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MSC2010: Final Public Version [Oct. 2009] MSC2010 This document is a printed form the Final Public Version of MSC2010 produced jointly by the editorial staﬀs of Mathematical Reviews (MR) and Zentralblatt f¨ur Mathematik (Zbl) in consultation with the mathematical community. The goals of this revision of the Mathematics Subject Classiﬁcation (MSC) were set out in the announcement of it and call for comments by the Executive Editor of MR and the Chief Editor of Zbl in August 2006. This document results from the MSC revision process that has been going on since then. MSC2010 will be fully deployed from July 2010. The editors of MR and Zbl deploying this revision therefore ask for feedback on remaining errors to help in this work, which should be given, preferably, on the Web site at http://msc2010.org or, if the internet is not available, through e-mail to feedback@msc2010.org. They are grateful for the many suggestions that were received previously which have much inﬂuenced what we have. How to use the Mathematics Subject Classiﬁcation [MSC] The main purpose of the classiﬁcation of items in the mathematical literature using the Mathematics Subject Classiﬁcation scheme is to help users ﬁnd the items of present or potential interest to them as readily as possible—in products derived from the Mathematical Reviews Database (MRDB), in Zentralblatt MATH (ZMATH), or anywhere else where this classiﬁcation scheme is used. An item in the mathematical literature should be classiﬁed so as to attract the attention of all those possibly interested in it. The item may be something which falls squarely within one clear area of the MSC, or it may involve several areas. Ideally, the MSC codes attached to an item should represent the subjects to which the item contains a contribution. The classiﬁcation should serve both those closely concerned with speciﬁc subject areas, and those familiar enough with subjects to apply their results and methods elsewhere, inside or outside of mathematics. It will be extremely useful for both users and classiﬁers to familiarize themselves with the entire classiﬁcation system and thus to become aware of all the classiﬁcations of possible interest to them. simply the MSC code that describes Every item in the MRDB or ZMATH receives precisely one primary classiﬁcation, which is its principal contribution. When an item contains several principal contributions to diﬀerent areas, the primary classiﬁcation should cover the most important among them. A paper or book may be assigned one or several secondary classiﬁcation numbers to cover any remaining principal contributions, ancillary results, motivation or origin of the matters discussed, intended or potential ﬁeld of application, or other signiﬁcant aspects worthy of notice. The principal contribution is meant to be the one including the most important part of the work actually done in the item. For example, a paper whose main overall content is the solution of a problem in graph theory, which arose in computer science and whose solution is (perhaps) at present only of interest to computer scientists, would have a primary classiﬁcation in 05C (Graph Theory) with one or more secondary classiﬁcations in 68 (Computer Science); conversely, a paper whose overall content lies mainly in computer science should receive a primary classiﬁcation in 68, even if it makes heavy use of graph theory and proves several new graph-theoretic results along the way. There are two types of cross-references given at the end of many of the entries in the MSC. The ﬁrst type is in braces: “{For A, see X}”; if this appears in section Y, it means that contributions described by A should usually be assigned the classiﬁcation code X, not Y. The other type of cross-reference merely points out related classiﬁcations; it is in brackets: “[See also . . . ]”, “[See mainly . . . ]”, etc., and the classiﬁcation codes listed in the brackets may, but need not, be included in the classiﬁcation codes of a paper, or they may be used in place of the classiﬁcation where the cross-reference is given. The classiﬁer must judge which classiﬁcation is the most appropriate for the paper at hand. 00–XX 00–01 00–02 00Axx 00A05 00A06 Mathematics for nonmathematicians (engineering, social sciences, GENERAL Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) General and miscellaneous speciﬁc topics General mathematics etc.) Problem books Recreational mathematics [See also 97A20] Popularization of mathematics Bibliographies External book reviews Dictionaries and other general reference works Formularies Philosophy of mathematics [See also 03A05] 00A07 00A08 00A09 00A15 00A17 00A20 00A22 00A30 00A35 Methodology of mathematics, didactics [See also 97Cxx, 97Dxx] 00A65 Mathematics and music 00A66 Mathematics and visual arts, visualization 00A67 Mathematics and architecture 00A69 General applied mathematics {For physics, see 00A79 and Sections 70 through 86} Theory of mathematical modeling General methods of simulation Dimensional analysis Physics (use more speciﬁc entries from Sections 70 through 86 when possible) 00A71 00A72 00A73 00A79 Conference proceedings and collections of papers Collections of abstracts of lectures Collections of articles of general interest Collections of articles of miscellaneous speciﬁc content Proceedings of conferences of general interest Proceedings of conferences of miscellaneous speciﬁc interest Festschriften Volumes of selected translations 00A99 Miscellaneous topics 00Bxx 00B05 00B10 00B15 00B20 00B25 00B30 00B50 00B55 Miscellaneous volumes of translations 00B60 00B99 Collections of reprinted articles [See also 01A75] None of the above, but in this section 01–XX 01–00 01–01 HISTORY AND BIOGRAPHY [See also the classiﬁcation number–03 in the other sections] General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Proceedings, conferences, collections, etc. Computational methods History of mathematics and mathematicians General histories, source books Ethnomathematics, general Paleolithic, Neolithic Indigenous cultures of the Americas Other indigenous cultures (non-European) Indigenous European cultures (pre-Greek, etc.) Egyptian Babylonian Greek, Roman China Japan Southeast Asia Islam (Medieval) India 01–02 01–06 01–08 01Axx 01A05 01A07 01A10 01A12 01A13 01A15 01A16 01A17 01A20 01A25 01A27 01A29 01A30 01A32 01A35 Medieval 01A40 01A45 01A50 01A55 01A60 01A61 01A65 01A67 01A70 01A72 01A73 01A74 01A75 15th and 16th centuries, Renaissance 17th century 18th century 19th century 20th century Twenty-ﬁrst century Contemporary Future prospectives Biographies, obituaries, personalia, bibliographies Schools of mathematics Universities Other institutions and academies Collected or selected works; reprintings or translations of classics [See also 00B60] Sociology (and profession) of mathematics 01A80 Historiography 01A85 01A90 Bibliographic studies 01A99 Miscellaneous topics 03–XX 03–00 03–01 03–02 03–03 MATHEMATICAL LOGIC AND FOUNDATIONS General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

