General General mathematics Mathematics for nonmathematicians (engineering, social sciences, etc.) Problem books Recreational mathematics [See also 97A20] Bibliographies External book reviews Dictionaries and other general reference works Formularies Philosophy of mathematics [See also 03A05] Methodology of mathematics, didactics [See also 97Cxx, 97Dxx] General applied mathematics {For physics, see 00A79 and Sections 70 through 86} Theory of mathematical modeling General methods of simulation Dimensional analysis Physics (use more specific entries from Sections 70 through 86 when possible) Miscellaneous topics Collections of abstracts of lectures Collections of articles of general interest Collections of articles of miscellaneous specific content Proceedings of conferences of general interest Proceedings of conferences of miscellaneous specific interest Festschriften Volumes of selected translations Miscellaneous volumes of translations Collections of reprinted articles [See also 01A75]    History and biography [See also the classification number -03 General histories, source books Ethnomathematics, general Paleolithic, Neolithic Indigenous cultures of the Americas Other indigenous cultures (non-European) Indigenous European cultures (pre-Greek, etc.) Egyptian Babylonian Greek, Roman China Japan Southeast Asia Islam (Medieval) India Medieval 15th and 16th centuries, Renaissance 17th century 18th century 19th century 20th century Twenty-first century Contemporary Future prospectives Biographies, obituaries, personalia, bibliographies Schools of mathematics Universities Other institutions and academies Collected or selected works; reprintings or translations of classics [See also 00B60] Sociology (and profession) of mathematics Historiography Bibliographic studies Miscellaneous topics    Mathematical logic and foundations Classical propositional logic Classical first-order logic Higher-order logic and type theory Subsystems of classical logic (including intuitionistic logic) Abstract deductive systems Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] Foundations of classical theories (including reverse mathematics) [See also 03F35] Mechanization of proofs and logical operations [See also 68T15] Combinatory logic and lambda-calculus [See also 68N18] Logic of knowledge and belief Temporal logic Modal logic {For knowledge and belief see 03B42; for temporal logic see 03B44; for provability logic see also 03F45} Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} Probability and inductive logic [See also 60A05] Many-valued logic Fuzzy logic; logic of vagueness [See also 68T27, 68T37, 94D05] Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.) Intermediate logics Other nonclassical logic Logic of natural languages [See also 68T50, 91F20] Logic in computer science [See also 68-XX] Other applications of logic None of the above, but in this section Equational classes, universal algebra [See also 08Axx, 08Bxx, 18C05] Basic properties of first-order languages and structures Quantifier elimination, model completeness and related topics Finite structures [See also 68Q15, 68Q19] Denumerable structures Ultraproducts and related constructions Model-theoretic forcing Other model constructions Categoricity and completeness of theories Interpolation, preservation, definability Classification theory, stability and related concepts Models with special properties (saturated, rigid, etc.) Properties of classes of models Set-theoretic model theory Effective and recursion-theoretic model theory [See also 03D45] Model-theoretic algebra [See also 08C10, 12Lxx, 13L05] Models of arithmetic and set theory [See also 03Hxx] Model theory of ordered structures; o-minimality Models of other mathematical theories Other classical first-order model theory Logic on admissible sets Other infinitary logic Logic with extra quantifiers and operators [See also 03B42, 03B44, 03B45, 03B48] Second- and higher-order model theory Nonclassical models (Boolean-valued, sheaf, etc.) Abstract model theory Applications of model theory [See also 03C60] None of the above, but in this section Thue and Post systems, etc. Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15] Turing machines and related notions [See also 68Q05] Complexity of computation [See also 68Q15, 68Q17] Recursive functions and relations, subrecursive hierarchies Recursively (computably) enumerable sets and degrees Other Turing degree structures Other degrees and reducibilities Undecidability and degrees of sets of sentences Word problems, etc. [See also 06B25, 08A50, 20F10, 68R15] Theory of numerations, effectively presented structures [See also 03C57; for intuitionistic and similar approaches see 03F55] Recursive equivalence types of sets and structures, isols Hierarchies Computability and recursion theory on ordinals, admissible sets, etc. Higher-type and set recursion theory Inductive definability Abstract and axiomatic computability and recursion theory Applications of computability