also 42Cxx; for aspects of combinatorics, see 05Axx; for number-theoretic aspects, see 11-XX; for representation theory, see 22Exx] 34-XX Ordinary differential equations 35-XX Partial differential equations 37-XX Dynamical systems and ergodic theory [See also 26A18, 28Dxx, /+ 34Cxx, 34Dxx, 58Jxx, 46Lxx, 70-XX] /++ 39-XX Finite differences and functional equations 40-XX Sequences, series, summability 41-XX Approximations and expansions {For all approximation theory in the complex domain, see 30Exx, 30E05 and 30E10; for all trigonometric /+ 30Exx, approximation and interpolation, see 42Axx, 42A10 and 42A15; for /+42Axx numerical approximation, see 65Dxx} 42-XX Fourier analysis 43-XX Abstract harmonic analysis {For other analysis on topological and Lie groups, see 22Exx} 44-XX Integral transforms, operational calculus {For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For numerical methods, see 65R10} 45-XX Integral equations 46-XX Functional analysis {For manifolds modeled on topological linear spaces, see 57N20, 58Bxx} 47-XX Operator theory 49-XX Calculus of variations and optimal control; optimization [See also 34H05, 65Kxx, 90Cxx, 93-XX] 51-XX Geometry {For algebraic geometry, see 14-XX} 52-XX Convex and discrete geometry /~ convex sets and discrete geometry 53-XX Differential geometry {For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx} 54-XX General topology {For the topology of manifolds of all dimensions, see 57Nxx} 55-XX Algebraic topology 57-XX Manifolds and cell complexes {For complex manifolds, see 32C10} 58-XX Global analysis, analysis on manifolds [See also 32-XX, 32Cxx, /+ 32-XX 32Fxx, 46-XX, 47Hxx, 53Cxx] {For geometric integration theory, see 49Qxx} /~ 49Q15 60-XX Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX, 92-XX, 93-XX, 94-XX. For numerical results, see 65U05 62-XX Statistics {For numerical methods, see 65U05} 65-XX Numerical analysis 68-XX Computer science {For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area} 70-XX Mechanics of particles and systems {For relativistic mechanics, see 83-XX, 83A05 and 83C10; for statistical mechanics, see 82-XX} /+ 83-XX 73-XX / - * (Mechanic of solids) 74-XX Mechanics of deformable solids /++ 76-XX Fluid mechanics {For general continuum mechanics, see 74axx, or other parts of 74-XX} /~ 73Bxx or other parts of 73-XX} 78-XX Optics, electromagnetic theory {For quantum optics, see 81V80} 80-XX Classical thermodynamics, heat transfer {For thermodynamics of solids, see 74A15} /~ 73B30) 81-XX Quantum Theory /~ quantum mechanics 82-XX Statistical mechanics, /~ statistical physics, structure of matter 83-XX Relativity and gravitational theory 85-XX Astronomy and astrophysics {For celestial mechanics, see 70F15} 86-XX Geophysics [See also 73N05, 76U05, 76V05] /~ [See also 73N20, 76U05, 76V05] 90-XX Operations research, programming /~ economics, operations research, programming, games 91-XX Game theory, economics, social and behavioral sciences /++ 92-XX Biology and other natural sciences /- , behavioral sciences 93-XX Systems theory; control {For optimal control, see 49-XX} 94-XX Information and communication, circuits 97-XX Mathematics education /++ ============ Complete list 00-XX General 00-01 Instructional exposition (textbooks, tutorial papers, etc.) 00-02 Research exposition (monographs, survey articles) 00Axx General and miscellaneous specific topics 00A05 General mathematics 00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.) 00A07 Problem books 00A08 Recreational mathematics 00A15 Bibliographies 00A17 External book reviews /++ 00A20 Dictionaries and other general reference works 00A22 Formularies 00A30 Philosophy of mathematics [See also 03A05] 00A35 Methodology of mathematics, didactics 00A69 General applied mathematics {For physics, see 00A79 and Sections 70 through 86} 00A71 Theory of mathematical modeling 00A72 General methods of simulation 00A73 Dimensional analysis 00A79 Physics (use more specific entries from Sections 70 through 86 when possible) 00A99 Miscellaneous topics 00Bxx Conference proceedings and collections of papers 00B05 Collections of abstracts of lectures 00B10 Collections of articles of general interest 00B15 Collections of articles of miscellaneous specific content 00B20 Proceedings of conferences of general interest 00B25 Proceedings of conferences of miscellaneous specific interest 00B30 Festschriften 00B50 Volumes of selected translations 00B55 Miscellaneous volumes of translations 00B60 Collections of reprinted articles [See also 01A75] 01-XX History and biography [See also the classification number -03 in the other sections] 01-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 01-01 Instructional exposition (textbooks, tutorial papers, etc.) 01-02 Research exposition (monographs, survey articles) 01-06 Proceedings, conferences, collections, etc. 01-08 Computational methods 01Axx History of mathematics and mathematicians 01A05 General histories, source books 01A07 Ethnomathematics, general 01A10 Paleolithic, Neolithic 01A12 Indigenous cultures of the Americas 01A13 Other indigenous cultures (non-European) 01A15 Indigenous European cultures (pre-Greek, etc.) 01A16 Egyptian 01A17 Babylonian 01A20 Greek, Roman 01A25 China 01A27 Japan 01A29 Southeast Asia 01A30 Islam (Medieval) 01A32 India 01A35 Medieval 01A40 15th and 16th centuries, Renaissance 01A45 17th century 01A50 18th century 01A55 19th century 01A60 20th century 01A61 Twenty-first century /+ 01A65 Contemporary 01A67 Future prospectives /+ 01A70 Biographies, obituaries, personalia, bibliographies 01A72 Schools of mathematics 01A73 Universities 01A74 Other institutions and academies 01A75 Collected or selected works; reprintings or translations of classics [See also 00B60] 01A80 Sociology (and profession) of mathematics 01A85 Historiography 01A90 Bibliographic studies 01A99 Miscellaneous topics 03-XX Mathematical logic and foundations 03-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 03-01 Instructional exposition (textbooks, tutorial papers, etc.) 03-02 Research exposition (monographs, survey articles) 03-03 Historical (must be assigned at least one classification number from Section 01) 03-04 Explicit machine computation and programs (not the theory of computation or programming) 03-06 Proceedings, conferences, collections, etc. 03A05 Philosophical and critical {For philosophy of mathematics, see also 00A30} 03Bxx General logic 03B05 Classical propositional logic 03B10 Classical first-order logic 03B15 Higher-order logic and type theory 03B20 Subsystems of classical logic (including intuitionistic logic) 03B22 Abstract deductive systems 03B25 Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] 03B30 Foundations /~foundation and axiomatics of classical theories (including reverse mathematics) [See also 03F35] 03B35 Mechanization of proofs and logical operations [See also 68T15] 03B40 Combinatory logic and lambda-calculus [See also 68N18] /++ 03B42 Logic of knowledge and belief /++ 03B44 Temporal logic /++ 03B45 Modal logic /~ modal and tense logic {For knowledge and belief see 03B42; for temporal logic see 03B44; /++ for provability logic see also 03F45} /~ {for provability logic see also 03F40} 03B46 relevance and entailment /-- 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} 03B48 Probability and inductive logic [See also 60A05] 03B50 Many-valued logic 03B52 Fuzzy logic; logic of vagueness /+ logic of vagueness [See also 68T27, 68T37, 94D05] /+ 68T27, 68T37; 03B53 Logics admitting inconsistency /~ paraconsistent logic (paraconsistent logics, discussive logics, etc.) /++ 03B55 Intermediate logics 03B60 Other nonclassical logic 03B65 Logic of natural languages [See also 68T50, 91F20] /~ [See also 68S05, 92K20] 03B70 Logic in computer science [See also 68-XX] /~ logic of programming [$ 68Q55, 68Q60] 03B80 Other applications of logic 03B99 None of the above, but in this section 03Cxx Model theory 03C05 Equational classes, universal algebra [See also 08Axx, 18C05] /++ 03C07 Basic properties of first-order languages and structures 03C10 Quantifier elimination, model completeness /+ , model completeness and related topics 03C13 Finite structures [See also 68Q15, 68Q19] 03C15 Denumerable structures 03C20 Ultraproducts and related constructions 03C25 Model-theoretic forcing 03C30 Other model constructions 03C35 Categoricity and completeness of theories 03C40 Interpolation, preservation, definability 03C45 Classification theory, /++ stability and related concepts 03C50 Models with special properties (saturated, rigid, etc.) 03C52 Properties of classes of models 03C55 Set-theoretic model theory 03C57 Effective and /++ recursion-theoretic model theory [See also 03D45] 03C60 Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] 03C62 Models of arithmetic and set theory [See also 03Hxx] 03C64 Model theory of ordered structures; o-minimality 03C65 Models of other mathematical theories 03C68 Other classical first-order model theory 03C70 Logic on admissible sets 03C75 Other infinitary logic 03C80 Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48] /~ [See also 03B45] 03C85 Second- and higher-order model theory 03C90 Nonclassical models (Boolean-valued, sheaf, etc.) 03C95 Abstract model theory 03C98 Applications of model theory [See also 03C60] /++ 03C99 None of the above, but in this section 03Dxx Computability and recursion theory /+ Computability and 03D03 Thue and Post systems, etc. 03D05 Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15] /~ [See also 68Qxx] 03D10 Turing machines and related notions [See also 68Q05] 03D15 Complexity of computation [See also 68Q15, 68Q17] /+ [See also 68Q15] 03D20 Recursive functions and relations, subrecursive hierarchies 03D25 Recursively (computably) enumerable sets and degrees /+ (computably) 03D28 Other Turing degree structures /++ 03D30 Other degrees and reducibilities 03D35 Undecidability and degrees of sets of sentences 03D40 Word problems, etc. [See also 06B25, 08A50, 20F10] 03D45 Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55] /++ 03D50 Recursive equivalence types of sets and structures, isols 03D55 Hierarchies 03D60 Computability and recursion theory on ordinals, /+ computability admissible sets, etc. 03D65 Higher-type and set recursion theory 03D70 Inductive definability 03D75 Abstract and axiomatic computability and /+ computability and recursion theory 03D80 Applications of computability and recursion theory 03D99 None of the above, but in this section 03Exx Set theory /- [See also 04-XX] 03E02 Partition relations /+ 03E04 Ordered sets and their cofinalities; pcf theory) [?] 03E05 Other combinatorial set theory /~ combinatorial set theory [$ 04a20] 03E10 Ordinal and cardinal numbers /- [$ 04A10] 03E15 Descriptive set theory [See also 28A05, 54H05] /- $ 04a10 03E17 Cardinal characteristics of the continuum /++ 03E20 Other classical set theory (including functions, realtions, [?] and set algebra) /++ 03E25 Axiom of choice and related propositions /- [...] 03E30 Axiomatics of classical set theory and its fragments 03E35 Consistency and independence results 03E40 Other aspects of forcing and Boolean-valued models 03E45 Inner models, including /++ constructibility, ordinal definability, and core models /~ related notions 03E47 Other notions of set-theoretic definability 03E50 Continuum hypothesis and Martin's axiom /- [..] 03E55 Large cardinals 03E60 Determinacy principles /~ determinacy and related principles /- which contradict the axiom of choice 03E65 Other hypotheses and axioms 03E70 Nonclassical and second-order set theories 03E72 Fuzzy set theory /~ fuzzy sets [see mainly 04A72] 03E75 Applications of set theory /+ 03E99 None of the above, but in this section 03Fxx Proof theory and constructive mathematics 03F03 Proof theory, general 03F05 Cut-elimination and normal-form theorems 03F07 Structure of proofs 03F10 Functionals in proof theory 03F15 Recursive ordinals and ordinal notations 03F20 Complexity of proofs 03F25 Relative consistency and interpretations 03F30 First-order arithmetic and fragments 03F35 Second- and higher-order arithmetic and fragments [See also 03B30] /- , 03E70 03F40 Gödel numberings in proof theory 03F45 Provability logics and related algebras (e.g., /++ diagonalizable algebras) [See also 03B45, 03G25, 06E25] /++ 03F50 Metamathematics of constructive systems 03F52 Linear logic and other substructural logics [See also 03B47] /++ 03F55 Intuitionistic mathematics 03F60 Constructive and recursive analysis [See also 03B30, 03D45, 26E40, 46S30, 47S30] /+ 03B30, 03D45, 03F65 Other constructive mathematics [See also 03D45] /~ [$ 26E40, 46S30, 47S30] 03F99 None of the above, but in this section 03Gxx Algebraic logic 03G05 Boolean algebras [See also 06Exx] 03G10 Lattices and related structures [See also 06Bxx] 03G12 Quantum logic [See also 06C15, 81P10] /+ 06C15 , 03G15 Cylindric and polyadic algebras; relation algebras 03G20 Lukasiewicz and Post algebras [See also 06D25, 06D30] 03G25 Other algebras related to logic [See also 03F45, 06E25, 06F35] /~ [$ 06F35] 03G30 Categorical logic, topoi [See also 18B25, 18C05, 18C10] /~ [$ 18B25] 03G99 None of the above, but in this section 03Hxx Nonstandard models [See also 03C62] 03H05 Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] 03H10 Other applications of nonstandard models (economics, physics, etc.) 03H15 Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05] 03H99 None of the above, but in this section 05-XX Combinatorics {For finite fields, see 11Txx} 05-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 05-01 Instructional exposition (textbooks, tutorial papers, etc.) 05-02 Research exposition (monographs, survey articles) 05-03 Historical (must be assigned at least one classification number from Section 01) 05-04 Explicit machine computation and programs (not the theory of computation or programming) 05-06 Proceedings, conferences, collections, etc. 05Axx Classical combinatorial problems 05A05 Combinatorial choice problems (subsets, representatives, permutations) 05A10 Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx] 05A15 Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16 Asymptotic enumeration 05A17 Partitions of integers [See also 11P81, 11P82, 11P83] 05A18 Partitions of sets 05A19 Combinatorial identities 05A20 Combinatorial inequalities 05A30 $q$-calculus and related topics [See also 03Dxx] 05A40 Umbral calculus 05A99 None of the above, but in this section 05Bxx Designs and configurations {For applications of design theory, see 94C30} 05B05 Block designs [See also 51E05, 62K10] 05B07 Triple systems 05B10 Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13] 05B15 Orthogonal arrays, Latin squares, Room squares 05B20 Matrices (incidence, Hadamard, etc.) 05B25 Finite geometries [See also 51D20, 51Exx] 05B30 Other designs, configurations [See also 51E30] 05B35 Matroids, geometric lattices [See also 52B40, 90C27] 05B40 Packing and covering [See also 11H31, 52C15, 52C17] 05B45 Tessellation and tiling problems [See also 52C20, 52C22] 05B50 Polyominoes 05B99 None of the above, but in this section 05Cxx Graph theory {For applications of graphs, see 68Q90, 68R10, 90C35, 94C15} 05C05 Trees 05C07 Degree sequences /++ 05C10 Topological graph theory, imbedding [See also 57M15, 57M25] 05C12 Distance in graphs 05C15 Chromatic theory of graphs and maps 05C17 Perfect graphs /++ 05C20 Directed graphs (digraphs), tournaments 05C25 Graphs and groups [See also 20F32] 05C30 Enumeration of graphs and maps 05C35 Extremal problems [See also 90C35] 05C38 Paths and cycles [See also 90B10] 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs 05C50 Graphs and matrices 05C55 Generalized Ramsey theory 05C60 Isomorphism problems (reconstruction conjecture, etc.) /- perfect graphs, 05C62 Graph representations (geometric and intersection /++ representations, etc.) /++ 05C65 Hypergraphs 05C69 Dominating sets, independent sets, cliques /++ 05C70 Factorization, matching, covering and packing 05C75 Structural characterization of types of graphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.) 