03–XX MSC2010: FINAL PUBLIC VERSION [Oct. 2009] S2 03–04 03–06 03Axx 03A05 03A10 03A99 03Bxx 03B05 03B10 03B15 03B20 03B22 03B25 03B30 03B47 Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Philosophical aspects of logic and foundations Philosophical and critical {For philosophy of mathematics, see also 00A30} Logic in the philosophy of science None of the above, but in this section General logic Classical propositional logic Classical ﬁrst-order logic Higher-order logic and type theory Subsystems of classical logic (including intuitionistic logic) Abstract deductive systems Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] Foundations of classical theories (including reverse mathematics) [See also 03F35] 03B35 Mechanization of proofs and logical operations [See also 68T15] 03B40 03B42 03B44 03B45 Modal logic (including the logic of norms) {For knowledge and belief, Combinatory logic and lambda-calculus [See also 68N18] Logics of knowledge and belief (including belief change) Temporal logic see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45} Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} Probability and inductive logic [See also 60A05] Other model constructions Categoricity and completeness of theories Interpolation, preservation, deﬁnability Classiﬁcation theory, stability and related concepts [See also 03C48] Abstract elementary classes and related topics [See also 03C45] Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05] Basic properties of ﬁrst-order languages and structures Quantiﬁer elimination, model completeness and related topics Finite structures [See also 68Q15, 68Q19] Denumerable structures Ultraproducts and related constructions Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05] Paraconsistent logics Intermediate logics Other nonclassical logic Combined logics Logic of natural languages [See also 68T50, 91F20] Logic in computer science [See also 68–XX] Other applications of logic None of the above, but in this section 03B48 03B50 Many-valued logic 03B52 03B53 03B55 03B60 03B62 03B65 03B70 03B80 03B99 03Cxx Model theory 03C05 03C07 03C10 03C13 03C15 03C20 03C25 Model-theoretic forcing 03C30 03C35 03C40 03C45 03C48 03C50 Models with special properties (saturated, rigid, etc.) 03C52 03C55 03C57 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 Other classical ﬁrst-order model theory 03C68 Logic on admissible sets 03C70 03C75 Other inﬁnitary logic Logic with extra quantiﬁers and operators [See also 03B42, 03B44, 03C80 03B45, 03B48] Second- and higher-order model theory Nonclassical models (Boolean-valued, sheaf, etc.) Abstract model theory Applications of model theory [See also 03C60] None of the above, but in this section Computability and recursion theory Thue and Post systems, etc. Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15] Turing machines and related notions [See also 68Q05] Complexity of computation (including implicit computational complexity) [See also 68Q15, 68Q17] Recursive functions and relations, subrecursive hierarchies Recursively (computably) enumerable sets and degrees Other Turing degree structures Properties of classes of models Set-theoretic model theory Eﬀective and recursion-theoretic model theory [See also 03D45] 03C85 03C90 03C95 03C98 03C99 03Dxx 03D03 03D05 03D10 03D15 03D20 03D25 03D28 Other degrees and reducibilities Algorithmic randomness and dimension [See also 68Q30] Undecidability and degrees of sets of sentences 03D30 03D32 03D35 03D40 Word problems, etc. [See also 06B25, 08A50, 20F10, 68R15] 03D45 Theory of numerations, eﬀectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55] Recursive equivalence types of sets and structures, isols Hierarchies Computability and recursion theory on ordinals, admissible sets, etc. Higher-type and set recursion theory Inductive deﬁnability Abstract and axiomatic computability and recursion theory Computation over the reals {For constructive aspects, see 03F60} Applications of computability and recursion theory None of the above, but in this section Set theory Partition relations Ordered sets and their coﬁnalities; pcf theory Other combinatorial set theory Ordinal and cardinal numbers Descriptive set theory [See also 28A05, 54H05] Cardinal characteristics of the continuum Other classical set theory (including functions, relations, and set algebra) Axiom of choice and related propositions Axiomatics of classical set theory and its fragments Consistency and independence results Other aspects of forcing and Boolean-valued models Inner models, including constructibility, ordinal deﬁnability, and core models Other notions of set-theoretic deﬁnability Continuum hypothesis and Martin’s axiom [See also 03E57] Large cardinals Generic absoluteness and forcing axioms [See also 03E50] Determinacy principles Other hypotheses and axioms Nonclassical and second-order set theories Fuzzy set theory Applications of set theory None of the above, but in this section Proof theory and constructive mathematics Proof theory, general Cut-elimination and normal-form theorems Structure of proofs Functionals in proof theory Recursive ordinals and ordinal notations Complexity of proofs Relative consistency and interpretations First-order arithmetic and fragments Second- and higher-order arithmetic and fragments [See also 03B30] G¨odel numberings and issues of incompleteness Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25] Metamathematics of constructive systems Linear logic and other substructural logics [See also 03B47] Intuitionistic mathematics Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30] Other constructive mathematics [See also 03D45] None of the above, but in this section Algebraic logic Boolean algebras [See also 06Exx] Lattices and related structures [See also 06Bxx] Quantum logic [See also 06C15, 81P10] Cylindric and polyadic algebras; relation algebras Lukasiewicz and Post algebras [See also 06D25, 06D30] Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35] Abstract algebraic logic Categorical logic, topoi [See also 18B25, 18C05, 18C10] None of the above, but in this section Nonstandard models [See also 03C62] Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] Other applications of nonstandard models (economics, physics, etc.) Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05] None of the above, but in this section 03D50 03D55 03D60 03D65 03D70 03D75 03D78 03D80 03D99 03Exx 03E02 03E04 03E05 03E10 03E15 03E17 03E20 03E25 03E30 03E35 03E40 03E45 03E47 03E50 03E55 03E57 03E60 03E65 03E70 03E72 03E75 03E99 03Fxx 03F03 03F05 03F07 03F10 03F15 03F20 03F25 03F30 03F35 03F40 03F45 03F50 03F52 03F55 03F60 03F65 03F99 03Gxx 03G05 03G10 03G12 03G15 03G20 03G25 03G27 03G30 03G99 03Hxx 03H05 03H10 03H15 03H99 [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