and recursion theory None of the above, but in this section Proof theory, general Cut-elimination and normal-form theorems Structure of proofs Functionals in proof theory Recursive ordinals and ordinal notations Complexity of proofs Relative consistency and interpretations First-order arithmetic and fragments Second- and higher-order arithmetic and fragments [See also 03B30] Gödel numberings in proof theory Provability logics and related algebras (e.g., diagonalizable algebras) [See also 03B45, 03G25, 06E25] Metamathematics of constructive systems Linear logic and other substructural logics [See also 03B47] Intuitionistic mathematics Constructive and recursive analysis [See also 03B30, 03D45, 26E40, 46S30, 47S30] Other constructive mathematics [See also 03D45] None of the above, but in this section Boolean algebras [See also 06Exx] Lattices and related structures [See also 06Bxx] Quantum logic [See also 06C15, 81P10] Cylindric and polyadic algebras; relation algebras Lukasiewicz and Post algebras [See also 06D25, 06D30] Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35] Categorical logic, topoi [See also 18B25, 18C05, 18C10] None of the above, but in this section Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] Other applications of nonstandard models (economics, physics, etc.) Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05] None of the above, but in this section    Combinatorics {For finite fields, see 11Txx} Block designs [See also 51E05, 62K10] Triple systems Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13] Orthogonal arrays, Latin squares, Room squares Matrices (incidence, Hadamard, etc.) Finite geometries [See also 51D20, 51Exx] Other designs, configurations [See also 51E30] Matroids, geometric lattices [See also 52B40, 90C27] Packing and covering [See also 11H31, 52C15, 52C17] Tessellation and tiling problems [See also 52C20, 52C22] Polyominoes None of the above, but in this section Trees Degree sequences Topological graph theory, imbedding [See also 57M15, 57M25] Distance in graphs Coloring of graphs and hypergraphs Perfect graphs Directed graphs (digraphs), tournaments Signed, gain and biased graphs Graphs and groups [See also 20F65] Enumeration of graphs and maps Extremal problems [See also 90C35] Paths and cycles [See also 90B10] Connectivity Eulerian and Hamiltonian graphs Graphs and matrices Generalized Ramsey theory Isomorphism problems (reconstruction conjecture, etc.) Graph representations (geometric and intersection representations, etc.) Hypergraphs Dominating sets, independent sets, cliques Factorization, matching, covering and packing Structural characterization of types of graphs Graph labelling (graceful graphs, bandwidth, etc.) Random graphs Graph minors Graph algorithms [See also 68R10, 68W05] Applications None of the above, but in this section Extremal set theory Ramsey theory Transversal (matching) theory Probabilistic methods None of the above, but in this section Symmetric functions Tableaux, representations of the symmetric group [See also 20C30] Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Group actions on designs, geometries and codes Group actions on posets and homology groups of posets [See also 06A11] Association schemes, strongly regular graphs Orthogonal polynomials [See also 33C45, 33C50, 33D45] None of the above, but in this section    Order, lattices, ordered algebraic structures [See also 18B35] Total order Partial order, general Combinatorics of partially ordered sets Algebraic aspects of posets [See also 05E25] Semilattices [See also 20M10; for topological semilattices see 22A26] Galois correspondences, closure operators None of the above, but in this section Structure theory Ideals, congruence relations Representation theory Varieties of lattices Complete lattices, completions Free lattices, projective lattices, word problems [See also 03D40, 08A50, 20F10] Topological lattices, order topologies [See also 06F30, 22A26, 54F05, 54H12] Continuous lattices and posets, applications [See also 06B30, 06D10, 06F30, 18B35, 22A26, 68Q55] None of the above, but in this section Modular lattices, Desarguesian lattices Semimodular lattices, geometric lattices Complemented lattices, orthocomplemented lattices and posets [See also 03G12, 81P10] Complemented modular lattices, continuous geometries None of the above, but in this section Structure and representation theory Complete distributivity Pseudocomplemented lattices Heyting algebras [See also 03G25] Frames, locales {For topological questions see 54-XX} Post algebras [See also 03G20] De Morgan algebras, Lukasiewicz algebras [See also 03G20] MV-algebras Lattices and duality Fuzzy lattices (soft algebras) and related topics None of the above, but in this section Structure theory Chain conditions, complete algebras Stone space and related constructions Ring-theoretic properties [See also 16E50, 16G30] Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45] Boolean functions [See also 94C10] None