05C80 Random graphs 05C83 Graph minors 05C85 Graph algorithms [See also 68W05, 68R10] /~ 68Q20, 68R10] 05C90 Applications 05C99 None of the above, but in this section 05Dxx Extremal combinatorics 05D05 Extremal set theory 05D10 Ramsey theory 05D15 Transversal (matching) theory 05D40 Probabilistic methods /++ 05D99 None of the above, but in this section 05Exx Algebraic combinatorics 05E05 Symmetric functions 05E10 Tableaux, representations of the symmetric group [See also 20C30] 05E15 Combinatorial problems concerning the classical groups [See also 22E45, 33C80] 05E20 Group actions on designs, geometries and codes 05E25 Group actions on posets and homology groups of posets [See also 06A11] /~ [$ 06A11] 05E30 Association schemes, strongly regular graphs 05E35 Orthogonal polynomials [See also 33C45, 33C50, 33D45] 05E99 None of the above, but in this section 06-XX Order, lattices, ordered algebraic structures [See also 18B35] 06-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 06-01 Instructional exposition (textbooks, tutorial papers, etc.) 06-02 Research exposition (monographs, survey articles) 06-03 Historical (must be assigned at least one classification number from Section 01) 06-04 Explicit machine computation and programs (not the theory of computation or programming) 06-06 Proceedings, conferences, collections, etc. 06Axx Ordered sets /~ order sets 06A05 Total order 06A06 Partial order, general 06A07 Combinatorics of partially ordered sets 06A08 shellable posets, Cohen-Macaulay posets [$ 52B20] /-- 06A09 cohomology of posets [$ 52B20] /-- 06A11 Algebraic aspects of posets [See also 05E25] /++ 06A12 Semilattices [See also 20M10; for topological semilattices see 22A26] /++ 06A15 Galois correspondences, closure operators 06A23 complete lattices, completions /-- 06A99 None of the above, but in this section 06Bxx Lattices [See also 03G10] 06B05 Structure theory 06B10 Ideals, congruence relations 06B15 Representation theory 06B20 Varieties of lattices 06B23 Complete lattices, completions 06B25 Free lattices, projective lattices, word problems [See also 03D40, 08A50, 20F10] 06B30 Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] 06B35 Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] /~ 68Q10] 06B99 None of the above, but in this section 06Cxx Modular lattices, complemented lattices 06C05 Modular lattices, Desarguesian lattices 06C10 Semimodular lattices, geometric lattices 06C15 Complemented lattices, orthocomplemented lattices and posets [See also 03G12, 81P10] 06C20 Complemented modular lattices, continuous geometries 06C99 None of the above, but in this section 06Dxx Distributive lattices 06D05 Structure and representation theory 06D10 Complete distributivity 06D15 Pseudocomplemented lattices 06D20 Heyting algebras [See also 03G25] /~ [$ 03Gxx] 06D25 Post algebras [See also 03G20] 06D30 De Morgan algebras, Lukasiewicz algebras [See also 03G20] 06D22 Frames, locales {For topological questions see 54-XX} /++ 06D35 MV-algebras /++ 06D50 Lattices and duality /++ 06D72 Fuzzy lattices (soft algebras) and related topics /++ 06D99 None of the above, but in this section 06Exx Boolean algebras (Boolean rings) [See also 03G05] 06E05 Structure theory 06E10 Chain conditions, complete algebras 06E15 Stone space and related constructions 06E20 Ring-theoretic properties [See also 16E50, 16G30] 06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G35, 03F45] /++ 06E30 Boolean functions [See also 94C10] 06E99 None of the above, but in this section 06Fxx Ordered structures 06F05 Ordered semigroups and monoids [See also 20Mxx] /+ and monoids 06F07 Quantales /++ 06F10 Noether lattices 06F15 Ordered groups [See also 20F60] 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] 06F25 Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80} 06F30 Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12] 06F35 BCK-algebras, BCI-algebras [See also 03G25] 06F99 None of the above, but in this section 08-XX General algebraic systems 08-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 08-01 Instructional exposition (textbooks, tutorial papers, etc.) 08-02 Research exposition (monographs, survey articles) 08-03 Historical (must be assigned at least one classification number from Section 01) 08-04 Explicit machine computation and programs (not the theory of computation or programming) 08-06 Proceedings, conferences, collections, etc. 08Axx Algebraic structures [See also 03C05] 08A02 Relational systems, laws of composition 08A05 Structure theory 08A30 Subalgebras, congruence relations 08A35 Automorphisms, endomorphisms 08A40 Operations, polynomials, primal algebras 08A45 Equational compactness 08A50 Word problems [See also 03D40, 06B25, 20F10, 68R15] 08A55 Partial algebras 08A60 Unary algebras 08A62 Finitary algebras 08A65 Infinitary algebras 08A68 Heterogeneous algebras 08A70 Applications of universal algebra in computer science 08A72 Fuzzy algebraic structures /++ 08A99 None of the above, but in this section 08Bxx Varieties 08B05 Equational logic, Malcev (Maltsev) conditions 08B10 Congruence modularity, congruence distributivity 08B15 Lattices of varieties 08B20 Free algebras 08B25 Products, amalgamated products, and other kinds of limits and colimits [See also 18A30] 08B26 Subdirect products and subdirect irreducibility 08B30 Injectives, projectives 08B99 None of the above, but in this section 08Cxx Other classes of algebras 08C05 Categories of algebras [See also 18C05] 08C10 Axiomatic model classes [See also 03Cxx, in particular 03C60] 08C15 Quasivarieties 08C99 None of the above, but in this section 11-XX Number theory 11-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 11-01 Instructional exposition (textbooks, tutorial papers, etc.) 11-02 Research exposition (monographs, survey articles) 11-03 Historical (must be assigned at least one classification number from Section 01) 11-04 Explicit machine computation and programs (not the theory of computation or programming) 11-06 Proceedings, conferences, collections, etc. 11Axx Elementary number theory {For analogues in number fields, see 11R04} 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11A07 Congruences; primitive roots; residue systems 11A15 Power residues, reciprocity 11A25 Arithmetic functions; related numbers; inversion formulas 11A41 Primes 11A51 Factorization; primality 11A55 Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15} 11A63 Radix representation; digital problems {For metric results, see 11K16} 11A67 Other representations 11A99 None of the above, but in this section 11Bxx Sequences and sets 11B05 Density, gaps, topology 11B13 Additive bases [See also 05B10] 11B25 Arithmetic progressions [See also 11N13] 11B34 Representation functions 11B37 Recurrences {For applications to special functions, see 33-XX} 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B50 Sequences (mod m) 11B57 Farey sequences; the sequences ${1^k, 2^k, ... }$ 11B65 Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 11B75 Other combinatorial number theory 11B83 Special sequences and polynomials 11B85 Automata sequences 11B99 None of the above, but in this section 11Cxx Polynomials and matrices 11C08 Polynomials [See also 13F20] 11C20 Matrices, determinants [See also 15A36] 11C99 None of the above, but in this section 11Dxx Diophantine equations [See also 11Gxx, 14Gxx] 11D04 Linear equations 11D09 Quadratic and bilinear equations 11D25 Cubic and quartic equations 11D41 Higher degree equations; Fermat's equation 11D45 Counting solutions of Diophantine equations /++ 11D57 Multiplicative and norm form equations 11D59 Thue-Mahler equations /++ 11D61 Exponential equations 11D68 Rational numbers as sums of fractions 11D72 Equations in many variables [See also 11P55] 11D75 Diophantine inequalities [See also 11J25] 11D79 Congruences in many variables 11D85 Representation problems [See also 11P55] 11D88 $p$-adic and power series fields 11D99 None of the above, but in this section 11Exx Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63} 11E04 Quadratic forms over general fields 11E08 Quadratic forms over local rings and fields 11E10 Forms over real fields 11E12 Quadratic forms over global rings and fields 11E16 General binary quadratic forms 11E20 General ternary and quaternary quadratic forms; forms of more than two variables 11E25 Sums of squares and representations by other particular quadratic forms 11E39 Bilinear and Hermitian forms 11E41 Class numbers of quadratic and Hermitian forms 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) 11E57 Classical groups [See also 14Lxx, 20Gxx] 11E70 $K$-theory of quadratic and Hermitian forms 11E72 Galois cohomology of linear algebraic groups [See also 20G10] 11E76 Forms of degree higher than two 11E81 Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24] 11E88 Quadratic spaces; Clifford algebras [See also 15A63, 15A66] 11E95 $p$-adic theory 11E99 None of the above, but in this section 11Fxx Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14-XX, 22Exx, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} 11F03 Modular and automorphic functions 11F06 Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] 11F11 Modular forms, one variable 11F12 Automorphic forms, one variable 11F20 Dedekind eta function, Dedekind sums 11F22 Relationship to Lie algebras and finite simple groups 11F23 Relations with algebraic geometry and topology 11F25 Hecke-Petersson operators, differential operators (one variable) 11F27 Theta series; Weil representation 11F30 Fourier coefficients of automorphic forms 11F32 Modular correspondences, etc. 11F33 Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F41 Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11F46 Siegel modular groups and their modular and automorphic forms 11F50 Jacobi forms /++ 11F52 Modular forms associated to Drinfel'd modules /++ 11F55 Other groups and their modular and automorphic forms (several variables) 11F60 Hecke-Petersson operators, differential operators (several variables) 11F66 Dirichlet series and functional equations in connection with modular forms 11F67 Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F72 Spectral theory; Selberg trace formula 11F75 Cohomology of arithmetic groups 11F80 Galois representations 11F85 $p$-adic theory, local fields [See also 14G20, 22E50] 11F99 None of the above, but in this section 11Gxx Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14-XX, 14Gxx, 14Kxx] 11G05 Elliptic curves over global fields [See also 14H52] 11G07 Elliptic curves over local fields [See also 14G20, 14H52] 11G09 Drinfeld modules; higher-dimensional motives, etc. [See also 14L05] 11G10 Abelian varieties of dimension $\gtr 1$ [See also 14Kxx] 11G15 Complex multiplication and moduli of abelian varieties [See also 14K22] 11G16 Elliptic and modular units [See also 11R27] 11G18 Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11G20 Curves over finite and local fields [See also 14H25] 11G25 Varieties over finite and local fields [See also 14G15, 14G20] 11G30 Curves of arbitrary genus or genus $\ne 1$ over global fields [See also 14H25] 11G35 Varieties over global fields [See also 14G25] 11G40 $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 11G45 Geometric class field theory [See also 11R37, 14C35, 19F05] 11G50 Heights /++ 11G55 Polylogarithms and relations with $K$-theory /++ 11G99 None of the above, but in this section 11Hxx Geometry of numbers {For applications in coding theory, see 94B75} 11H06 Lattices and convex bodies [See also 11P21, 52C05, 52C07] /~ 19F05] 11H16 Nonconvex bodies 11H31 Lattice packing and covering [See also 05B40, 52C15, 52C17] 11H46 Products of linear forms 11H50 Minima of forms 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 11H56 Automorphism groups of lattices 11H60 Mean value and transfer theorems 11H71 Relations with coding theory 11H99 None of the above, but in this section 11Jxx Diophantine approximation, transcendental number theory [See also 11K60] 11J04 Homogeneous approximation to one number 11J06 Markov and Lagrange spectra and generalizations 11J13 Simultaneous homogeneous approximation, linear forms 11J17 Approximation by numbers from a fixed field 11J20 Inhomogeneous linear forms 11J25 Diophantine inequalities [See also 11D75] 11J54 Small fractional parts of polynomials and generalizations 11J61 Approximation in non-Archimedean valuations 11J68 Approximation to algebraic numbers 11J70 Continued fractions and generalizations [See also 11A55, 11K50] 11J71 Distribution modulo one [See also 11K06] 11J72 Irrationality; linear independence over a field 11J81 Transcendence (general theory) 11J82 Measures of irrationality and of transcendence 11J83 Metric theory 11J85 Algebraic independence; Gelfond's method 11J86 Linear forms in logarithms; Baker's method 11J89 Transcendence theory of elliptic and abelian functions 11J91 Transcendence theory of other special functions 11J93 Transcendence theory of Drinfel'd and $t$-modules /++ 11J95 Results involving abelian varieties /++ 11J97 Analogues of methods in Nevanlinna theory (work of Vojta et al.) /++ 11J99 None of the above, but in this section 11Kxx Probabilistic theory: distribution modulo $1$; metric theory of algorithms 11K06 General theory of distribution modulo $1$ [See also 11J71] 11K16 Normal numbers, radix expansions, etc. [See also 11A63] 11K31 Special sequences 11K36 Well-distributed sequences and other variations 11K38 Irregularities of distribution, discrepancy [See also 11Nxx] 11K41 Continuous, $p$-adic and abstract analogues 11K45 Pseudo-random numbers; Monte Carlo methods 11K50 Metric theory of continued fractions [See also 11A55, 11J70] 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx] 11K60 Diophantine approximation [See also 11Jxx] 11K65 Arithmetic functions [See also 11Nxx] 11K70 Harmonic analysis and almost periodicity 11K99 None of the above, but in this section 11Lxx Exponential sums and character sums {For finite fields, see 11Txx} 11L03 Trigonometric and exponential sums, general 11L05 Gauss and Kloosterman sums; generalizations 11L07 Estimates on exponential sums 11L10 Jacobsthal and Brewer sums; other complete character sums 11L15 Weyl sums 11L20 Sums over primes 11L26 Sums over arbitrary intervals 11L40 Estimates on character sums 11L99 None of the above, but in this section 11Mxx Zeta and $L$-functions: analytic theory 11M06 $zeta (s)$ and $L(s, chi)$ 11M20 Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11M26 Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M35 Hurwitz and Lerch zeta functions 11M36 Selberg zeta functions and regularized determinants /++ 11M38 Zeta and $L$-fi=unctions in characteristic $p$ /++ 11M41 Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45. For algebro-geometric methods, see 14G10} [See also 11E45, 11F66, 11F70, 11F72} 11M45 Tauberian theorems [See also 40E05] /+ [...] 