S3 05–XX 05–00 05–01 05–02 05–03 05–04 05–06 05Axx 05A05 05A10 05A15 05A16 05A17 05A18 05A19 05A20 05A30 05A40 05A99 05Bxx 05B05 05B07 05B10 COMBINATORICS {For ﬁnite ﬁelds, see 11Txx} General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Enumerative combinatorics {For enumeration in graph theory, see 05C30} Permutations, words, matrices Factorials, binomial coeﬃcients, combinatorial functions [See also 11B65, 33Cxx] Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] Asymptotic enumeration Partitions of integers [See also 11P81, 11P82, 11P83] Partitions of sets Combinatorial identities, bijective combinatorics Combinatorial inequalities q-calculus and related topics [See also 33Dxx] Umbral calculus None of the above, but in this section Designs and conﬁgurations {For applications of design theory, see 94C30} Block designs [See also 51E05, 62K10] Triple systems Diﬀerence sets (number-theoretic, group-theoretic, etc.) [See also 11B13] Orthogonal arrays, Latin squares, Room squares 05C05 05C07 05C10 Finite geometries [See also 51D20, 51Exx] Other designs, conﬁgurations [See also 51E30] 05B15 05B20 Matrices (incidence, Hadamard, etc.) 05B25 05B30 05B35 Matroids, geometric lattices [See also 52B40, 90C27] Packing and covering [See also 11H31, 52C15, 52C17] 05B40 05B45 Tessellation and tiling problems [See also 52C20, 52C22] Polyominoes 05B50 None of the above, but in this section 05B99 Graph theory {For applications of graphs, see 68R10, 81Q30, 81T15, 05Cxx 82B20, 82C20, 90C35, 92E10, 94C15} Trees Vertex degrees [See also 05E30] Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] Distance in graphs Coloring of graphs and hypergraphs Perfect graphs Directed graphs (digraphs), tournaments Flows in graphs Signed and weighted graphs Graphs and abstract algebra (groups, rings, ﬁelds, etc.) [See also 20F65] Enumeration in graph theory Graph polynomials Extremal problems [See also 90C35] Paths and cycles [See also 90B10] Connectivity Density (toughness, etc.) Eulerian and Hamiltonian graphs Graphs and linear algebra (matrices, eigenvalues, etc.) Graph designs and isomomorphic decomposition [See also 05B30] Generalized Ramsey theory [See also 05D10] Games on graphs [See also 91A43, 91A46] Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) Graph representations (geometric and intersection representations, etc.) For graph drawing, see also 68R10 Inﬁnite graphs Hypergraphs Dominating sets, independent sets, cliques Factorization, matching, partitioning, covering and packing Fractional graph theory, fuzzy graph theory Structural characterization of families of graphs Graph operations (line graphs, products, etc.) Graph labelling (graceful graphs, bandwidth, etc.) Random graphs [See also 60B20] Random walks on graphs Small world graphs, complex networks [See also 90Bxx, 91D30] 05C63 05C65 05C69 05C70 05C72 05C75 05C76 05C78 05C80 05C81 05C82 05C12 05C15 05C17 05C20 05C21 05C22 05C25 05C30 05C31 05C35 05C38 05C40 05C42 05C45 05C50 05C51 05C55 05C57 05C60 05C62 MSC2010: FINAL PUBLIC VERSION [Oct. 2009] 06Exx 05C83 05C85 05C90 05C99 05Dxx 05D05 05D10 05D15 05D40 05D99 05Exx 05E05 05E10 05E15 05E18 05E30 05E40 05E45 05E99 06–XX 06–00 06–01 06–02 06–03 06–04 06–06 06Axx 06A05 06A06 06A07 06A11 06A12 06A15 06A75 06A99 06Bxx 06B05 06B10 06B15 06B20 06B23 06B25 06B30 06B35 Graph minors Graph algorithms [See also 68R10, 68W05] Applications [See also 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15] None of the above, but in this section Extremal combinatorics Extremal set theory Ramsey theory [See also 05C55] Transversal (matching) theory Probabilistic methods None of the above, but in this section Algebraic combinatorics Symmetric functions and generalizations Combinatorial aspects of representation theory [See also 20C30] Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80] Group actions on combinatorial structures Association schemes, strongly regular graphs Combinatorial aspects of commutative algebra Combinatorial aspects of simplicial complexes None of the above, but in this section ORDER, LATTICES, ORDERED ALGEBRAIC STRUCTURES [See also 18B35] General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Ordered sets Total order Partial order, general Combinatorics of partially ordered sets Algebraic aspects of posets Semilattices [See also 20M10; for topological semilattices see 22A26] Galois correspondences, closure operators Generalizations of ordered sets None of the above, but in this section Lattices [See also 03G10] Structure theory Ideals, congruence relations Representation theory Varieties of lattices Complete lattices, completions Free lattices, projective lattices, word problems [See also 03D40, 08A50, 20F10] Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] Generalizations of lattices None of the above, but in this section 06B75 06B99 06Cxx Modular lattices, complemented lattices 06C05 Modular lattices, Desarguesian lattices Semimodular lattices, geometric lattices 06C10 06C15 Complemented lattices, orthocomplemented lattices and posets [See also 03G12, 81P10] Complemented modular lattices, continuous geometries None of the above, but in this section Distributive lattices Structure and representation theory Complete distributivity Pseudocomplemented lattices Heyting algebras [See also 03G25] Frames, locales {For topological questions see 54–XX} Post algebras [See also 03G20] De Morgan algebras, Lukasiewicz algebras [See also 03G20] Lattices and duality Fuzzy lattices (soft algebras) and related topics Other generalizations of distributive lattices None of the above, but in this section Boolean algebras (Boolean rings) [See also 03G05] Structure theory Chain conditions, complete algebras Stone spaces (Boolean spaces) and related structures Ring-theoretic properties [See also 16E50, 16G30] 06C20 06C99 06Dxx 06D05 06D10 06D15 06D20 06D22 06D25 06D30 06D35 MV-algebras 06D50 06D72 06D75 06D99 06Exx 06E05 06E10 06E15 06E20 [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