of the above, but in this section Ordered semigroups and monoids [See also 20Mxx] Quantales Noether lattices Ordered groups [See also 20F60] Ordered abelian groups, Riesz groups, ordered linear spaces [See also 46A40] Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80} Topological lattices, order topologies [See also 06B30, 22A26, 54F05, 54H12] BCK-algebras, BCI-algebras [See also 03G25] None of the above, but in this section    General algebraic systems Relational systems, laws of composition Structure theory Subalgebras, congruence relations Automorphisms, endomorphisms Operations, polynomials, primal algebras Equational compactness Word problems [See also 03D40, 06B25, 20F10, 68R15] Partial algebras Unary algebras Finitary algebras Infinitary algebras Heterogeneous algebras Applications of universal algebra in computer science Fuzzy algebraic structures None of the above, but in this section Equational logic, Malcev (Maltsev) conditions Congruence modularity, congruence distributivity Lattices of varieties Free algebras Products, amalgamated products, and other kinds of limits and colimits [See also 18A30] Subdirect products and subdirect irreducibility Injectives, projectives None of the above, but in this section Categories of algebras [See also 18C05] Axiomatic model classes [See also 03Cxx, in particular 03C60] Quasivarieties None of the above, but in this section    Number theory Multiplicative structure; Euclidean algorithm; greatest common divisors Congruences; primitive roots; residue systems Power residues, reciprocity Arithmetic functions; related numbers; inversion formulas Primes Factorization; primality Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] Radix representation; digital problems {For metric results, see 11K16} Other representations None of the above, but in this section Density, gaps, topology Additive bases [See also 05B10] Arithmetic progressions [See also 11N13] Representation functions Recurrences {For applications to special functions, see 33-XX} Fibonacci and Lucas numbers and polynomials and generalizations Sequences (mod $m$) Farey sequences; the sequences ${1^k, 2^k, \cdots}$ Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] Bernoulli and Euler numbers and polynomials Bell and Stirling numbers Other combinatorial number theory Special sequences and polynomials Automata sequences None of the above, but in this section Polynomials [See also 13F20] Matrices, determinants [See also 15A36] None of the above, but in this section Linear equations Quadratic and bilinear equations Cubic and quartic equations Higher degree equations; Fermat's equation Counting solutions of Diophantine equations Multiplicative and norm form equations Thue-Mahler equations Exponential equations Rational numbers as sums of fractions Equations in many variables [See also 11P55] Diophantine inequalities [See also 11J25] Congruences in many variables Representation problems [See also 11P55] $p$-adic and power series fields None of the above, but in this section Quadratic forms over general fields Quadratic forms over local rings and fields Forms over real fields Quadratic forms over global rings and fields General binary quadratic forms General ternary and quaternary quadratic forms; forms of more than two variables Sums of squares and representations by other particular quadratic forms Bilinear and Hermitian forms Class numbers of quadratic and Hermitian forms Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) Classical groups [See also 14Lxx, 20Gxx] $K$-theory of quadratic and Hermitian forms Galois cohomology of linear algebraic groups [See also 20G10] Forms of degree higher than two Algebraic theory of quadratic forms; Witt groups and rings [See also 19G12, 19G24] Quadratic spaces; Clifford algebras [See also 15A63, 15A66] $p$-adic theory None of the above, but in this section Elliptic curves over global fields [See also 14H52] Elliptic curves over local fields [See also 14G20, 14H52] Drinfeld modules; higher-dimensional motives, etc. [See also 14L05] Abelian varieties of dimension $\gtr 1$ [See also 14Kxx] Complex multiplication and moduli of abelian varieties [See also 14K22] Elliptic and modular units [See also 11R27] Arithmetic aspects of modular and Shimura varieties [See also 14G35] Curves over finite and local fields [See also 14H25] Varieties over finite and local fields [See also 14G15, 14G20] Curves of arbitrary genus or genus $\ne 1$ over global fields [See also 14H25] Varieties over global fields [See also 14G25] $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] Geometric class field theory [See also 11R37, 14C35, 19F05] Heights [See also 14G40] Polylogarithms and relations with $K$-theory None of the above, but in this section Lattices and convex bodies [See also 11P21, 52C05, 52C07] Nonconvex bodies Lattice packing and covering [See also 05B40, 52C15, 52C17] Products of linear forms Minima of forms Quadratic forms (reduction theory, extreme forms, etc.) Automorphism