11M99 None of the above, but in this section 11Nxx Multiplicative number theory 11N05 Distribution of primes 11N13 Primes in progressions [See also 11B25] 11N25 Distribution of integers with specified multiplicative constraints 11N30 Turan theory [See also 30Bxx] 11N32 Primes represented by polynomials; other multiplicative structure of polynomial values 11N35 Sieves 11N36 Applications of sieve methods 11N37 Asymptotic results on arithmetic functions 11N45 Asymptotic results on counting functions for algebraic and topological structures 11N56 Rate of growth of arithmetic functions 11N60 Distribution functions associated with additive and positive multiplicative functions 11N64 Other results on the distribution of values or the characterization of arithmetic functions 11N69 Distribution of integers in special residue classes 11N75 Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx] 11N80 Generalized primes and integers 11N99 None of the above, but in this section 11Pxx Additive number theory; partitions 11P05 Waring's problem and variants 11P21 Lattice points in specified regions 11P32 Goldbach-type theorems; other additive questions involving primes 11P55 Applications of the Hardy-Littlewood method [See also 11D85] 11P70 Inverse problems of additive number theory /++ 11P81 Elementary theory of partitions [See also 05A17] 11P82 Analytic theory of partitions 11P83 Partitions; congruences and congruential restrictions 11P99 None of the above, but in this section 11Rxx Algebraic number theory: global fields {For complex multiplication, see 11G15} 11R04 Algebraic numbers; rings of algebraic integers 11R06 PV-numbers and generalizations; other special algebraic numbers 11R09 Polynomials (irreducibility, etc.) 11R11 Quadratic extensions 11R16 Cubic and quartic extensions 11R18 Cyclotomic extensions 11R20 Other abelian and metabelian extensions 11R21 Other number fields 11R23 Iwasawa theory 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants 11R32 Galois theory 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10 11R34 Galois cohomology [See also 12Gxx, 16H05, 19A31] 11R37 Class field theory 11R39 Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55] 11R42 Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R44 Distribution of prime ideals [See also 11N05] 11R45 Density theorems 11R47 Other analytic theory [See also 11Nxx] 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11R54 Other algebras and orders, and their zeta and $L$-functions [See also 11S45, 16H05, 16Kxx] 11R56 Adele rings and groups 11R58 Arithmetic theory of algebraic function fields [See also 14-XX] 11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.) /++ 11R65 Class groups and Picard groups of orders 11R70 $K$-theory of global fields [See also 19Fxx] 11R80 Totally real and totally positive fields [See also 12J15] 11R99 None of the above, but in this section 11Sxx Algebraic number theory: local and $p$-adic fields 11S05 Polynomials 11S15 Ramification and extension theory 11S20 Galois theory 11S23 Integral representations 11S25 Galois cohomology [See also 12Gxx, 16H05] 11S31 Class field theory; $p$-adic formal groups [See also 14L05] 11S37 Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50] 11S40 Zeta functions and $L$-functions [See also 11M41, 19F27] 11S45 Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16H05, 16Kxx 11S70 $K$-theory of local fields [See also 19Fxx] 11S80 Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 11S85 Other nonanalytic theory 11S90 Prehomogeneous vector spaces /++ 11S99 None of the above, but in this section 11Txx Finite fields and commutative rings (number-theoretic aspects) 11T06 Polynomials 11T22 Cyclotomy 11T23 Exponential sums 11T24 Other character sums and Gauss sums 11T30 Structure theory 11T55 Arithmetic theory of polynomial rings over finite fields 11T60 Finite upper-half planes 11T71 Algebraic coding theory; cryptography 11T99 None of the above, but in this section 11Uxx Connections with logic 11U05 Decidability [See also 03B25] 11U07 Ultraproducts [See also 03C20] 11U09 Model theory [See also 03Cxx] 11U10 Nonstandard arithmetic [See also 03H15] 11U99 None of the above, but in this section 11Yxx Computational number theory [See also 11-04] 11Y05 Factorization 11Y11 Primality 11Y16 Algorithms; complexity [See also 68Q25] 11Y35 Analytic computations 11Y40 Algebraic number theory computations 11Y50 Computer solution of Diophantine equations 11Y55 Calculation of integer sequences 11Y60 Evaluation of constants 11Y65 Continued fraction calculations 11Y70 Values of arithmetic functions; tables 11Y99 None of the above, but in this section 11Z05 Miscellaneous applications of number theory /- 11Z50 + 11Z05 12-XX Field theory and polynomials 12-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 12-01 Instructional exposition (textbooks, tutorial papers, etc.) 12-02 Research exposition (monographs, survey articles) 12-03 Historical (must be assigned at least one classification number from Section 01) 12-04 Explicit machine computation and programs (not the theory of computation or programming) 12-06 Proceedings, conferences, collections, etc. 12Dxx Real and complex fields 12D05 Polynomials: factorization 12D10 Polynomials: location of zeros (algebraic theorems) {For the analytic theory, see 26C10, 30C15} 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx] 12D99 None of the above, but in this section 12Exx General field theory 12E05 Polynomials (irreducibility, etc.) 12E10 Special polynomials 12E12 Equations 12E15 Skew fields, division rings [See also 11R52, 11R54, 11S45, 16Kxx] 12E20 Finite fields (field-theoretic aspects) 12E25 Hilbertian fields; Hilbert's irreducibility theorem 12E30 Field arithmetic /++ 12E99 None of the above, but in this section 12Fxx Field extensions 12F05 Algebraic extensions 12F10 Separable extensions, Galois theory 12F12 Inverse Galois theory 12F15 Inseparable extensions 12F20 Transcendental extensions 12F99 None of the above, but in this section 12Gxx Homological methods (field theory) 12G05 Galois cohomology [See also 16K50, 16H05] /- 13A20 /+ 16K50 12G10 Cohomological dimension 12G99 None of the above, but in this section 12Hxx Differential and difference algebra 12H05 Differential algebra [See also 13Nxx] 12H10 Difference algebra [See also 39Axx] 12H20 Abstract differential equations [See also 34Gxx] 12H25 $p$-adic differential equations [See also 11S80, 14G20, 34Gxx] 12H99 None of the above, but in this section 12Jxx Topological fields 12J05 Normed fields 12J10 Valued fields 12J12 Formally $p$-adic fields 12J15 Ordered fields 12J17 Topological semifields 12J20 General valuation theory 12J25 Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10] 12J27 Krasner-Tate algebras {See mainly 32P05} [See also 46S10, 47S10] 12J99 None of the above, but in this section 12Kxx Generalizations of fields 12K05 Near-fields [See also 16Y30] 12K10 Semifields [See also 16Y60] 12K99 None of the above, but in this section 12Lxx Connections with logic 12L05 Decidability [See also 03B25] 12L10 Ultraproducts [See also 03C20] 12L12 Model theory [See also 03C60] 12L15 Nonstandard arithmetic [See also 03H15] 12L99 None of the above, but in this section 12Y05 Computational aspects of field theory and polynomials 13-XX Commutative rings and algebras 13-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 13-01 Instructional exposition (textbooks, tutorial papers, etc.) 13-02 Research exposition (monographs, survey articles) 13-03 Historical (must be assigned at least one classification number from Section 01) 13-04 Explicit machine computation and programs (not the theory of computation or programming) 13-06 Proceedings, conferences, collections, etc. 13Axx General commutative ring theory 13A02 Graded rings [See also 16W50] 13A05 Divisibility 13A10 Radical theory 13A15 Ideals; multiplicative ideal theory 13A18 Valuations and their generalizations 13A20 Brauer groups [$ 12Gxx, 16H05] /-- 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics /+ , analytic spread 13A35 Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$ ; tight closure [See also 13Mxx, 13B22] /~ [$13Mxx] 13A50 Actions of groups on commutative rings; /++ invariant theory [See also 14L25] /- 14D25 /+ 14L25 13A99 None of the above, but in this section 13Bxx Ring extensions and related topics 13B02 Extension theory 13B05 Galois theory (commutative rings) 13B10 Morphisms /- and derivations 13B21 Integral dependence 13B22 Integral closure of rings and ideals; /+ of rings and ideals integrally closed rings, related rings (Japanese, etc.) 13B24 Going up; going down; going between 13B25 Polynomials over commutative rings 13B30 Quotients and localization 13B35 Completion [See also 13J10] 13B40 Étale and flat extensions ; /+ and flat Henselization; Artin approximation [See also 13J15, 14B12] 13B99 None of the above, but in this section 13Cxx Theory of modules and ideals 13C05 Structure, classification theorems 13C10 Projective and free modules and ideals [See also 19A13] /~ 18G05, 19A13] 13C11 Injective and flat modules and ideals /- [$ 18G05] 13C12 Torsion modules and ideals 13C13 Other special types 13C14 Cohen-Macaulay modules [See also 13H10] 13C15 Dimension theory, depth, related rings (catenary, etc.) 13C20 Class groups 13C40 Linkage, complete intersections and determinantal ideals [See also 14M12] 13C99 None of the above, but in this section 13Dxx Homological methods /~(co)homological methods {For noncommutative rings, see 16Exx ; /++ for general categories, see 18Gxx} /++ 13D02 Syzygies and resolutions /+ and resolutions 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, /++ dihedral, etc.) /++ 13D05 Homological dimension /~ (co)homological dimension 13D07 Homological functors on modules (Tor, Ext, etc.) /++ 13D10 Deformations and infinitesimal methods [See also 14B12, 14D15, 16S80, 32Gxx] 13D15 Grothendieck groups, $K$-theory [See also 14C35, 18F30, 19-XX] 13D22 Homological conjectures (intersection theorems) /++ 13D25 Complexes 13D30 Torsion theory [See also 13C12, 18E40] 13D40 Hilbert-Samuel and Hilbert-Kunz functions; /+ and Hilbert-Kunz Poincaré series 13D45 Local cohomology [See also 14B15] 13D99 None of the above, but in this section 13Exx Chain conditions, finiteness conditions 13E05 Noetherian rings and modules 13E10 Artinian rings and modules, finite-dimensional algebras 13E15 Rings and modules of finite generation or presentation; number of generators /++ 13E99 None of the above, but in this section 13Fxx Arithmetic rings and other special rings [See also 12-XX] 13F05 Dedekind, Prüfer and Krull rings and their /+ and Krull generalizations 13F07 Euclidean rings and generalizations 13F10 Principal ideal rings 13F15 Factorial rings, unique factorization domains [See also 14M05] 13F20 Polynomial rings and ideals ; rings of integer-valued polynomials /* [See also 11C08] 13F25 Formal power series rings [See also 13J05] 13F30 Valuation rings 13F40 Excellent rings 13F45 Seminormal rings 13F50 Rings with straightening laws, Hodge algebras 13F55 Face and Stanley-Reisner rings; simplicial complexes /++ 13F99 None of the above, but in this section 13G05 Integral domains 13Hxx Local rings and semilocal rings 13H05 Regular local rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13H15 Multiplicity theory and related topics [See also 14C17] 13H99 None of the above, but in this section 13Jxx Topological rings and modules [See also 16W60, 16W80] 13J05 Power series rings [See also 13F25] 13J07 Analytical algebras and rings [See also 32B05] 13J10 Complete rings, completion [See also 13B35] 13J15 Henselian rings [See also 13B40] 13J20 Global topological rings 13J25 Ordered rings [See also 06F25] 13J30 Real algebra [See also 12D12, 14Pxx] /++ 13J99 None of the above, but in this section 13K05 Witt vectors and related rings 13L05 Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] 13Mxx Finite commutative rings {For number-theoretic aspects, see 11Txx} 13M05 Structure 13M10 Polynomials (commutative rings) 13M99 None of the above, but in this section 13Nxx Differential algebra [See also 12H05, 14F10] 13N05 Modules of differentials [See also 16S32] 13N10 Rings of differential operators and their modules /+ and their modules [See also 16S32, 32C38] 13N15 Derivations /++ 13N99 None of the above, but in this section 13Pxx Computational aspects of commutative algebra [See also 68Q40] 13P05 Polynomials, factorization [See also 12Y05] 13P10 Polynomial ideals, Gröbner bases 13P99 None of the above, but in this section 14-XX Algebraic geometry 14-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 14-01 Instructional exposition (textbooks, tutorial papers, etc.) 14-02 Research exposition (monographs, survey articles) 14-03 Historical (must be assigned at least one classification number from Section 01) 14-04 Explicit machine computation and programs (not the theory of computation or programming) 14-06 Proceedings, conferences, collections, etc. 14Axx Foundations 14A05 Relevant commutative algebra [See also 13-XX] 14A10 Varieties and morphisms /+ and morphisms 14A15 Schemes and morphisms /+ and morphisms 14A20 Generalizations (algebraic spaces, stacks) /- motifs /+ stacks 14A22 Noncommutative algebraic geometry /- algebraic supervarieties [$ 14M30, 32C11, 58A50] 14A25 Elementary questions 14A99 None of the above, but in this section 14Bxx Local theory [See also 32Sxx] /- 32Bxx /+ 32Sxx 14B05 Singularities [See also 14E15, 14H20, 32Sxx, 58C27] 14B07 Deformations of singularities [See also 14D15, 32Sxx] 14B10 Infinitesimal methods [See also 13D10] 14B12 Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10] 14B15 Local cohomology [See also 13D45, 32C36] 14B20 Formal neighborhoods 14B25 Local structure of morphisms : étale, flat, etc. 14B99 None of the above, but in this section 14Cxx Cycles and subschemes 14C05 Parametrization (Chow and Hilbert schemes) 14C10 equivalence relations /-- 14C15 Chow groups and rings 14C17 Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14C20 Divisors, linear systems, invertible sheaves 14C21 Pencils, nets, webs [See also 53A60] 14C22 Picard groups 14C25 Algebraic cycles 14C30 Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], /~ [$ 32J25], Hodge conjecture 14C34 Torelli problem [See also 32G20] /++ 14C35 Applications of methods of algebraic $K$-theory [See also 19Exx] /~ [$ 14F05, 18F25, 19E15] 14C40 Riemann-Roch theorems [See also 19E20, 19L10] /- 14F12, 14C99 None of the above, but in this section 14Dxx Families, fibrations 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) /~ Picard-Fuchs theory, etc.) 14D06 Fibrations, degenerations /++ 14D07 Variation of Hodge structures 14D10 Arithmetic ground fields (finite, local, global) 14D15 Formal methods; deformations [See also 13D10, 14B07, 16S80 32Gxx] 14D20 Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14D21 Applications of vector bundles and moduli /++ spaces in mathematical physics (twistor /++ theory, instantons, quantum field theory) /++ 14D22 Fine and coarse moduli spaces 14D25 geometric invariants [$ 14L30] /-- 14D99 None of the above, but in this section 14Exx Birational geometry /~mappings and correspondences 14E05 Rational and birational maps /~ rational maps, birational correspondences 14E07 Birational automorphisms, Cremona group and generalizations [See also 32G20] /+ [...] 