06Exx 06E25 06E30 06E75 06E99 06Fxx 06F05 06F07 06F10 06F15 06F20 06F25 06F30 06F35 06F99 08–XX 08–00 08–01 08–02 08–03 08–04 08B26 08B30 08B99 08Cxx 08C05 08C10 08C15 08C20 08C99 11–XX 11–00 11–01 11–02 11–03 11–04 11–06 11Axx GENERAL ALGEBRAIC SYSTEMS General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Algebraic structures [See also 03C05] Relational systems, laws of composition Structure theory Subalgebras, congruence relations Automorphisms, endomorphisms Operations, polynomials, primal algebras Equational compactness 08–06 08Axx 08A02 08A05 08A30 08A35 08A40 08A45 08A50 Word problems [See also 03D40, 06B25, 20F10, 68R15] 08A55 08A60 08A62 08A65 08A68 08A70 08A72 08A99 08Bxx 08B05 08B10 08B15 08B20 08B25 Partial algebras Unary algebras Finitary algebras Inﬁnitary algebras Heterogeneous algebras Applications of universal algebra in computer science Fuzzy algebraic structures None of the above, but in this section Varieties [See also 03C05] Equational logic, Malcev (Maltsev) conditions Congruence modularity, congruence distributivity Lattices of varieties Free algebras Products, amalgamated products, and other kinds of limits and colimits [See also 18A30] Subdirect products and subdirect irreducibility Injectives, projectives None of the above, but in this section Other classes of algebras Categories of algebras [See also 18C05] Axiomatic model classes [See also 03Cxx, in particular 03C60] Quasivarieties Natural dualities for classes of algebras [See also 06E15, 18A40, 22A30] None of the above, but in this section NUMBER THEORY General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Elementary number theory {For analogues in number ﬁelds, see 11R04} 11A05 Multiplicative structure; Euclidean algorithm; greatest common 11A07 11A15 11A25 11A41 divisors Congruences; primitive roots; residue systems Power residues, reciprocity Arithmetic functions; related numbers; inversion formulas Primes MSC2010: FINAL PUBLIC VERSION [Oct. 2009] S4 Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45] Boolean functions [See also 94C10] Generalizations of Boolean algebras None of the above, but in this section Ordered structures Ordered semigroups and monoids [See also 20Mxx] Quantales Noether lattices Ordered groups [See also 20F60] Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] Ordered rings, algebras, modules {For ordered ﬁelds, see 12J15; see also 13J25, 16W80} Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12] BCK-algebras, BCI-algebras [See also 03G25] None of the above, but in this section 11A51 11A55 11A63 Factorization; primality Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] Radix representation; digital problems {For metric results, see 11K16} Other representations None of the above, but in this section Sequences and sets Density, gaps, topology Additive bases, including sumsets [See also 05B10] Arithmetic progressions [See also 11N13] Arithmetic combinatorics; higher degree uniformity Representation functions Recurrences {For applications to special functions, see 33–XX} Fibonacci and Lucas numbers and polynomials and generalizations Sequences (mod m) Farey sequences; the sequences 1k, 2k,··· Binomial coeﬃcients; factorials; q-identities [See also 05A10, 05A30] Bernoulli and Euler numbers and polynomials Bell and Stirling numbers Other combinatorial number theory Special sequences and polynomials Automata sequences None of the above, but in this section Polynomials and matrices Polynomials [See also 13F20] 11A67 11A99 11Bxx 11B05 11B13 11B25 11B30 11B34 11B37 11B39 11B50 11B57 11B65 11B68 11B73 11B75 11B83 11B85 11B99 11Cxx 11C08 11C20 Matrices, determinants [See also 15B36] 11C99 11Dxx 11D04 11D07 11D09 11D25 11D41 11D45 11D57 Multiplicative and norm form equations 11D59 11D61 11D68 11D72 11D75 11D79 11D85 11D88 11D99 11Exx None of the above, but in this section Diophantine equations [See also 11Gxx, 14Gxx] Linear equations The Frobenius problem Quadratic and bilinear equations Cubic and quartic equations Higher degree equations; Fermat’s equation Counting solutions of Diophantine equations Thue-Mahler equations Exponential equations Rational numbers as sums of fractions Equations in many variables [See also 11P55] Diophantine inequalities [See also 11J25] Congruences in many variables Representation problems [See also 11P55] p-adic and power series ﬁelds None of the above, but in this section Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63} Quadratic forms over general ﬁelds Quadratic forms over local rings and ﬁelds Forms over real ﬁelds Quadratic forms over global rings and ﬁelds General binary quadratic forms General ternary and quaternary quadratic forms; forms of more than two variables Sums of squares and representations by other particular quadratic forms Bilinear and Hermitian forms Class numbers of quadratic and Hermitian forms Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) Classical groups [See also 14Lxx, 20Gxx] K-theory of quadratic and Hermitian forms Galois cohomology of linear algebraic groups [See also 20G10] Forms of degree higher than two Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24] Quadratic spaces; Cliﬀord algebras [See also 15A63, 15A66] p-adic theory None of the above, but in this section Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} Modular and automorphic functions Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40] Holomorphic modular forms of integral weight Automorphic forms, one variable Dedekind eta function, Dedekind sums Relationship to Lie algebras and ﬁnite simple groups Relations with algebraic geometry and topology 11E04 11E08 11E10 11E12 11E16 11E20 11E25 11E39 11E41 11E45 11E57 11E70 11E72 11E76 11E81 11E88 11E95 11E99 11Fxx 11F03 11F06 11F11 11F12 11F20 11F22 11F23 [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

S5 MSC2010: FINAL PUBLIC VERSION [Oct. 2009] 11Pxx 11F25 11F27 11F30 11F32 11F33 11F37 11F41 11F46 11F50 11F52 11F55 11F60 11F66 11F67 11F68 11F70 11F72 11F75 11F80 11F85 11F99 11Gxx 11G05 11G07 11G09 11G10 11G15 11G16 11G18 11G20 11G25 11G30 11G32 11G35 11G40 Hecke-Petersson operators, diﬀerential operators (one variable) Theta series; Weil representation; theta correspondences Fourier coeﬃcients of automorphic forms Modular correspondences, etc. Congruences for modular and p-adic modular forms [See also 14G20, 22E50] Forms of half-integer weight; nonholomorphic modular forms Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms Jacobi forms Modular forms associated to Drinfeld modules Other groups and their modular and automorphic forms (several variables) Hecke-Petersson operators, diﬀerential operators (several variables) Langlands L-functions; one variable Dirichlet series and functional equations Special values of automorphic L-series, periods of modular forms, cohomology, modular symbols Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series Representation-theoretic methods; automorphic representations over local and global ﬁelds Spectral theory; Selberg trace formula Cohomology of arithmetic groups Galois representations p-adic theory, local ﬁelds [See also 14G20, 22E50] None of the above, but in this section Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14Gxx, 14Kxx] Elliptic curves over global ﬁelds [See also 14H52] Elliptic curves over local ﬁelds [See also 14G20, 14H52] Drinfeld modules; higher-dimensional motives, etc. [See also 14L05] Abelian varieties of dimension > 1 [See also 14Kxx] Complex multiplication and moduli of abelian varieties [See also 14K22] Elliptic and modular units [See also 11R27] Arithmetic aspects of modular and Shimura varieties [See also 14G35] Curves over ﬁnite and local ﬁelds [See also 14H25] Varieties over ﬁnite and local ﬁelds [See also 14G15, 14G20] Curves of arbitrary genus or genus = 1 over global ﬁelds [See also 14H25] Dessins d’enfants, Bely˘ı theory Varieties over global ﬁelds [See also 14G25] L-functions of varieties over global ﬁelds; Birch-Swinnerton-Dyer conjecture [See also 14G10] Arithmetic mirror symmetry [See also 14J33] Geometric class ﬁeld theory [See also 11R37, 14C35, 19F05] Heights [See also 14G40, 37P30] Polylogarithms and relations with K-theory None of the above, but in this section Geometry of numbers {For applications in coding theory, see 94B75} Lattices and convex bodies [See also 11P21, 52C05, 52C07] Nonconvex bodies Lattice packing and covering [See also 05B40, 52C15, 52C17] Products of linear forms 11G42 11G45 11G50 11G55 11G99 11Hxx 11H06 11H16 11H31 11H46 11H50 Minima of forms Quadratic forms (reduction theory, extreme forms, etc.) 11H55 11H56 Automorphism groups of lattices 11H60 Mean value and transfer theorems 11H71 11H99 11Jxx Relations with coding theory None of the above, but in this section Diophantine approximation, transcendental number theory [See also 11K60] Homogeneous approximation to one number Markov and Lagrange spectra and generalizations Simultaneous homogeneous approximation, linear forms Approximation by numbers from a ﬁxed ﬁeld Inhomogeneous linear forms Diophantine inequalities [See also 11D75] Small fractional parts of polynomials and generalizations Approximation in non-Archimedean valuations Approximation to algebraic numbers Continued fractions and generalizations [See also 11A55, 11K50] Distribution modulo one [See also 11K06] Irrationality; linear independence over a ﬁeld Transcendence (general theory) Measures of irrationality and of transcendence 11J04 11J06 11J13 11J17 11J20 11J25 11J54 11J61 11J68 11J70 11J71 11J72 11J81 11J82 11N35 11N36 11N37 11N45 11N56 11N60 11N64 11N69 11N75 11J83 11J85 11J86 11J87 11J89 11J91 11J93 11J95 11J97 11J99 11Kxx 11K06 11K16 Metric theory Algebraic independence; Gelfond’s method Linear forms in logarithms; Baker’s method Schmidt Subspace Theorem and applications Transcendence theory of elliptic and abelian functions Transcendence theory of other special functions Transcendence theory of Drinfeld and t-modules Results involving abelian varieties Analogues of methods in Nevanlinna theory (work of Vojta et al.) None of the above, but in this section Probabilistic theory: distribution modulo 1; metric theory of algorithms General theory of distribution modulo 1 [See also 11J71] Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. [See also 11A63] Special sequences 11K31 11K36 Well-distributed sequences and other variations 11K38 11K41 11K45 11K50 Metric theory of continued fractions [See also 11A55, 11J70] 11K55 Metric theory of other algorithms and expansions; measure and Irregularities of distribution, discrepancy [See also 11Nxx] Continuous, p-adic and abstract analogues Pseudo-random numbers; Monte Carlo methods Hausdorﬀ dimension [See also 11N99, 28Dxx] Diophantine approximation [See also 11Jxx] Arithmetic functions [See also 11Nxx] Harmonic analysis and almost periodicity None of the above, but in this section Exponential sums and character sums {For ﬁnite ﬁelds, see 11Txx} Trigonometric and exponential sums, general Gauss and Kloosterman sums; generalizations Estimates on exponential sums Jacobsthal and Brewer sums; other complete character sums 11K60 11K65 11K70 11K99 11Lxx 11L03 11L05 11L07 11L10 11L15 Weyl sums Sums over primes 11L20 Sums over arbitrary intervals 11L26 Estimates on character sums 11L40 None of the above, but in this section 11L99 Zeta and L-functions: analytic theory 11Mxx ζ(s) and L(s, χ) 11M06 Real zeros of L(s, χ); results on L(1, χ) 11M20 11M26 Nonreal zeros of ζ(s) and L(s, χ); Riemann and other hypotheses 11M32 Multiple Dirichlet series and zeta functions and multizeta values 11M35 11M36 Hurwitz and Lerch zeta functions Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas Zeta and L-functions in characteristic p Other Dirichlet series and zeta functions {For local and global ground ﬁelds, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} Tauberian theorems [See also 40E05] Relations with random matrices Relations with noncommutative geometry None of the above, but in this section 11M38 11M41 11M45 11M50 11M55 11M99 11Nxx Multiplicative number theory 11N05 11N13 11N25 11N30 11N32 Distribution of primes Primes in progressions [See also 11B25] Distribution of integers with speciﬁed multiplicative constraints Tur´an theory [See also 30Bxx] Primes represented by polynomials; other multiplicative structure of polynomial values Sieves Applications of sieve methods Asymptotic results on arithmetic functions Asymptotic results on counting functions for algebraic and topological structures Rate of growth of arithmetic functions Distribution functions associated with additive and positive multiplicative functions Other results on the distribution of values or the characterization of arithmetic functions Distribution of integers in special residue classes Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx] Generalized primes and integers None of the above, but in this section Additive number theory; partitions Lattice points in speciﬁed regions Goldbach-type theorems; other additive questions involving primes 11N80 11N99 11Pxx 11P05 Waring’s problem and variants 11P21 11P32 [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