14E08 Rationality questions [See also 14J50, 14L27] 14E09 automorphisms [$ 14J50, 14L27] /-- 14E15 Global theory and resolution of singularities /~ of singularities, resolution [See also 14B05, 32S20, 32S45] 14E20 Coverings /- , fundamental group 14E22 Ramification problems [See also 11S15] 14E25 Embeddings 14E30 Minimal model program (Mori theory, extremal rays) /~ minimal models 14E35 results in dimensions <=3 /-- 14E40 local structure of maps: etale, flat, etc. [$ 13-XX, 14F20] /-- 14E99 None of the above, but in this section 14Fxx (Co)homology theory [See also 13Dxx] 14F05 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] /+ 32Lxx, 14F07 Weierstrass points in one and several variables; gap sheaves /-- 14F10 Differentials and other special sheaves [See also 32C38] /++ 14F17 Vanishing theorems [See also 32L20] 14F20 Étale and other Grothendieck /~ Grothendieck cohomology and topologies and cohomologies /~ topology [$ 18F10] 14F22 Brauer groups of schemes [See also 16K50] /++ 14F25 Classical real and complex cohomology 14F30 $p$-adic cohomology, crystalline cohomology 14F32 intersection (co)homology [$ 32S60] /-- 14F35 Homotopy theory; fundamental groups [See also 14E20, 14H30] /++ 14F40 de Rham cohomology [See also 14C30, 32C35, 32L10] 14F42 Motivic cohomology /++ 14F43 Other algebro-geometric (co)homologies (e.g., /++ intersection, equivariant, Lawson (co)homologies) /++ 14F45 Topological properties 14F99 None of the above, but in this section 14Gxx Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] 14G05 Rational points /- rationality questions 14G10 Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields 14G20 Local ground fields /~ p-adic ground fields 14G22 Rigid analytic geometry /++ 14G25 Global ground fields 14G27 Other nonalgebraically closed ground fields /+ other 14G32 Universal profinite groups (relationship to moduli /++ spaces, projective and moduli towers) /++ 14G35 Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 14G40 Arithmetic varieties and schemes; Arakelov theory; heights /++ 14G50 Applications to coding theory and cryptography /++ 14G99 None of the above, but in this section 14Hxx Curves 14H05 Algebraic functions; function fields [See also 11R58] 14H10 Families, moduli (algebraic) 14H15 Families, moduli (analytic) [See also 30F10, 32Gxx] 14H20 Singularities, local rings [See also 13Hxx] 14H25 Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx] 14H30 Coverings, fundamental group [See also 14E20, 14F35] 14H35 correspondences [$ 14Exx] /-- 14H37 Automorphisms 14H40 Jacobians, Prym varieties [See also 32G20] /+ , Prym varieties 14H42 Theta functions; Schottky problem /- 14H41 /+ 14H42 [See also 14K25, 32G20] 14H45 Special curves and curves of low genus 14H50 Plane and space curves /+ plane and 14H51 Special divisors (gonality, Brill-Noether theory) /++ 14H52 Elliptic curves [See also 11G05, 11G07, 14Kxx] 14H55 Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx] 14H60 Vector bundles on curves and their moduli /+ and their moduli [See also 14F05] 14H70 Relationship to integrable systems /++ 14H81 Relationship to physics /++ 14H99 None of the above, but in this section 14Jxx Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx} 14J10 Families, moduli, classification: algebraic theory 14J15 Moduli, classification: analytic theory; relations with modular forms /++ [See also 32G13, 32J15] 14J17 Singularities /- of surfaces 14J20 Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] 14J25 Special surfaces {For Hilbert modular surfaces, see 14G35} 14J26 Rational and ruled surfaces 14J27 Elliptic surfaces 14J28 $K3$ surfaces and Enriques surfaces 14J29 Surfaces of general type 14J30 $3$-folds /- special 14J32 Calabi-Yau manifolds, mirror symmetry /++ 14J35 $4$-folds /~ special 4-folds [$ 14E05] 14J40 $n$-folds (n>4) 14J45 Fano varieties 14J50 Automorphisms of surfaces and higher-dimensional varieties /- [$ 14E09] 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14F05, 32Lxx] /- , and their moduli 14J70 Hypersurfaces 14J80 Topology of surfaces (Donaldson polynomials, /++ Seiberg-Witten invariants) /++ 14J81 Relationship to physics /++ 14J99 None of the above, but in this section 14Kxx Abelian varieties and schemes 14K02 Isogeny 14K05 Algebraic theory 14K10 Algebraic moduli, classification 14K12 Subvarieties 14K15 Arithmetic ground fields [See also 11Dxx, 11Fxx, 11Gxx, 14Gxx] 14K20 Analytic theory; abelian integrals and differentials 14K22 Complex multiplication [See also 11G15] 14K25 Theta functions 14K30 Picard schemes, higher Jacobians [See also 14H40, 32G20] 14K99 None of the above, but in this section 14Lxx Algebraic groups /~ group schemes {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45} 14L05 Formal groups, $p$-divisible groups [See also 55N22] /++ 14L10 Group varieties 14L15 Group schemes 14L17 Affine algebraic groups, hyperalgebra constructions [See also 17B45, 18D35] 14L25 Geometric invariant theory /++ 14L27 automorphism groups [$ 14E09] /-- 14L30 Group actions on varieties or schemes (quotients) [See also 14L25] /~ 14D25] 14L35 Classical groups (geometric aspects) [See also 20Gxx, 51N30] 14L40 Other algebraic groups (geometric aspects) 14L99 None of the above, but in this section 14Mxx Special varieties 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13C14, 13F45, 13H10] 14M06 Linkage [See also 13C40] 14M07 Low codimension problems [See also 14Cxx] 14M10 Complete intersections [See also 13C40] 14M12 Determinantal varieties [See also 13C40] 14M15 Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] 14M17 Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] 14M20 Rational and unirational varieties /+ and unirational 14M25 Toric varieties, Newton polyhedra 14M30 Supervarieties [See also 32C11, 58A50] /- 14A22, 14M99 None of the above, but in this section 14Nxx Projective and enumerative geometry /~ classical methods and problems [See also 51-XX] 14N05 Projective techniques [See also 51N35] 14N10 Enumerative problems (combinatorial problems) 14N15 Classical problems, Schubert calculus /++ 14N20 Configurations of linear subspaces /++ 14N25 Varieties of low degree /++ 14N30 Adjunction problems /++ 14N35 Gromov-Witten invariants, quantum cohomology /++ 14N99 None of the above, but in this section 14Pxx Real algebraic and real analytic geometry 14P05 Real algebraic sets [See also 12Dxx] 14P10 Semialgebraic sets and related spaces 14P15 Real analytic and semianalytic sets [See also 32B20, 32C05] 14P20 Nash functions and manifolds [See also 32C07, 58A07] 14P25 Topology of real algebraic varieties 14P99 None of the above, but in this section 14Qxx Computational aspects in algebraic geometry [See also 12-04, 13Pxx, 68Q40] /+ 13Pxx, 14Q05 Curves 14Q10 Surfaces, hypersurfaces 14Q15 Higher-dimensional varieties 14Q20 Effectivity 14Q99 None of the above, but in this section 14Rxx Affine geometry /++++ 14R05 Classification of affine varieties 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14R15 Jacobian problem 14R20 Group actions on affine varieties 14R25 Affine fibrations 14R99 None of the above, but in this section 15-XX Linear and multilinear algebra; matrix theory /- (finite and infinite) 15-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 15-01 Instructional exposition (textbooks, tutorial papers, etc.) 15-02 Research exposition (monographs, survey articles) 15-03 Historical (must be assigned at least one classification number from Section 01) 15-04 Explicit machine computation and programs (not the theory of computation or programming) 15-06 Proceedings, conferences, collections, etc. 15A03 Vector spaces, linear dependence, rank 15A04 Linear transformations, semilinear transformations 15A06 Linear equations 15A09 Matrix inversion, generalized inverses 15A12 Conditioning of matrices [See also 65F35] 15A15 Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 15A18 Eigenvalues, singular values, and eigenvectors 15A21 Canonical forms, reductions, classification 15A22 Matrix pencils [See also 47A56] 15A23 Factorization of matrices 15A24 Matrix equations and identities 15A27 Commutativity 15A29 Inverse problems /++ 15A30 Algebraic systems of matrices [See also 16S50, 20Gxx, 20Hxx] 15A33 Matrices over special rings (quaternions, finite fields, etc.) 15A36 Matrices of integers [See also 11C20] 15A39 Linear inequalities 15A42 Inequalities involving eigenvalues and eigenvectors 15A45 Miscellaneous inequalities involving matrices 15A48 Positive matrices and their generalizations; cones of matrices 15A51 Stochastic matrices 15A52 Random matrices 15A54 Matrices over function rings in one or more variables 15A57 Other types of matrices (Hermitian, skew-Hermitian, etc.) 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 15A63 Quadratic and bilinear forms, inner products {See mainly 11Exx} 15A66 Clifford algebras, spinors 15A69 Multilinear algebra, tensor products 15A72 Vector and tensor algebra, theory of invariants [See also 13A50, 14L25] /- 14D25 /+ 14L25 15A75 Exterior algebra, Grassmann algebras 15A78 Other algebras built from modules 15A90 Applications of matrix theory to physics 15A99 Miscellaneous topics 16-XX Associative rings and algebras {For the commutative case, see 13-XX} 16-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 16-01 Instructional exposition (textbooks, tutorial papers, etc.) 16-02 Research exposition (monographs, survey articles) 16-03 Historical (must be assigned at least one classification number from Section 01) 16-04 Explicit machine computation and programs (not the theory of computation or programming) 16-06 Proceedings, conferences, collections, etc. 16-08 computational methods /-- 16Bxx General and miscellaneous 16B50 Category-theoretic methods and results (except as in 16D90, 16E10) [See also 18-XX] 16B70 Applications of logic [See also 03Cxx] 16B99 None of the above, but in this section 16Dxx Modules, bimodules and ideals 16D10 General module theory 16D15 1-sided ideals /-- 16D20 Bimodules 16D25 Ideals /- 2-sided 16D30 Infinite-dimensional /~ maximal and prime 2-sided ideals [$ 16N60, 16D60], simple rings (except as in 16Kxx) 16D40 Free, projective, and flat modules and ideals [See also 19A13] /- 18G05, 16D50 Injective modules, self-injective rings [See also 16L60] /- , 18G05 16D60 Simple and semisimple modules, primitive rings and ideals 16D70 Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation 16D80 Other classes of modules and ideals [See also 16G60] 16D90 Module categories [See also 16Exx, 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 16D99 None of the above, but in this section 16Exx Homological methods /+ and results [$ 18Gxx] {For commutative rings, see 13Dxx; /++ for general categories, see 18Gxx} /++ 16E05 Syzygies, resolutions, complexes /++ 16E10 Homological dimension 16E20 Grothendieck groups, $K$-theory, etc. [See also 18F30, 19-XX] 16E30 Homological functors on modules (Tor, Ext, etc.) /++ (...) 16E40 (Co)homology of rings and algebras /~ Hochschild and other homology and (e.g. Hochschild, cyclic, dihedral, etc.) /~ cohomology theories for rings 16E45 Differential graded algebras and applications /++ 16E50 von Neumann regular rings and generalizations 16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. 16E65 Homological conditions on rings (generalizations of /++ regular, Gorenstein, Cohen-Macaulay rings, etc.) /++ 16E70 other rings of low global or flat dimension /-- 16E99 None of the above, but in this section 16Gxx Representation theory of rings and algebras 16G10 Representations of Artinian rings 16G20 Representations of quivers and partially ordered sets 16G30 Representations of orders, lattices, algebras over commutative rings [See also 16H05] 16G50 Cohen-Macaulay modules 16G60 Representation type (finite, tame, wild, etc.) 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 16G99 None of the above, but in this section 16H05 Orders and arithmetic, separable algebras, Azumaya algebras [See also 11R52, 11R54, 11S45] /- 13A20, 16Kxx Division rings and semisimple Artin rings [See also 12E15, 15A30] 16K20 Finite-dimensional { /- for Brower group theory, see 12Gxx, 13A20; For crossed products, see 16S35} 16K40 Infinite-dimensional and general 16K50 Brauer groups [See also 14F22] /++ 16K99 None of the above, but in this section 16Lxx Local rings and generalizations 16L30 Noncommutative local and semilocal rings, perfect rings 16L60 Quasi-Frobenius rings [See also 16D50] 16L99 None of the above, but in this section 16Nxx Radicals and radical properties of rings 16N20 Jacobson radical, quasimultiplication 16N40 Nil and nilpotent radicals, sets, ideals, rings 16N60 Prime and semiprime rings [See also 16D60, 16U10] - 16D30, 16N80 General radicals and rings {For radicals in module categories, see 16S90} 16N99 None of the above, but in this section 16Pxx Chain conditions, growth conditions, and other forms of finiteness 16P10 Finite rings and finite-dimensional algebras {For semisimple, see 16K20; for commutative, see 11Txx, 13Mxx} 16P20 Artinian rings and modules 16P40 Noetherian rings and modules 16P50 Localization and Noetherian rings [See also 16U20] 16P60 Chain conditions on annihilators and summands: Goldie type conditions [See also 16U20], Krull dimension 16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence 16P90 Growth rate, Gelfand-Kirillov dimension 16P99 None of the above, but in this section 16Rxx Rings with polynomial identity 16R10 $T$-ideals, identities, varieties of rings and algebras 16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16R30 Trace rings and invariant theory 16R40 Identities other than those of matrices over commutative rings 16R50 Other kinds of identities (generalized polynomial, rational, involution) 16R99 None of the above, but in this section 16Sxx Rings and algebras arising under various constructions 16S10 Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 16S20 Centralizing and normalizing extensions 16S30 Universal enveloping algebras of Lie algebras {See mainly 17B35} 16S32 Rings of differential operators [See also 13N10, 32C38] 16S34 Group rings [See also 20C05, 20C07], Laurent polynomial rings 16S35 Twisted and skew group rings, crossed products 16S36 Ordinary and skew polynomial rings and semigroup rings [See also 20M25] 16S37 Quadratic and Koszul algebras /++ 16S38 Rings arising from non-commutative algebraic geometry /++ 16S40 Smash products of general Hopf actions [See also 16W30] 16S50 Endomorphism rings; matrix rings [See also 15-XX] 16S60 Rings of functions, subdirect products, sheaves of rings 16S70 Extensions of rings by ideals 16S80 Deformations of rings [See also 14D15] 16S90 Maximal ring of quotients, torsion theories, radicals on module categories [See