11Pxx MSC2010: FINAL PUBLIC VERSION [Oct. 2009] S6 11R04 11R06 11R34 11R37 11R39 11R42 11R44 11R45 11R47 11R52 11R54 11P55 11P70 11P81 11P82 11P83 11P84 11P99 11Rxx 11R09 11R11 11R16 11R18 11R20 11R21 11R23 11R27 11R29 11R32 11R33 Applications of the Hardy-Littlewood method [See also 11D85] Inverse problems of additive number theory, including sumsets Elementary theory of partitions [See also 05A17] Analytic theory of partitions Partitions; congruences and congruential restrictions Partition identities; identities of Rogers-Ramanujan type None of the above, but in this section Algebraic number theory: global ﬁelds {For complex multiplication, see 11G15} Algebraic numbers; rings of algebraic integers PV-numbers and generalizations; other special algebraic numbers; Mahler measure Polynomials (irreducibility, etc.) Quadratic extensions Cubic and quartic extensions Cyclotomic extensions Other abelian and metabelian extensions Other number ﬁelds Iwasawa theory Units and factorization Class numbers, class groups, discriminants Galois theory Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] Galois cohomology [See also 12Gxx, 19A31] Class ﬁeld theory Langlands-Weil conjectures, nonabelian class ﬁeld theory [See also 11Fxx, 22E55] Zeta functions and L-functions of number ﬁelds [See also 11M41, 19F27] Distribution of prime ideals [See also 11N05] Density theorems Other analytic theory [See also 11Nxx] Quaternion and other division algebras: arithmetic, zeta functions Other algebras and orders, and their zeta and L-functions [See also 11S45, 16Hxx, 16Kxx] Ad`ele rings and groups Arithmetic theory of algebraic function ﬁelds [See also 14–XX] Cyclotomic function ﬁelds (class groups, Bernoulli objects, etc.) Class groups and Picard groups of orders K-theory of global ﬁelds [See also 19Fxx] Totally real ﬁelds [See also 12J15] None of the above, but in this section Algebraic number theory: local and p-adic ﬁelds Polynomials Ramiﬁcation and extension theory Galois theory Integral representations Galois cohomology [See also 12Gxx, 16H05] Class ﬁeld theory; p-adic formal groups [See also 14L05] Langlands-Weil conjectures, nonabelian class ﬁeld theory [See also 11Fxx, 22E50] Zeta functions and L-functions [See also 11M41, 19F27] Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16Hxx, 16Kxx] K-theory of local ﬁelds [See also 19Fxx] Other analytic theory (analogues of beta and gamma functions, p- adic integration, etc.) Non-Archimedean dynamical systems [See mainly 37Pxx] 11S82 Other nonanalytic theory 11S85 Prehomogeneous vector spaces 11S90 None of the above, but in this section 11S99 Finite ﬁelds and commutative rings (number-theoretic aspects) 11Txx Polynomials 11T06 Cyclotomy 11T22 Exponential sums 11T23 Other character sums and Gauss sums 11T24 Structure theory 11T30 Arithmetic theory of polynomial rings over ﬁnite ﬁelds 11T55 Finite upper half-planes 11T60 Algebraic coding theory; cryptography 11T71 None of the above, but in this section 11T99 Connections with logic 11Uxx Decidability [See also 03B25] 11U05 11U07 Ultraproducts [See also 03C20] 11U09 Model theory [See also 03Cxx] 11U10 11U99 11R56 11R58 11R60 11R65 11R70 11R80 11R99 11Sxx 11S05 11S15 11S20 11S23 11S25 11S31 11S37 Nonstandard arithmetic [See also 03H15] None of the above, but in this section 11S40 11S45 11S70 11S80 11Yxx 11Y05 11Y11 11Y16 11Y35 11Y40 11Y50 11Y55 11Y60 11Y65 11Y70 11Y99 11Zxx 11Z05 11Z99 12–XX 12–00 12–01 12–02 12–03 12–04 12–06 12Dxx 12D05 12D10 12D15 12D99 12Exx 12E05 12E10 12E12 12E15 12E20 12E25 12E30 12E99 12Fxx 12F05 12F10 12F12 12F15 12F20 12F99 12Gxx 12G05 12G10 12G99 12Hxx 12H05 12H10 12H20 12H25 12H99 12Jxx 12J05 12J10 12J12 12J15 12J17 12J20 12J25 12J27 12J99 12Kxx 12K05 12K10 12K99 12Lxx 12L05 12L10 12L12 12L15 12L99 Computational number theory [See also 11–04] Factorization Primality Algorithms; complexity [See also 68Q25] Analytic computations Algebraic number theory computations Computer solution of Diophantine equations Calculation of integer sequences Evaluation of constants Continued fraction calculations Values of arithmetic functions; tables None of the above, but in this section Miscellaneous applications of number theory Miscellaneous applications of number theory None of the above, but in this section FIELD THEORY AND POLYNOMIALS General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Real and complex ﬁelds Polynomials: factorization Polynomials: location of zeros (algebraic theorems) {For the analytic theory, see 26C10, 30C15} Fields related with sums of squares (formally real ﬁelds, Pythagorean ﬁelds, etc.) [See also 11Exx] None of the above, but in this section General ﬁeld theory Polynomials (irreducibility, etc.) Special polynomials Equations Skew ﬁelds, division rings [See also 11R52, 11R54, 11S45, 16Kxx] Finite ﬁelds (ﬁeld-theoretic aspects) Hilbertian ﬁelds; Hilbert’s irreducibility theorem Field arithmetic None of the above, but in this section Field extensions Algebraic extensions Separable extensions, Galois theory Inverse Galois theory Inseparable extensions Transcendental extensions None of the above, but in this section Homological methods (ﬁeld theory) Galois cohomology [See also 14F22, 16Hxx, 16K50] Cohomological dimension None of the above, but in this section Diﬀerential and diﬀerence algebra Diﬀerential algebra [See also 13Nxx] Diﬀerence algebra [See also 39Axx] Abstract diﬀerential equations [See also 34Mxx] p-adic diﬀerential equations [See also 11S80, 14G20] None of the above, but in this section Topological ﬁelds Normed ﬁelds Valued ﬁelds Formally p-adic ﬁelds Ordered ﬁelds Topological semiﬁelds General valuation theory [See also 13A18] Non-Archimedean valued ﬁelds [See also 30G06, 32P05, 46S10, 47S10] Krasner-Tate algebras [See mainly 32P05; see also 46S10, 47S10] None of the above, but in this section Generalizations of ﬁelds Near-ﬁelds [See also 16Y30] Semiﬁelds [See also 16Y60] None of the above, but in this section Connections with logic Decidability [See also 03B25] Ultraproducts [See also 03C20] Model theory [See also 03C60] Nonstandard arithmetic [See also 03H15] None of the above, but in this section [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