also 13D30, 18E40] {For radicals of rings, see 16Nxx} 16S99 None of the above, but in this section 16Uxx Conditions on elements /-- (including elements of matrix rings, etc.) 16U10 Integral domains 16U20 Řre rings, multiplicative sets, Řre localization 16U30 Divisibility, noncommutative UFDs 16U50 algebraicity and local finiteness [$ 16N40] /-- 16U60 Units, groups of units, general linear groups 16U70 Center, normalizer (invariant elements) 16U80 Generalizations of commutativity 16U99 None of the above, but in this section 16Wxx Rings and algebras with additional structure 16W10 Rings with involution: Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] 16W20 Automorphisms and endomorphisms /- actions of groups and semigroups /- and their fixed rings 16W22 Actions of groups and semigroups; invariant theory /++ 16W25 Derivations, actions of Lie algebras 16W30 Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30, 16S40]; /- 16S32, 16S34, 16S35, 16S36, rings, modules, etc. on which these act 16W35 Ring-theoretic aspects of quantum groups [See also /-- 17B37, 20G42] /-- 16W50 Graded rings and modules 16W55 ``Super'' (or ``skew'') structure [See also 17A70, /- 17B70, 17C70] {For exterior algebras, see 15A75; for Clifford algebras, see 11E88, 15A66} 16W60 Valuations, completions, /~ filtrations and valuations, /- eralizations of Euclidean algorithm, completions, formal power series and related constructions [See also 13Jxx] 16W70 Filtered rings; filtrational and graded techniques /-- 16W80 Topological and ordered rings and modules [See also 13Jxx] 16W99 None of the above, but in this section 16Yxx Generalizations {For nonassociative rings, see 17-XX} 16Y30 Near-rings [See also 12K05] 16Y60 Semirings [See also 12K10] 16Y99 None of the above, but in this section 16Z05 Computational aspects of associative rings /++ 17-XX Nonassociative rings and algebras 17-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 17-01 Instructional exposition (textbooks, tutorial papers, etc.) 17-02 Research exposition (monographs, survey articles) 17-03 Historical (must be assigned at least one classification number from Section 01) 17-04 Explicit machine computation and programs (not the theory of computation or programming) 17-06 Proceedings, conferences, collections, etc. 17-08 Computational methods 17Axx General nonassociative rings 17A01 General theory 17A05 Power-associative 17A15 Noncommutative Jordan algebras 17A20 Flexible algebras A25 nodal algebras /-- 17A30 Algebras satisfying other identities 17A32 Leibniz algebras /-- 17A35 Division algebras 17A36 Automorphisms, derivations, other operators 17A40 Ternary compositions 17A42 Other $n$-ary compositions 17A45 Quadratic algebras (but not quadratic Jordan algebras) 17A50 Free algebras 17A60 Structure theory 17A65 Radical theory 17A70 Superalgebras /~ super algebras 17A75 Composition algebras 17A80 Valued algebras 17A99 None of the above, but in this section 17Bxx Lie algebras and Lie superalgebras /+ and Lie superalgebras {For Lie groups, see 22Exx} 17B01 Identities, free Lie (super)algebras /+ {super} 17B05 Structure theory 17B10 Representations, algebraic theory (weights) 17B15 Representations, analytic theory 17B20 Simple, semisimple, reductive (super)algebras (roots) /+ {super} 17B25 Exceptional (super)algebras /+ {super) 17B30 Solvable, nilpotent (super)algebras /+ {super} 17B35 Universal enveloping algebras [See also 16S30] 17B37 Quantum groups (quantized enveloping algebras) and /+ (...) related deformations [See also 16W35, 20G42, /- 16W30, /+ 16W35, 20G42, 81R50, 82B23] 17B40 Automorphisms, derivations, other operators 17B45 Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx] 17B50 Modular Lie (super)algebras /+ {super} 17B55 Homological methods in Lie (super)algebras /+ {super} 17B56 Cohomology of Lie (super)algebras /+ {super} 17B60 Lie (super)algebras associated with other structures /+ {super} (associative, Jordan, etc.) [See also 15A30, 16W10, 17C40, 17C50] 17B62 Lie bialgebras /++ 17B63 Poisson algebras /++ 17B65 Infinite-dimensional Lie (super)algebras [See also /+ {super} 22E65] 17B66 Lie algebras of vector fields and related algebras 17B67 Kac-Moody algebras (structure and representation theory) 17B68 Virasoro and related algebras 17B69 Vertex operators 17B70 Graded Lie (super)algebras /+ {super} 17B75 Color Lie (super)algebras /++ 17B80 Applications to integrable systems /++ 17B81 Applications to physics 17B99 None of the above, but in this section 17Cxx Jordan algebras (algebras, triples and pairs) 17C05 Identities and free Jordan structures 17C10 Structure theory 17C17 Radicals 17C20 Simple, semisimple algebras 17C27 Idempotents, Peirce decompositions 17C30 Associated groups, automorphisms 17C36 Associated manifolds 17C37 Associated geometries 17C40 Exceptional Jordan structures 17C50 Jordan structures associated with other structures [See also 16W10] 17C55 Finite-dimensional structures 17C60 Division algebras 17C65 Jordan structures on Banach spaces and algebras [See also 46H70, 46L70] 17C70 Super structures 17C90 Applications to physics 17C99 None of the above, but in this section 17Dxx Other nonassociative rings and algebras 17D05 Alternative rings 17D10 Malcev (Maltsev) rings and algebras 17D15 Right alternative rings 17D20 $(gamma, delta)$-rings, including $(1,-1)$-rings 17D25 Lie-admissible algebras 17D92 Genetic algebras 17D99 None of the above, but in this section 18-XX Category theory; abstract homological algebra /+ abstract {See 13Dxx for commutative rings, 16Exx for associative /++ rings, 20Jxx for groups, 57Txx for topological groups /++ and related structures; see also 55Nxx and 55Uxx for /++ algebraic topology} /++ 18-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 18-01 Instructional exposition (textbooks, tutorial papers, etc.) 18-02 Research exposition (monographs, survey articles) 18-03 Historical (must be assigned at least one classification number from Section 01) 18-04 Explicit machine computation and programs (not the theory of computation or programming) 18-06 Proceedings, conferences, collections, etc. 18Axx General theory of categories and functors 18A05 Definitions, generalizations 18A10 Graphs, diagram schemes, precategories /- neocategories [See also 20L05] /~ [$20Lxx] 18A15 Foundations, relations to logic and deductive systems [See especially 03-XX] / [$ 03-XX] 18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms /- factorization (bicategories) 18A22 Special properties of functors (faithful, full, etc.) 18A23 Natural morphisms, dinatural morphisms 18A25 Functor categories, comma categories 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 18A32 Factorization of morphisms, /- (via images, coimages, dominions, codominions) substructures, quotient structures, congruences, amalgams 18A35 Categories admitting limits (complete categories), functors preserving limits, /~ commuting with limits, completions /~ continuous functors, completions 18A40 Adjoint functors (representable functors, universal constructions, reflective subcategories, reflections, etc.), constructions of adjoints (Kan extensions, etc.) 18A99 None of the above, but in this section 18Bxx Special categories 18B05 Category of sets, characterizations [See also 03-XX] 18B10 Category of relations, additive relations 18B15 Embedding theorems, universal categories /- imbedding /+ embedding [See also 18E20] 18B20 Categories of machines, automata, operative categories [See also 03D05, 68Qxx] 18B25 Topoi [See also 03G30] 18B30 Categories of topological spaces and continuous mappings [See also 54-XX] 18B35 Preorders, orders and lattices (viewed as categories) [See also 06-XX] 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx] 18B99 None of the above, but in this section 18Cxx Categories and theories /~ categories and algebraic theories 18C05 Equational categories [See also 03C05, 08C05] 18C10 Theories (e.g. algebraic theories), structure, and semantics [See also 03G30] 18C15 Triples (=standard construction, monad or triad), algebras for a triple, homology and derived functors for triples [See also 18Gxx] 18C20 Algebras and Kleisli categories associated with monads 18C30 Sketches and generalizations /++ 18C35 Accessible and locally presentable categories /++ 18C50 Categorical semantics of formal languages /++ 18C99 None of the above, but in this section 18Dxx Categories with structure 18D05 Double categories, $2$-categories, bicategories and generalizations /~ hypercategories 18D10 Monoidal categories (=multiplicative categories), symmetric monoidal categories, braided categories /++ [See also 19D23] 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) 18D20 Enriched categories (over closed or monoidal categories) 18D25 Strong functors, strong adjunctions 18D30 Fibered categories 18D35 Structured objects in a category (group objects, etc.) 18D50 Operads [See also 55P48] /++ 18D99 None of the above, but in this section 18Exx Abelian categories 18E05 Preadditive, additive categories 18E10 Exact categories, abelian categories 18E15 Grothendieck categories 18E20 Embedding theorems [See also 18B15] /- imbedding /+ embedding 18E25 Derived functors and satellites 18E30 Derived categories, triangulated categories 18E35 Localization of categories 18E40 Torsion theories, radicals [See also 13D30, 16S90] 18E99 None of the above, but in this section 18Fxx Categories and geometry 18F05 Local categories and functors 18F10 Grothendieck topologies [See also 14F20] 18F15 Abstract manifolds and fiber bundles [See also 55Rxx, 57Pxx] 18F20 Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30] 18F25 Algebraic $K$-theory and $L$-theory [See also 11Exx, 11R70, 11S70, 12-XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67] 18F30 Grothendieck groups [See also 13D15, 16E20, 19Axx] 18F99 None of the above, but in this section 18Gxx Abstract homological algebra [See also 13Dxx, 16Exx, /+ abstract 20Jxx, 55Nxx, 55Uxx, 57Txx] /~ 55Uxx] 18G05 Projectives and injectives [See also 13C10, 13C11, 16D40, 16D50] 18G10 Resolutions; derived functors [See also 13D02, 16E05, 18E25] /~ [$ 18E25] 18G15 Ext and Tor, generalizations, Künneth formula [See also 55U25] 18G20 Homological dimension [See also 13D05, 16E10] 18G25 Relative homological algebra, projective classes 18G30 Simplicial sets, simplicial objects (in a category) [See also 55U10] 18G35 Chain complexes [See also 18E30, 55U15] 18G40 Spectral sequences, hypercohomology [See also 55Txx] 18G50 Nonabelian homological algebra 18G55 Nonabelian homotopical algebra 18G60 Other (co)homology theories /~ ... (cyclic, dihedral, etc.) [See also 19D55, 46L80, 58B30, 58J20, 58J22] /+ 58J20, 18G99 None of the above, but in this section 19-XX K-theory [See also 16E20, 18F25] 19-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 19-01 Instructional exposition (textbooks, tutorial papers, etc.) 19-02 Research exposition (monographs, survey articles) 19-03 Historical (must be assigned at least one classification number from Section 01) 19-04 Explicit machine computation and programs (not the theory of computation or programming) 19-06 Proceedings, conferences, collections, etc. 19Axx Grothendieck groups and $K_0$ [See also 13D15, 18F30] 19A13 Stability for projective modules [See also 13C10] 19A15 Efficient generation 19A22 Frobenius induction, Burnside and representation rings 19A31 $K_0$ of group rings and orders 19A49 $K_0$ of other rings 19A99 None of the above, but in this section 19Bxx Whitehead groups and $K_1$ 19B10 Stable range conditions 19B14 Stability for linear groups 19B28 $K_1$ of group rings and orders [See also 57Q10] 19B37 Congruence subgroup problems [See also 20H05] 19B99 None of the above, but in this section 19Cxx Steinberg groups and $K_2$ 19C09 Central extensions and Schur multipliers 19C20 Symbols, presentations and stability of $K_2$ /+ and stability 19C30 $K_2$ and the Brauer group 19C40 Excision for $K_2$ 19C99 None of the above, but in this section 19Dxx Higher algebraic $K$-theory 19D06 $Q$- and plus-constructions 19D10 Algebraic $K$-theory of spaces 19D23 Symmetric monoidal categories [See also 18D10] 19D25 Karoubi-Villamayor-Gersten $K$-theory 19D35 Negative $K$-theory, NK and Nil 19D45 Higher symbols, Milnor $K$-theory 19D50 Computations of higher $K$-theory of rings [See also 13D15, 16E20] 19D55 $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 19D99 None of the above, but in this section ------------------- 19Exx $K$-theory in geometry 19E08 $K$-theory of schemes [See also 14C35] 19E15 Algebraic cycles and motivic cohomology /+ and mothivic cohomology [See also 14C25, 14C35] 19E20 Relations with cohomology theories [See also 14Fxx] 19E99 None of the above, but in this section 19Fxx $K$-theory in number theory [See also 11R70, 11S70] 19F05 Generalized class field theory [See also 11G45] 19F15 Symbols and arithmetic [See also 11R37] 19F27 Etale cohomology, higher regulators, zeta and $L$-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10] 19F99 None of the above, but in this section 19Gxx $K$-theory of forms [See also 11Exx] 19G05 Stability for quadratic modules 19G12 Witt groups of rings [See also 11E81] 19G24 $L$-theory of group rings [See also 11E81] 19G38 Hermitian $K$-theory, relations with $K$-theory of rings 19G99 None of the above, but in this section 19Jxx Obstructions from topology 19J05 Finiteness and other obstructions in $K_0$ 19J10 Whitehead (and related) torsion 19J25 Surgery obstructions [See also 57R67] 19J35 Obstructions to group actions 19J99 None of the above, but in this section 19Kxx $K$-theory and operator algebras {See mainly 46L80, and also 46M20} 19K14 $K_0$ as an ordered group, traces 19K33 EXT and $K$-homology [See also 55N22] 19K35 Kasparov theory ($KK$-theory) [See also 58J22] /~ [$ 58G12] 19K56 Index theory [See also 58J20, 58J22] /~ [$ 58G12] 19K99 None of the above, but in this section 19Lxx Topological $K$-theory [See also 55N15, 55R50, 55S25] 19L10 Riemann-Roch theorems, Chern characters 19L20 $J$-homomorphism, Adams operations [See also 55Q50] 19L41 Connective $K$-theory, cobordism [See also 55N22] 19L47 Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] 19L64 Computations, geometric applications 19L99 None of the above, but in this section 19M05 Miscellaneous applications of $K$-theory 20-XX Group theory and generalizations 20-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 20-01 Instructional exposition (textbooks, tutorial papers, etc.) 20-02 Research exposition (monographs, survey articles) 20-03 Historical (must be assigned at least one classification number from Section 01) 20-04 Explicit machine computation and programs (not the theory of computation or programming) 20-06 Proceedings, conferences, collections, etc. 20Axx Foundations 20A05 Axiomatics and elementary properties 20A10 Metamathematical considerations {For word problems, see 20F10} 20A15 Applications of logic to group theory 20A99 None of the above, but in this section 20Bxx Permutation groups 20B05 General theory for finite groups 20B07 General theory for infinite groups 20B10 Characterization theorems 20B15 Primitive groups 20B20 Multiply transitive finite groups 20B22 Multiply transitive infinite groups 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX] 