S7 12Yxx 12Y05 12Y99 13–XX 13–00 13–01 13–02 13–03 13–04 13–06 13Axx 13A02 13A05 13A15 13A18 13A30 13A35 13A50 13B25 13B30 13B35 13B40 13B99 13Cxx 13C05 13C10 13C11 13C12 13C13 13C14 13C15 13C20 13C40 13D02 13D03 13D05 13D07 13D09 13D10 13D15 13D22 13D30 13D40 13D45 13D99 13Exx 13E05 13E10 13E15 13E99 13Fxx 13F05 13F07 13F10 13A99 13Bxx 13B02 13B05 13B10 Morphisms 13B21 13B22 13C60 Module categories 13C99 13Dxx COMMUTATIVE ALGEBRA General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. General commutative ring theory Graded rings [See also 16W50] Divisibility; factorizations [See also 13F15] Ideals; multiplicative ideal theory Valuations and their generalizations [See also 12J20] Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22] Actions of groups on commutative rings; invariant theory [See also 14L24] None of the above, but in this section Ring extensions and related topics Extension theory Galois theory Integral dependence; going up, going down Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] Rings of fractions and localization [See also 16S85] Completion [See also 13J10] ´Etale and ﬂat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] None of the above, but in this section Theory of modules and ideals Structure, classiﬁcation theorems Projective and free modules and ideals [See also 19A13] Injective and ﬂat modules and ideals Torsion modules and ideals Other special types Cohen-Macaulay modules [See also 13H10] Dimension theory, depth, related rings (catenary, etc.) Class groups [See also 11R29] Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] None of the above, but in this section Homological methods {For noncommutative rings, see 16Exx; for general categories, see 18Gxx} Syzygies, resolutions, complexes (Co)homology of commutative rings and algebras (e.g., Hochschild, Andr´e-Quillen, cyclic, dihedral, etc.) Homological dimension Homological functors on modules (Tor, Ext, etc.) Derived categories Deformations and inﬁnitesimal methods [See also 14B10, 14B12, 14D15, 32Gxx] Grothendieck groups, K-theory [See also 14C35, 18F30, 19Axx, 19D50] Homological conjectures (intersection theorems) Torsion theory [See also 13C12, 18E40] Hilbert-Samuel and Hilbert-Kunz functions; Poincar´e series Local cohomology [See also 14B15] None of the above, but in this section Chain conditions, ﬁniteness conditions Noetherian rings and modules Artinian rings and modules, ﬁnite-dimensional algebras Rings and modules of ﬁnite generation or presentation; number of generators None of the above, but in this section Arithmetic rings and other special rings Dedekind, Pr¨ufer, Krull and Mori rings and their generalizations Euclidean rings and generalizations Principal ideal rings MSC2010: FINAL PUBLIC VERSION [Oct. 2009] 14Bxx Computational aspects of ﬁeld theory and polynomials Computational aspects of ﬁeld theory and polynomials None of the above, but in this section 13F15 13F20 Rings deﬁned by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05] Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] Formal power series rings [See also 13J05] Valuation rings [See also 13A18] 13F25 13F30 13F35 Witt vectors and related rings 13F40 13F45 13F50 13F55 13F60 13F99 13Gxx 13G05 13G99 13Hxx 13H05 13H10 Excellent rings Seminormal rings Rings with straightening laws, Hodge algebras Stanley-Reisner face rings; simplicial complexes [See also 55U10] Cluster algebras None of the above, but in this section Integral domains Integral domains None of the above, but in this section Local rings and semilocal rings Regular local rings Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13H15 Multiplicity theory and related topics [See also 14C17] 13H99 13Jxx 13J05 13J07 13J10 13J15 13J20 13J25 13J30 13J99 13Lxx 13L05 13L99 13Mxx 13M05 13M10 13M99 13Nxx 13N05 Modules of diﬀerentials 13N10 None of the above, but in this section Topological rings and modules [See also 16W60, 16W80] Power series rings [See also 13F25] Analytical algebras and rings [See also 32B05] Complete rings, completion [See also 13B35] Henselian rings [See also 13B40] Global topological rings Ordered rings [See also 06F25] Real algebra [See also 12D15, 14Pxx] None of the above, but in this section Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] None of the above, but in this section Finite commutative rings {For number-theoretic aspects, see 11Txx} Structure Polynomials None of the above, but in this section Diﬀerential algebra [See also 12H05, 14F10] Rings of diﬀerential operators and their modules [See also 16S32, 32C38] Derivations None of the above, but in this section Computational aspects and applications [See also 14Qxx, 68W30] Polynomials, factorization [See also 12Y05] Gr¨obner bases; other bases for ideals and modules (e.g., Janet and border bases) Solving polynomial systems; resultants Computational homological algebra [See also 13Dxx] Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) None of the above, but in this section ALGEBRAIC GEOMETRY General reference works (handbooks, dictionaries, bibliographies, etc.) Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) Historical (must also be assigned at least one classiﬁcation number from Section 01) Explicit machine computation and programs (not the theory of computation or programming) Proceedings, conferences, collections, etc. Foundations Relevant commutative algebra [See also 13–XX] Varieties and morphisms Schemes and morphisms Generalizations (algebraic spaces, stacks) Noncommutative algebraic geometry [See also 16S38] Elementary questions None of the above, but in this section Local theory Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] Deformations of singularities [See also 14D15, 32S30] Inﬁnitesimal methods [See also 13D10] Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10] Local cohomology [See also 13D45, 32C36] Formal neighborhoods Local structure of morphisms: ´etale, ﬂat, etc. [See also 13B40] 13N15 13N99 13Pxx 13P05 13P10 13P15 13P20 13P25 13P99 14–XX 14–00 14–01 14–02 14–03 14–04 14–06 14Axx 14A05 14A10 14A15 14A20 14A22 14A25 14A99 14Bxx 14B05 14B07 14B10 14B12 14B15 14B20 14B25 [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.] [Source Date: Monday 12 October 2009 21:56]