20B27 Infinite automorphism groups [See also 12F10] 20B30 Symmetric groups 20B35 Subgroups of symmetric groups 20B40 Computational methods 20B99 None of the above, but in this section 20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)] 20C05 Group rings of finite groups and their modules [See also 16S34] 20C07 Group rings of infinite groups and their modules [See also 16S34] 20C08 Hecke algebras and their representations /++ 20C10 Integral representations of finite groups 20C11 $p$-adic representations of finite groups 20C12 Integral representations of infinite groups 20C15 Ordinary representations and characters 20C20 Modular representations and characters 20C25 Projective representations and multipliers 20C30 Representations of finite symmetric groups 20C32 Representations of infinite symmetric groups 20C33 Representations of finite groups of Lie type 20C34 Representations of sporadic groups 20C35 Applications of group representations to physics 20C40 Computational methods 20C99 None of the above, but in this section 20Dxx Abstract finite groups 20D05 Classification of simple and nonsolvable groups 20D06 Simple groups: alternating groups and groups of Lie type [See also 20Gxx, 22Exx] 20D08 Simple groups: sporadic groups 20D10 Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks [See also 20F17] 20D15 Nilpotent groups, $p$-groups 20D20 Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20D25 Special subgroups (Frattini, Fitting, etc.) 20D30 Series and lattices of subgroups 20D35 Subnormal subgroups 20D40 Products of subgroups 20D45 Automorphisms 20D50 covering of subgroups /-- 20D60 Arithmetic and combinatorial problems 20D99 None of the above, but in this section 20Exx Structure and classification of infinite or finite groups 20E05 Free nonabelian groups 20E06 Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07 Subgroup theorems; subgroup growth 20E08 Groups acting on trees 20E10 Quasivarieties and varieties of groups 20E15 Chains and lattices of subgroups, subnormal subgroups [See also 20F22] 20E18 Limits, profinite groups 20E22 Extensions, wreath products, and other compositions [See also 20J05] 20E25 Local properties 20E26 Residual properties and generalizations 20E28 Maximal subgroups 20E32 Simple groups [See also 20D05] 20E34 General structure theorems 20E36 General theorems concerning automorphisms of groups 20E42 Groups with a $BN$-pair; buildings [See also 51E24] 20E45 Conjugacy classes /++ 20E99 None of the above, but in this section 20Fxx Special aspects of infinite or finite groups 20F05 Generators, relations, and presentations 20F06 Cancellation theory; application of van Kampen diagrams [See also 57M05] 20F10 Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 68Qxx] 20F12 Commutator calculus 20F14 Derived series, central series, and generalizations 20F16 Solvable groups, supersolvable groups 20F17 Formations of groups, Fitting classes [See also 20D10] 20F18 Nilpotent groups 20F19 Generalizations of solvable and nilpotent groups 20F22 Other classes of groups defined by subgroup chains 20F24 FC-groups and their generalizations 20F28 Automorphism groups of groups [See also 20E36] 20F29 Representations of groups as automorphism groups of algebraic systems 20F32 geometric group theory [$ 05C25, 20Exx, 20Gxx] /-- 20F34 Fundamental groups and their automorphisms [See also 57M05, 57Sxx, 22E40] 20F36 Braid groups; Artin groups 20F38 Other groups related to topology or analysis 20F40 Associated Lie structures 20F45 Engel conditions 20F50 Periodic groups; locally finite groups 20F55 Reflection and Coxeter groups [See also 22E40] /+ reflection and 20F60 Ordered groups [See mainly 06F15] 20F65 Geometric group theory [See also 05C25, 20E08, 57Mxx] /++ 20F67 Hyperbolic groups and nonpositively curved groups /++ 20F69 Asymptotic properties of groups /++ 20F99 None of the above, but in this section 20Gxx Linear algebraic groups (classical groups) {For arithmetic theory, see 11E57, 11H06; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55} 20G05 Representation theory 20G10 Cohomology theory 20G15 Linear algebraic groups over arbitrary fields 20G20 Linear algebraic groups over the reals, the complexes, the quaternions 20G25 Linear algebraic groups over local fields and their integers 20G30 Linear algebraic groups over global fields and their integers 20G35 Linear algebraic groups over adčles and other rings and schemes 20G40 Linear algebraic groups over finite fields 20G42 Quantum groups (quantized function algebras) and their /++ representations [ See also 17B37, 16W35] /++ 20G45 Applications to physics /- ; explicit representations 20G99 None of the above, but in this section 20Hxx Other groups of matrices [See also 15A30] 20H05 Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20] 20H10 Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx] 20H15 Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] /+ expecially and 20H20 Other matrix groups over fields 20H25 Other matrix groups over rings 20H30 Other matrix groups over finite fields 20H99 None of the above, but in this section 20Jxx Connections with homological algebra and category theory 20J05 Homological methods in group theory 20J06 Cohomology of groups 20J10 groups arising as cohomology groups /-- 20J15 Category of groups 20J99 None of the above, but in this section 20Kxx Abelian groups 20K01 Finite abelian groups 20K10 Torsion groups, primary groups and generalized primary groups 20K15 Torsion free groups, finite rank 20K20 Torsion free groups, infinite rank 20K21 Mixed groups 20K25 Direct sums, direct products, etc. 20K26 indecomposable groups /-- 20K27 Subgroups 20K30 Automorphisms, homomorphisms, endomorphisms, etc. 20K35 Extensions 20K40 Homological and categorical methods 20K45 Topological methods [See also 22A05, 22B05] 20K99 None of the above, but in this section 20L05 Groupoids (i.e. small categories in which all /- 20Lxx /+ 20L05 morphisms are isomorphisms {For sets with a single binary operation, see 20N02; for topological groupoids, see 22A22, 58H05} 20L05 general theory /-- 20L10 connections with group theory /-- 20L13 mappings of groupoids /-- 20L15 connections with topology /-- 20L17 connections with category theory /-- 20L99 none of the above, but in this section /-- 20Mxx Semigroups 20M05 Free semigroups, generators and relations, word problems /~problems 20M07 Varieties of semigroups 20M10 General structure theory 20M11 Radical theory 20M12 Ideal theory 20M14 Commutative semigroups 20M15 Mappings of semigroups 20M17 Regular semigroups 20M18 Inverse semigroups 20M19 Orthodox semigroups 20M20 Semigroups of transformations, etc. [See also 47D03, 47H20, 54H15] 20M25 Semigroup rings, multiplicative semigroups of rings [See also 16S36, 16Y60] 20M30 Representation of semigroups; actions of semigroups on sets /++ 20M35 Semigroups in automata theory, linguistics, etc. [See also 03D05, 68Qxx, 68S05] 20M50 Connections of semigroups with homological algebra and category theory 20M99 None of the above, but in this section 20Nxx Other generalizations of groups 20N02 Sets with a single binary operation (groupoids) 20N05 Loops, quasigroups [See also 05Bxx] 20N07 mappings of loops /-- 20N10 Ternary systems (heaps, semiheaps, heapoids, etc.) 20N15 $n$-ary systems 20N20 Hypergroups 20N25 Fuzzy groups [See also 04A72] 20N99 None of the above, but in this section 20P05 Probabilistic methods in group theory 22-XX Topological groups, Lie groups {For transformation groups see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis see 43-XX} 22-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 22-01 Instructional exposition (textbooks, tutorial papers, etc.) 22-02 Research exposition (monographs, survey articles) 22-03 Historical (must be assigned at least one classification number from Section 01) 22-04 Explicit machine computation and programs (not the theory of computation or programming) 22-06 Proceedings, conferences, collections, etc. 22Axx Topological and differentiable /+ and differentiable algebraic systems {For topological rings and fields see 12Jxx, 13Jxx, 16W80; for dual spaces of operator algebras and topological groups see 47D35} 22A05 Structure of general topological groups 22A10 Analysis on general topological groups 22A15 Structure of topological semigroups 22A20 Analysis on topological semigroups 22A22 Topological groupoids (including differentiable and Lie groupoids) 22A25 Representations of general topological groups and semigroups 22A26 Topological semilattices, lattices and applications [See also 06B30, 06B35, 06F30] 22A30 Other topological algebraic systems and their representations 22A99 None of the above, but in this section 22Bxx Locally compact abelian groups (LCA groups) 22B05 General properties and structure of LCA groups 22B10 Structure of group algebras of LCA groups 22B99 None of the above, but in this section 22C05 Compact groups 22Dxx Locally compact groups and their algebras 22D05 General properties and structure of locally compact groups 22D10 Unitary representations of locally compact groups 22D12 Other representations of locally compact groups 22D15 Group algebras of locally compact groups 22D20 Representations of group algebras 22D25 $C^*$-algebras and $W$*-algebras in relation to group representations [See also 46Lxx] 22D30 Induced representations 22D35 Duality theorems 22D40 Ergodic theory on groups [See also 28Dxx, 43A60] 22D45 Automorphism groups of locally compact groups 22D99 None of the above, but in this section 22Exx Lie groups {For the topology of Lie groups and homogeneous spaces see 57-XX, 57Sxx, 57Txx; /+ 57-XX, for analysis thereon see 43-XX, 43A80, 43A85, 43A90} /+ 43-XX, 22E05 Local Lie groups [See also 34-XX, 35-XX, 58H05] /+ 22E10 General properties and structure of complex Lie groups [See also {32M05] 22E15 General properties and structure of real Lie groups 22E20 General properties and structure of other Lie groups 22E25 Nilpotent and solvable Lie groups 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 22E30 Analysis on real and complex Lie groups [See also 33C80, 43-XX] 22E35 Analysis on $p$-adic Lie groups [See also {11R56] 22E40 Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 22E41 Continuous cohomology [See also 57R32, 57Txx, 58H10] 22E43 Structure and representation of the Lorentz group 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods, For the purely algebraic theory [See 20G05] 22E46 Semisimple Lie groups and their representations 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also {17B35] 22E50 Representations of Lie and linear algebraic groups over local fields 22E55 Representations of Lie and linear algebraic groups over global fields and adčle rings [See also {20G05] 22E60 Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx} 22E65 Infinite-dimensional Lie groups and their Lie algebras [See also 17B65, 58B25, 58H05] 22E67 Loop groups and related constructions, group-theoretic treatment [See also 58D05] 22E70 Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10] 22E99 None of the above, but in this section 22Fxx Noncompact transformation groups /++++ 22F05 General theory of group and pseudogroup actions {For topological properties of spaces with an action see 57S20} 22F10 Measurable group actions {see also 28Dxx and 22D40] 22F30 Homogeneous spaces {For general actions on manifolds or preserving geometrical structures see 57; for discrete subgroups of Lie groups see especially 22E40} 22F50 Groups as automorphisms of other structures 26-XX Real functions [See also 54C30] 26-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 26-01 Instructional exposition (textbooks, tutorial papers, etc.) 26-02 Research exposition (monographs, survey articles) 26-03 Historical (must be assigned at least one classification number from Section 01) 26-04 Explicit machine computation and programs (not the theory of computation or programming) 26-06 Proceedings, conferences, collections, etc. 26Axx Functions of one variable 26A03 Foundations: limits and generalizations, elementary topology of the line 26A06 One-variable calculus 26A09 Elementary functions 26A12 Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48] 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 26A16 Lipschitz (Hölder) classes 26A18 Iteration [See also 37-XX, 39B12, 47H10, 54H25] /- 58F08, 58F13, /+ 37-XX 26A21 Classification of real functions; Baire classification of sets and functions [See also 03E15, /- 04A15 /+ 03E15 28A05, 54C50] 26A24 Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26A30 Singular functions, Cantor functions, functions with other special properties 26A33 Fractional derivatives and integrals 26A36 Antidifferentiation 26A39 Denjoy and Perron integrals, other special integrals 26A42 Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX] 26A45 Functions of bounded variation, generalizations 26A46 Absolutely continuous functions 26A48 Monotonic functions, generalizations 26A51 Convexity, generalizations 26A99 None of the above, but in this section 26Bxx Functions of several variables 26B05 Continuity and differentiation questions 26B10 Implicit function theorems, Jacobians, transformations with several variables 26B12 Calculus of vector functions 26B15 Integration: length, area, volume [See also 28A75, 51M25] 26B20 Integral formulas (Stokes, Gauss, Green, etc.) 26B25 Convexity, generalizations 26B30 Absolutely continuous functions, functions of bounded variation 26B35 Special properties of functions of several variables, Hölder conditions, etc. 26B40 Representation and superposition of functions 26B99 None of the above, but in this section 26Cxx Polynomials, rational functions 26C05 Polynomials: analytic properties, etc. [See also 12Dxx, 12Exx] 26C10 Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 26C15 Rational functions [See also 14Pxx] 26C99 None of the above, but in this section 26Dxx Inequalities {For maximal function inequalities, see 42B25; for functional inequalities, see 39B72; for probabilistic inequalities, see 60E15} 26D05 Inequalities for trigonometric functions and polynomials 26D07 Inequalities involving other types of functions 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals 26D20 Other analytical inequalities 26D99 None of the above, but in this section 26Exx Miscellaneous topics [See also 58Cxx] 26E05 Real-analytic functions [See also 32B05, 32C05] 26E10 $C^\infty$-functions, quasi-analytic functions [See also 58C25] 26E15 Calculus of functions on infinite-dimensional spaces [See also 46G05, 58Cxx] 26E20 Calculus of functions taking values in infinite-dimensional spaces [See also 46E40, 46G10, 58Cxx] 26E25 Set-valued functions [See also 28B20, 54C60] {For nonsmooth analysis, see 49J52, 58Cxx, 90Cxx} 26E30 Non-Archimedean analysis [See also 12J25] 26E35 Nonstandard analysis [See also 03H05, 28E05, 54J05] 26E40 Constructive real analysis [See also 03F60, 03F65] 26E50 Fuzzy real analysis [See also 03E72, 28E10] /- 04A72, /+ 03E72, 26E60 Means [See also 47A64] /++ 26E99 None of the above, but in this section 28-XX Measure and integration {For analysis on manifolds, see 58-XX} 28-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 28-01 Instructional exposition (textbooks, tutorial papers, etc.) 28-02 Research exposition (monographs, survey articles) 28-03 Historical (must be assigned at least one classification number from Section 01) 28-04 Explicit machine computation and programs (not the theory of computation or programming) 28-06 Proceedings, conferences, collections, etc. 28Axx Classical measure theory 28A05 Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 04A15, 26A21, 54H05] 28A10 Real- or complex-valued set functions 28A12 Contents, measures, outer measures, capacities 28A15 Abstract differentiation theory, differentiation of set functions [See also 26A24] 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 28A25 Integration with respect to measures and other set functions 28A33 Spaces of measures, convergence of measures [See also 46E27, 60Bxx] 28A35 Measures and integrals in product spaces 28A50 Integration and disintegration of measures 28A51 Lifting theory [See also 46G15] 28A60 Measures on Boolean rings, measure algebras [See also 54H10] /~ 54A10] 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28A78 Hausdorff and packing measures /+ and packing 28A80 Fractals [See also 37Fxx] /- 58Fxx /+ 37Fxx] 28A99 None of the above, but in this section 28Bxx Set functions, measures and integrals with values in abstract spaces 28B05 Vector-valued set functions, measures and integrals [See also 46G10] 28B10 Group- or semigroup-valued set functions, measures and integrals 28B15 Set functions, measures and integrals with values in ordered spaces 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 90A14] 28B99 None of the above, but in this section 28Cxx Set functions and measures on spaces with additional structure [See also 46G12, 58C35, 58D20] 28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures 28C10 Set functions and measures on topological groups, Haar measures, invariant measures [See also 22Axx, 43A05] 28C15 Set functions and measures on topological spaces (regularity of measures, etc.) 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 28C99 None of the above, but in this section 28Dxx Measure-theoretic ergodic theory [See also 11K50, 11K55, 22D40, 47A35, 54H20, 37Axx, 60Fxx, 60G10] /- 58Fxx /+ 37Axx 28D05 Measure-preserving transformations 28D10 One-parameter continuous families of measure-preserving transformations 28D15 General groups of measure-preserving transformations 28D20 Entropy and other invariants 28D99 None of the above, but in this section 28Exx Miscellaneous topics in measure theory 28E05 Nonstandard measure theory [See also 03H05, 26E35] 28E10 Fuzzy measure theory [See also 04A72, 26E50, 94D05] /~ [$ 03E72] 28E15 Other connections with logic and set theory 28E99 None of the above, but in this section 30-XX Functions of a complex variable {For analysis on manifolds, see 37-XX} /- 58-XX /+ 37-XX 30-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 30-01 Instructional exposition (textbooks, tutorial papers, etc.) 30-02 Research exposition (monographs, survey articles) 30-03 Historical (must be assigned at least one classification number from Section 01) 30-04 Explicit machine computation and programs (not the theory of computation or programming) 30-06 Proceedings, conferences, collections, etc. 30Axx General properties 30A05 Monogenic properties of complex functions (including polygenic and areolar monogenic functions) 30A10 Inequalities in the complex domain 30A99 None of the above, but in this section 30Bxx Series expansions 30B10 Power series (including lacunary series) 30B20 Random power series 30B30 Boundary behavior of power series, over-convergence 30B40 Analytic continuation 30B50 Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX] 30B60 Completeness problems, closure of a system of functions 30B70 Continued fractions [See also 11A55, 40A15] 30B99 None of the above, but in this section 30Cxx Geometric function theory 30C10 Polynomials 30C15 Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 30C20 Conformal mappings of special domains 30C25 Covering theorems in conformal mapping theory 30C30 Numerical methods in conformal mapping theory [See also 65E05] 30C35 General theory of conformal mappings 30C40 Kernel functions and applications 30C45 Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C50 Coefficient problems for univalent and multivalent functions 30C55 General theory of univalent and multivalent functions 30C62 Quasiconformal mappings in the plane 30C65 Quasiconformal mappings in $R^n$, other generalizations 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30C80 Maximum principle; Schwarz's lemma, Lindelof principle, analogues and generalizations; subordination 30C85 Capacity and harmonic measure in the complex plane [See also 31A15] 30C99 None of the above, but in this section 30Dxx Entire and meromorphic functions, and related topics 30D05 Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 39-XX, 37Fxx] /~ [$ 34A20, 39-XX, 58F05, 58F23] 30D10 Representations of entire functions by series and integrals 30D15 Special classes of entire functions and growth estimates 30D20 Entire functions, general theory 30D30 Meromorphic functions, general theory 30D35 Distribution of values, Nevanlinna theory 30D40 Cluster sets, prime ends, boundary behavior 30D45 Bloch functions, normal functions, normal families 30D50 Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part 30D55 ${H}^p$-classes 30D60 Quasi-analytic and other classes of functions 30D99 None of the above, but in this section 30Exx Miscellaneous topics of analysis in the complex domain 30E05 Moment problems, interpolation problems 30E10 Approximation in the complex domain 30E15 Asymptotic representations in the complex domain 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions [See also 45Exx] 30E25 Boundary value problems [See also 45Exx] 30E99 None of the above, but in this section 30Fxx Riemann surfaces 30F10 Compact Riemann surfaces and uniformization [See also 14H15, 32G15] 30F15 Harmonic functions on Riemann surfaces 30F20 Classification theory of Riemann surfaces 30F25 Ideal boundary theory 30F30 Differentials on Riemann surfaces 30F35 Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx] 30F40 Kleinian groups [See also 20H10] 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions) 30F50 Klein surfaces 30F60 Teichmüller theory [See also 32G15] 30F99 None of the above, but in this section 30Gxx Generalized function theory 30G06 Non-Archimedean function theory [See also 12J25]; nonstandard function theory [See also 03H05] 30G12 Finely holomorphic functions and topological function theory 30G15 topological function theory /-- 30G20 Generalizations of Bers or Vekua type (pseudoanalytic, $p$-analytic, etc.) 30G25 Discrete analytic functions 30G30 Other generalizations of analytic functions (including abstract-valued functions) 30G35 Functions of hypercomplex variables and generalized variables 30G99 None of the above, but in this section 30H05 Spaces and algebras of analytic functions [See also 32A38, 46Exx, 46J15] 31-XX Potential theory {For probabilistic potential theory, see 60J45} 31-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 31-01 Instructional exposition (textbooks, tutorial papers, etc.) 31-02 Research exposition (monographs, survey articles) 31-03 Historical (must be assigned at least one classification number from Section 01) 31-04 Explicit machine computation and programs (not the theory of computation or programming) 31-06 Proceedings, conferences, collections, etc. 31Axx Two-dimensional theory 31A05 Harmonic, subharmonic, superharmonic functions 31A10 Integral representations, integral operators, integral equations methods 31A15 Potentials and capacity, harmonic measure, extremal length [See also 30C85] 31A20 Boundary behavior (theorems of Fatou type, etc.) 31A25 Boundary value and inverse problems 31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation 31A35 Connections with differential equations 31A99 None of the above, but in this section 31Bxx Higher-dimensional theory 31B05 Harmonic, subharmonic, superharmonic functions 31B10 Integral representations, integral operators, integral equations methods 31B15 Potentials and capacities, extremal length 31B20 Boundary value and inverse problems 31B25 Boundary behavior 31B30 Biharmonic and polyharmonic equations and functions 31B35 Connections with differential equations 31B99 None of the above, but in this section 31Cxx Other generalizations 31C05 Harmonic, subharmonic, superharmonic functions 31C10 Pluriharmonic and plurisubharmonic functions [See also 32F05] 31C12 Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14] 31C15 Potentials and capacities 31C20 Discrete potential theory and numerical methods 31C25 Dirichlet spaces 31C35 Martin boundary theory [See also 60J50] 31C40 Fine potential theory 31C45 Other generalizations (nonlinear potential theory, etc.) 31C99 None of the above, but in this section 31D05 Axiomatic potential theory 32-XX Several complex variables and analytic spaces {For infinite-dimensional holomorphy, see also 46G20, 58B12} 32-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 32-01 Instructional exposition (textbooks, tutorial papers, etc.) 32-02 Research exposition (monographs, survey articles) 32-03 Historical (must be assigned at least one classification number from Section 01) 32-04 Explicit machine computation and programs (not the theory of computation or programming) 32-06 Proceedings, conferences, collections, etc. 32Axx Holomorphic functions of several complex variables 32A05 Power series, series of functions 32A07 Special domains (Reinhardt, Hartogs, circular, tube) /~ tube domains, etc.) 32A10 Holomorphic functions 32A12 Multifunctions /++ 32A15 Entire functions 32A17 Special families of functions /- (e.g. normal families) 32A18 Bloch functions, normal functions /++ 32A19 Normal families of functions, mappings /++ 32A20 Meromorphic functions 32A22 Nevanlinna theory (local); growth estimates; other inequalities {For geometric theory, see 32H25, 32H30} 32A25 Integral representations; canonical kernels (Szego, Bergman, etc.) /++ 32A26 Integral representations, constructed kernels (e.g. /++ Cauchy, Fantappič-type kernels) /++ 32A27 Local theory of residues [See also 32C30] 32A30 Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) {For functions of several hypercomplex variables, see 30G35} 32A35 ${H}^p$-spaces Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15] 32A36 Bergman spaces /++ 32A37 Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) /~ (VMOA) in n-dimensions) [See also 46Exx] 32A38 Algebras of holomorphic functions [See also 30H05, /++ 46J10, 46J15] /++ 32A40 Boundary behavior of holomorphic functions /+ of olomorphic functions 32A45 Hyperfunctions [See also 46F15] 32A50 Harmonic analysis of several complex variables [See /++ mainly 43-XX] /++ 32A55 Singular integrals /++ 32A60 Zero sets of holomorphic functions /++ 32A65 Banach algebra techniques [See mainly 46Jxx] /++ 32A70 Functional analysis techniques [See mainly 46Exx] /++ 32A99 None of the above, but in this section 32Bxx Local analytic geometry [See also 13-XX and 14-XX] 32B05 Analytic algebras and generalizations, preparation theorems 32B10 Germs of analytic sets, local parametrization /++ 32B15 Analytic subsets of affine space 32B20 Semi-analytic sets and subanalytic sets [See also 14P15] 32B25 Triangulation and related questions 32B99 None of the above, but in this section 32Cxx Analytic spaces /~ general theory of ... 32C05 Real-analytic manifolds, real-analytic spaces [See also 14Pxx, 58A07] 32C07 Real-analytic sets, complex Nash functions [See also 14P15, 14P20] 32C09 Embedding of real analytic manifolds /++ 32C11 Complex supergeometry [See also 14A22, 14M30, 58A50] 32C15 Complex spaces 32C16 CR-manifolds /-- 32C18 Topology of analytic spaces 32C20 Normal analytic spaces 32C22 Embedding of analytic spaces /++ 32C25 Analytic subsets and submanifolds 32C30 Integration on analytic sets and spaces, currents {For local theory, see 32A25 or 32A27} 32C35 Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30] 32C36 Local cohomology of analytic spaces 32C37 Duality theorems 32C38 Sheaves of differential operators and their modules, D-modules /++ [See also 14F10, 16S32, 35A27, 58J15] /- 58G07 /+ 58J15 32C55 The Levi problem in complex spaces; generalizations /++ 32C81 Applications to physics 32C99 None of the above, but in this section 32Dxx Analytic continuation 32D05 Domains of holomorphy 32D10 Envelopes of holomorphy 32D15 Continuation of analytic objects 32D20 Removable singularities 32D25 Riemann domains /++ 32D99 None of the above, but in this section 32Exx Holomorphic convexity 32E05 Holomorphically convex complex spaces, reduction theory 32E10 Stein spaces, Stein manifolds 32E20 Polynomial convexity 32E30 Holomorphic and polynomial approximation, Runge pairs, interpolation 32E35 Global boundary behavior of holomorphic functions 32E40 The Levi problem 32E99 None of the above, but in this section 32Fxx Geometric convexity /- partial differential operators 32F10 $q$-convexity, $q$-concavity 32F15 pseudoconvex domains /-- 32F17 Other notions of convexity /++ 32F18 Finite-type conditions /++ 32F20 $\overline\partial$-Neumann and /-- $\overline\partial_b$-Neumann problems [$ 35N15] /-- 32F25 real submanifolds in complex manifolds /-- 32F27 Topological consequences of geometric convexity /++ 32F30 pseudoconvex manifolds /-- 32F32 Analytical consequences of geometric convexity /++ (vanishing theorems, etc.) /++ 32F40 CR structures (tangential) CR operators and generalizations /-- 32F45 Invariant metrics and pseudodistances /++ 32F99 None of the above, but in this section 32Gxx Deformations of analytic structures 32G05 Deformations of complex structures [See also 13D10, 16S80, 58H10, 58H15] 32G07 Deformations of special (e.g. CR) structures 32G08 Deformations of fiber bundles 32G10 Deformations of submanifolds and subspaces 32G13 Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15] 32G15 Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 32G20 Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30] 32G34 Moduli and deformations for ordinary differential equations [See also 34A20] (e.g. Khnizhnik-Zamolodchikov equation) 32G81 Applications to physics 32G99 None of the above, but in this section 32Hxx Holomorphic mappings and correspondences 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions 32H04 Meromorphic mappings 32H12 Boundary uniqueness of mappings /++ 32H15 invariant metrics and pseudodistances /-- 32H20 hyperbolic complex manifolds /-- 32H25 Picard-type theorems and generalizations {For function-theoretic properties, see 32A22} 32H30 Value distribution theory in higher dimensions {For function-theoretic properties, see 32A22} 32H35 Proper mappings, finiteness theorems 32H40 Boundary regularity of mappings /- holomorphic maps /+ mappings 32H50 iterations problems /-- 32H99 None of the above, but in this section 32Jxx Compact analytic spaces {For Riemann surfaces, see 14Hxx, 30Fxx; for algebraic theory, see 14Jxx} 32J05 Compactification of analytic spaces 32J10 Algebraic dependence theorems 32J15 Compact surfaces 32J17 Compact $3$-folds 32J18 Compact $n$-folds /- (n>=4) 32J20 algebraicity criteria /-- 32J25 