14Bxx MSC2010: FINAL PUBLIC VERSION [Oct. 2009] S8 14B99 14Cxx 14C05 14C15 14C17 14C20 14C21 14C22 14C25 14C30 14C34 14C35 14C40 14C99 14Dxx 14D05 14D06 14D07 14D10 14D15 14D20 14D21 14D22 14D23 14D24 14D99 14Exx 14E05 14E07 14E08 14E15 14F10 14F17 14F18 14F20 14F22 14F25 14F30 14F35 14F40 14F42 14F43 14F45 14F99 14Gxx 14G05 14G10 14G15 14G17 14G20 14G22 14G25 14G27 14G32 Arcs and motivic integration Coverings [See also 14H30] Ramiﬁcation problems [See also 11S15] Embeddings 14E16 McKay correspondence 14E18 14E20 14E22 14E25 14E30 Minimal model program (Mori theory, extremal rays) 14E99 14Fxx 14F05 None of the above, but in this section (Co)homology theory [See also 13Dxx] Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] Diﬀerentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38] Vanishing theorems [See also 32L20] Multiplier ideals ´Etale and other Grothendieck topologies and (co)homologies Brauer groups of schemes [See also 12G05, 16K50] Classical real and complex (co)homology p-adic cohomology, crystalline cohomology Homotopy theory; fundamental groups [See also 14H30] de Rham cohomology [See also 14C30, 32C35, 32L10] Motivic cohomology; motivic homotopy theory [See also 19E15] Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Topological properties None of the above, but in this section Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx] Rational points Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture) Finite ground ﬁelds Positive characteristic ground ﬁelds Local ground ﬁelds Rigid analytic geometry Global ground ﬁelds Other nonalgebraically closed ground ﬁelds Universal proﬁnite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) 14G35 Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 14G40 Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40] 14G50 None of the above, but in this section Cycles and subschemes Parametrization (Chow and Hilbert schemes) (Equivariant) Chow groups and rings; motives Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] Divisors, linear systems, invertible sheaves Pencils, nets, webs [See also 53A60] Picard groups Algebraic cycles Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture Torelli problem [See also 32G20] Applications of methods of algebraic K-theory [See also 19Exx] Riemann-Roch theorems [See also 19E20, 19L10] None of the above, but in this section Families, ﬁbrations Structure of families (Picard-Lefschetz, monodromy, etc.) Fibrations, degenerations Variation of Hodge structures [See also 32G20] Arithmetic ground ﬁelds (ﬁnite, local, global) Formal methods; deformations [See also 13D10, 14B07, 32Gxx] Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum ﬁeld theory) [See also 32L25, 81Txx] Fine and coarse moduli spaces Stacks and moduli problems Geometric Langlands program: algebro-geometric aspects [See also 22E57] None of the above, but in this section Birational geometry Rational and birational maps Birational automorphisms, Cremona group and generalizations Rationality questions [See also 14M20] Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14J10 14J15 14J17 14J20 14J25 14J26 14J27 14J28 14J29 14J30 14J32 14J33 14J35 14J40 14J45 14J50 14J60 14G99 14Hxx 14H05 14H10 14H15 14H20 14H25 14H30 14H37 14H40 14H42 14H45 14H50 14H51 14H52 14H55 14H57 14H60 14H70 14H81 14H99 14Jxx None of the above, but in this section Curves Algebraic functions; function ﬁelds [See also 11R58] Families, moduli (algebraic) Families, moduli (analytic) [See also 30F10, 32G15] Singularities, local rings [See also 13Hxx, 14B05] Arithmetic ground ﬁelds [See also 11Dxx, 11G05, 14Gxx] Coverings, fundamental group [See also 14E20, 14F35] Automorphisms Jacobians, Prym varieties [See also 32G20] Theta functions; Schottky problem [See also 14K25, 32G20] Special curves and curves of low genus Plane and space curves Special divisors (gonality, Brill-Noether theory) Elliptic curves [See also 11G05, 11G07, 14Kxx] Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx] Dessins d’enfants theory {For arithmetic aspects, see 11G32} Vector bundles on curves and their moduli [See also 14D20, 14F05] Relationships with integrable systems Relationships with physics None of the above, but in this section Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx} Families, moduli, classiﬁcation: algebraic theory Moduli, classiﬁcation: analytic theory; relations with modular forms [See also 32G13] Singularities [See also 14B05, 14E15] Arithmetic ground ﬁelds [See also 11Dxx, 11G25, 11G35, 14Gxx] Special surfaces {For Hilbert modular surfaces, see 14G35} Rational and ruled surfaces Elliptic surfaces K3 surfaces and Enriques surfaces Surfaces of general type 3-folds [See also 32Q25] Calabi-Yau manifolds Mirror symmetry [See also 11G42, 53D37] 4-folds n-folds (n > 4) Fano varieties Automorphisms of surfaces and higher-dimensional varieties Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] Hypersurfaces Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) Relationships with physics None of the above, but in this section Abelian varieties and schemes Isogeny Algebraic theory Algebraic moduli, classiﬁcation [See also 11G15] Subvarieties Arithmetic ground ﬁelds [See also 11Dxx, 11Fxx, 11G10, 14Gxx] Analytic theory; abelian integrals and diﬀerentials Complex multiplication [See also 11G15] Theta functions [See also 14H42] Picard schemes, higher Jacobians [See also 14H40, 32G20] None of the above, but in this section Algebraic groups {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45} Formal groups, p-divisible groups [See also 55N22] Group varieties Group schemes Aﬃne algebraic groups, hyperalgebra constructions [See also 17B45, 18D35] Geometric invariant theory [See also 13A50] Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] Classical groups (geometric aspects) [See also 20Gxx, 51N30] Other algebraic groups (geometric aspects) None of the above, but in this section Special varieties Varieties deﬁned by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10] Linkage [See also 13C40] Low codimension problems Complete intersections [See also 13C40] Determinantal varieties [See also 13C40] Grassmannians, Schubert varieties, ﬂag manifolds [See also 32M10, 51M35] [Source Date: Monday 12 October 2009 21:56] 14J81 14J99 14Kxx 14K02 14K05 14K10 14K12 14K15 14K20 14K22 14K25 14K30 14K99 14Lxx 14L35 14L40 14L99 14Mxx 14M05 14J70 14J80 14L05 14L10 14L15 14L17 14L24 14L30 14M06 14M07 14M10 14M12 14M15 [Licence: This text is available under the Creative Commons Attribution-Noncommercial-Share Alike License: http://creativecommons.org/licenses/by-nc-sa/3.0/ Additional terms may apply.]

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