Transcendental methods of algebraic geometry [See also 14C30] 32J27 Compact Kähler manifolds: generalizations, classification 32J81 Applications to physics 32J99 None of the above, but in this section 32Kxx Generalizations of analytic spaces {(should also be assigned at least one other classification number in this section)} 32K05 Banach analytic spaces [See also 58Bxx] 32K07 Formal and graded complex spaces [See also 58C50] 32K15 Differentiable functions on analytic spaces, differentiable spaces [See also 58C25] 32K99 None of the above, but in this section 32Lxx Holomorphic fiber spaces [See also 55Rxx] 32L05 Holomorphic bundles and generalizations /- fiber bundles /+ bundles 32L07 Hermite-Einstein bundles; Kahler-Einstein bundles [$ 53C07] /-- 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results [See also 14F05, 18F20, 55N30] 32L15 Bundle convexity [See also 32F10] 32L20 Vanishing theorems 32L25 Twistor theory, double fibrations 32L30 holomorphic foliations [$ 58F18] /-- 32L81 Applications to physics 32L99 None of the above, but in this section 32Mxx Complex spaces with a group of automorphisms 32M05 Complex Lie groups, automorphism groups acting on /- of /+ acting on complex spaces [See also 22E10] 32M10 Homogeneous complex manifolds [See also 14M17, 57T15] 32M12 Almost homogeneous manifolds and spaces [See also 14M17] 32M15 Hermitian symmetric spaces, bounded symmetric domains Jordan algebras /++ [See also 22E10, 22E40, 53C35, 57T15] 32M17 Automorphism groups of ${\bf C}^n$$ and affine /++ manifolds /++ 32M25 Complex vector fields /++ 32M99 None of the above, but in this section 32Nxx Automorphic functions [See also 11Fxx, 20H10, 22E40, 30F35] 32N05 General theory of automorphic functions of several complex variables 32N10 Automorphic forms 32N15 Automorphic functions in symmetric domains 32N99 None of the above, but in this section 32P05 Non-Archimedean complex analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem) 32Qxx Complex manifolds /++++ 32Q05 Negative curvature manifolds 32Q10 Positive curvature manifolds 32Q15 Kähler manifolds 32Q20 Kähler-Einstein manifolds [See also 53Cxx 32Q25 Calabi-Yau theory 32Q28 Stein manifolds 32Q30 Uniformization 32Q35 Complex manifolds as subdomains of Euclidean space 32Q40 Embedding theorems 32Q45 (Kobayashi) Hyperbolic manifolds 32Q55 Topological aspects of complex manifolds 32Q57 Classification theorems 32Q60 Almost complex manifolds 32Q65 Pseudoholomorphic curves 32Q99 None of the above, but in this section 32Sxx Singularities 32S05 Local singularities [See also 14J17] /- 14B05] /+ 14J17] 32S10 Invariants of analytic local rings 32S15 Equisingularity (topological and analytic) [See also 14E15] 32S20 Global theory of singularities; cohomological properties [See also 14E15] 32S22 Relations with arrangements of hyperplanes [See also /++ 52C30] /++ 32S25 (Hyper-) surface singularities [See also 14J17] 32S30 Deformations of singularities; vanishing cycles [See also 14B07] 32S35 Mixed Hodge theory of singular varieties [See also 14C30, 14D07] 32S40 Monodromy; relations with differential equations and $D$-modules 32S45 Modifications; resolution of singularities [See also 14E15] 32S50 Topological aspects: Lefschetz theorems, topological classification, invariants 32S55 Milnor fibration; relations with knot theory [See also 57M25, 57Q45] 32S60 Stratifications; constructible sheaves; intersection cohomology [See also 58C27] 32S65 Singularities of holomorphic vector fields and foliations /++ 32S70 Other operations on singularities 32S99 None of the above, but in this section 32Txx Pseudoconvex domains /++++ 32T05 Domains of holomorphy 32T15 Strongly pseudoconvex domains 32T20 Worm domains 32T25 Finite type domains 32T27 Geometric and analytic invariants on weakly pseudoconvex boundaries 32T35 Exhaustion functions 32T40 Peak functions 32T99 None of the above, but in this section 32Uxx Pluripotential theory /++ 32U05 Plurisubharmonic functions and generalizations [See also 31C10 32U10 Plurisubharmonic exhaustion functions 32U15 Pluripotential theory 32U20 Capacity theory and generalizations 32U25 Lelong numbers 32U30 Removable sets 32U35 Pluricomplex Green functions 32U40 Currents 32U99 None of the above, but in this section 32Vxx $CR$ Manifolds /++++ 32V05 $CR$ structures, $CR$ operators, and generalizations 32V10 $CR$ functions 32V15 $CR$ manifolds as boundaries of domains 32V20 Analysis on $CR$ manifolds 32V25 Extension of functions and other analytic objects from $CR$ manifolds 32V30 Embeddings of $CR$ manifolds 32V35 Finite type conditions on $CR$ manifolds 32V40 Real submanifolds in complex manifolds 32V99 None of the above, but in this section 32Wxx Differential operators in several variables /++++ 32W05 $\overline\partial$ and $\overline\partial$-Neumann operators 32W10 $\overline\partial_b$ and $\overline\partial_b$-Neumann operators 32W15 $CR$ manifolds as boundaries of domains 32W20 Complex Monge-Ampčre operators 32W25 Pseudodifferential operators in several complex variables 32W30 Heat kernels in several complex variables 32W35 Finite type conditions on $CR$ manifolds 32W50 Other partial differential equations of complex analysis 32W99 None of the above, but in this section 33-XX Special functions {33-XX deals with the properties of functions as functions. For orthogonal functions see also 42Cxx; for aspects of combinatorics see 05Axx; for number-theoretic aspects see 11-XX; for representation theory see 22Exx} 33-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 33-01 Instructional exposition (textbooks, tutorial papers, etc.) 33-02 Research exposition (monographs, survey articles) 33-03 Historical (must be assigned at least one classification number from Section 01) 33-04 Explicit machine computation and programs (not the theory of computation or programming) 33-06 Proceedings, conferences, collections, etc. 33Bxx Elementary classical functions 33B10 Exponential and trigonometric functions 33B15 Gamma, beta and polygamma functions 33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) 33E30 Higher logarithm functions /++ 33B99 None of the above, but in this section 33Cxx Hypergeometric functions 33C05 Classical hypergeometric functions, $_2F_1$ 33C10 Bessel and Airy functions, cylinder functions, $_0F_1$ 33C15 Confluent hypergeometric functions, Whittaker functions, $_1F_1$ 33C20 Generalized hypergeometric series, $_pF_q$ 33C45 Orthogonal polynomials and functions of hypergeometric type /++ (Jacobi, Laguerre, Hermite, Askey scheme, etc.; /++ see 42C05 for general orthogonal polynomials and functions) /++ /- (Chebyshev, Legendre, Gegenbauer, Jacobi, Laguerre, Hermite, Hahn, etc.) 33C47 Other special orthogonal polynomials and functions /++ 33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one varaible 33C52 Orthogonal polynomials and functions associated with root systems /++ 33C55 Spherical harmonics /~ spherical functions, spherical harmonics, /- ultraspherical polynomials 33C60 Hypergeometric integrals and functions defined by them ($E$, $G$ and ${H}$ functions) 33C65 Appell, Horn and Lauricella functions 33C67 Hypergeometric functions associated with root systems /++ 33C70 Other hypergeometric functions and integrals in several variables 33C75 Elliptic integrals as hypergeometric functions 33C80 Connections with groups, algebras, and related topics /~ root systems and related topics 33C90 Applications 33C99 None of the above, but in this section 33Dxx Basic hypergeometric functions 33D05 $q$-gamma functions, $q$-beta functions and integrals 33D10 basic theta functions /-- 33D15 Basic hypergeometric functions in one variable, ${}_r\phi_s$ /++ 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) /~ in one and several variables 33D50 Orthogonal polynomials and functions in several /++ variables expressible in terms of basic hypergeometric /++ functions in one variable /++ 33D52 Basic orthogonal polynomials and functions associated /++ with root systems (Macdonald polynomials, etc.) /++ 33D60 Basic hypergeometric integrals and functions defined by them 33D65 Bibasic functions and multiple bases 33D67 Basic hypergeometric functions associated with root /++ systems /++ 33D70 Other basic hypergeometric functions and integrals in several variables 33D80 Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke /~ groups algebras, and related topics 33D90 Applications 33D99 None of the above, but in this section 33Exx Other special functions 33E05 Elliptic functions and integrals 33E10 Lamé, Mathieu, and spheroidal wave functions 33E12 Mittag-leffler functions and generalizations /++ 33E15 Other wave functions 33E17 Painlevé-type functions /++ 33E20 Other functions defined by series and integrals 33E30 Other functions coming from differential, difference and integral equations 33E50 Special functions in characteristic $p$ (gamma /++ functions etc.) /++ 33E99 None of the above, but in this section 33Fxx Computational aspects 33F05 Numerical approximation [See also 65D20] 33F10 Symbolic computation (Gosper and Zeilberger algorithms, etc.) [See also 68Q40] 33F99 None of the above, but in this section 34-XXv. Ordinary differential equations 34-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 34-01 Instructional exposition (textbooks, tutorial papers, etc.) 34-02 Research exposition (monographs, survey articles) 34-03 Historical (must be assigned at least one classification number from Section 01) 34-04 Explicit machine computation and programs (not the theory of computation or programming) 34-06 Proceedings, conferences, collections, etc. 34Axx General theory 34A05 Explicit solutions and reductions 34A09 Implicit equations, differential-algebraic equations [See also 65L80] 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 34A25 Analytical theory: series, transformations, transforms, operational calculus, etc. [See also 44-XX] 34A26 Geometric methods in differential equations 34A30 Linear equations and systems 34A34 Nonlinear equations and systems, general 34A35 Differential equations of infinite order 34A36 Discontinuous equations 34A37 Differential equations with impulses 34A40 Differential inequalities 34A45 Theoretical approximation of solutions {For numerical analysis, see 65Lxx} 34A55 Inverse problems 34A60 Differential inclusions [See also 49J24, 49K24] 34A99 None of the above, but in this section 34Bxx Boundary value problems {For ordinary differential operators, see 34Lxx} 34B05 Linear boundary value problems 34B10 Multipoint boundary value problems 34B15 Nonlinear boundary value problems 34B16 Singular nonlinear boundary value problems 34B18 Positive solutions of nonlinear boundary value problems 34B20 Weyl theory and its generalizations 34B24 Sturm-Liouville theory [See also 34Lxx] 34B27 Green functions 34B30 Special equations (Mathieu, Hill, Bessel, etc.) 34B37 Boundary value problems with impulses 34B40 Boundary values on infinite intervals 34B45 Boundary value problems on graphs and networks 34B60 Applications 34B99 None of the above, but in this section 34Cxx Qualitative theory [See also 37-XX] 34C05 Location of integral curves, singular points, limit cycles 34C10 Oscillation theory, zeros, disconjugacy and comparison theory 34C11 Growth, boundedness, comparison of solutions 34C12 Monotone systems 34C14 Symmetries, invariants 34C15 Nonlinear oscillations, coupled oscillators 34C20 Transformation and reduction of equations and systems, normal forms 34C23 Bifurcation [See mainly 37Gxx] 34C25 Periodic solutions 34C26 Relaxation oscillations 34C27 Almost periodic solutions 34C28 Complex behavior, chaotic systems [See mainly 37Dxx] 34C29 Averaging method 34C30 Manifolds of solutions 34C37 Homoclinic and heteroclinic solutions 34C40 Equations and systems on manifolds 34C41 Equivalence, asymptotic equivalence 34C45 Method of integral manifolds 34C55 Hysteresis 34C60 Applications 34C99 None of the above, but in this section 34Dxx Stability theory [See also 37C75, 93Dxx] 34D05 Asymptotic properties 34D08 Characteristic and Lyapunov exponents 34D09 Dichotomy, trichotomy 34D10 Perturbations 34D15 Singular perturbations 34D20 Lyapunov stability 34D23 Global stability 34D30 Structural stability and analogous concepts [See also 37C20] 34D35 Stability of manifolds of solutions 34D40 Ultimate boundedness 34D45 Attractors [See also 37C70, 37D50] 34D99 None of the above, but in this section 34Exx Asymptotic theory 34E05 Asymptotic expansions 34E10 Perturbations, asymptotics 34E13 Multiple scale methods 34E15 Singular perturbations, general theory 34E18 Methods of nonstandard analysis 34E20 Singular perturbations, turning point theory, WKB methods 34E99 None of the above, but in this section 34F05 Equations and systems with randomness [See also 34K50, 60H10, 93E03] 34Gxx Differential equations in abstract spaces [See also 37Kxx, 34Lxx, 47Dxx, 47Hxx, 47Jxx, 58D25] 34G10 Linear equations [See also 47Axx, 47Bxx, 47D06, 47D09] 34G20 Nonlinear equations [See also 47Hxx, 47Jxx] 34G25 Evolution inclusions 34G99 None of the above, but in this section 34H05 Control problems [See also 49J25, 49K25, 93C15] 34Kxx Functional-differential and differential-difference equations 34K05 General theory 34K06 Linear functional-differential equations 34K07 Theoretical approximation of solutions 34K10 Boundary value problems 34K11 Oscillation theory 34K12 Growth, boundedness, comparison of solutions 34K13 Periodic solutions 34K14 Almost-periodic solutions 34K17 Transformation and reduction of equations and systems, normal forms 34K18 Bifurcation theory 34K19 Invariant manifolds 34K20 Stability theory 34K23 Complex (chaotic) behavior of solutions 34K25 Asymptotic theory 34K26 Singular perturbations 34K28 Numerical approximation of solutions 34K29 Inverse problems 34K30 Equations in abstract spaces [See also 34Gxx] 34K35 Control problems [See also 49J25, 49K25, 93C15] 34K40 Neutral equations 34K50 Stochastic delay equations [See also 34F05, 60Hxx] 34K60 Applications 34K99 None of the above, but in this section 34Lxx Ordinary differential operators [See also 47E05] 34L05 General spectral theory 34L10 Eigenfunction expansions, completeness of eigenfunctions 34L15 Estimation of eigenvalues, upper and lower bounds 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions 34L25 Scattering theory 34L30 Nonlinear ordinary differential operators 34L40 Particular operators (Dirac, one-dimensional Schrödinger, etc.) 34L99 None of the above, but in this section 34Mxx Differential equations in the complex domain [See also 30Dxx, 32G34] 34M05 Entire and meromorphic solutions 34M10 Oscillation, growth of solutions 34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) 34M20 Nonanalytic aspects 34M25 Formal solutions, transform techniques 34M30 Asymptotics, summation methods 34M35 Singularities, monodromy, local behavior of solutions, normal forms 34M40 Stokes phenomena and connection problems (linear and nonlinear) 34M45 Differential equations on complex manifolds 34M50 Inverse problems (Riemann-Hilbert, inverse dfifferential Galois, etc.) 34M55 Painlevé and other special equations; classification, hierarchies; isomonodromic deformations 34M60 Singular perturbation problems in the complex domain (complex WKB, turning points, steepest descent) [See also