groups of lattices Mean value and transfer theorems Relations with coding theory None of the above, but in this section Homogeneous approximation to one number Markov and Lagrange spectra and generalizations Simultaneous homogeneous approximation, linear forms Approximation by numbers from a fixed field Inhomogeneous linear forms Diophantine inequalities [See also 11D75] Small fractional parts of polynomials and generalizations Approximation in non-Archimedean valuations Approximation to algebraic numbers Continued fractions and generalizations [See also 11A55, 11K50] Distribution modulo one [See also 11K06] Irrationality; linear independence over a field Transcendence (general theory) Measures of irrationality and of transcendence Metric theory Algebraic independence; Gelfond's method Linear forms in logarithms; Baker's method Transcendence theory of elliptic and abelian functions Transcendence theory of other special functions Transcendence theory of Drinfel'd and $t$-modules Results involving abelian varieties Analogues of methods in Nevanlinna theory (work of Vojta et al.) None of the above, but in this section Trigonometric and exponential sums, general Gauss and Kloosterman sums; generalizations Estimates on exponential sums Jacobsthal and Brewer sums; other complete character sums Weyl sums Sums over primes Sums over arbitrary intervals Estimates on character sums None of the above, but in this section $\zeta (s)$ and $L(s, \chi)$ Real zeros of $L(s, \chi)$; results on $L(1, \chi)$ Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses Hurwitz and Lerch zeta functions Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas Zeta and $L$-functions in characteristic $p$ Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} Tauberian theorems [See also 40E05] None of the above, but in this section Distribution of primes Primes in progressions [See also 11B25] Distribution of integers with specified multiplicative constraints Turán theory [See also 30Bxx] Primes represented by polynomials; other multiplicative structure of polynomial values Sieves Applications of sieve methods Asymptotic results on arithmetic functions Asymptotic results on counting functions for algebraic and topological structures Rate of growth of arithmetic functions Distribution functions associated with additive and positive multiplicative functions Other results on the distribution of values or the characterization of arithmetic functions Distribution of integers in special residue classes Applications of automorphic functions and forms to multiplicative problems [See also 11Fxx] Generalized primes and integers None of the above, but in this section Waring's problem and variants Lattice points in specified regions Goldbach-type theorems; other additive questions involving primes Applications of the Hardy-Littlewood method [See also 11D85] Inverse problems of additive number theory Elementary theory of partitions [See also 05A17] Analytic theory of partitions Partitions; congruences and congruential restrictions None of the above, but in this section Algebraic numbers; rings of algebraic integers PV-numbers and generalizations; other special algebraic numbers Polynomials (irreducibility, etc.) Quadratic extensions Cubic and quartic extensions Cyclotomic extensions Other abelian and metabelian extensions Other number fields Iwasawa theory Units and factorization Class numbers, class groups, discriminants Galois theory Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] Galois cohomology [See also 12Gxx, 16H05, 19A31] Class field theory Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55] Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] Distribution of prime ideals [See also 11N05] Density theorems Other analytic theory [See also 11Nxx] Quaternion and other division algebras: arithmetic, zeta functions Other algebras and orders, and their zeta and $L$-functions [See also 11S45, 16H05, 16Kxx] Adèle rings and groups Arithmetic theory of algebraic function fields [See also 14-XX] Cyclotomic function fields (class groups, Bernoulli objects, etc.) Class groups and Picard groups of orders $K$-theory of global fields [See also 19Fxx] Totally real and totally positive fields [See also 12J15] None of the above, but in this section Polynomials Ramification and extension theory Galois theory Integral representations Galois cohomology [See also 12Gxx, 16H05] Class field theory; $p$-adic formal groups [See also 14L05] Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50] Zeta functions and $L$-functions [See also 11M41, 19F27] Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16H05, 16Kxx] $K$-theory of local fields [See also 19Fxx] Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) Other nonanalytic theory Prehomogeneous vector spaces None of the above, but in this section Decidability [See also 03B25] Ultraproducts [See also 03C20] Model theory [See also 03Cxx] Nonstandard arithmetic [See also 03H15] None of the above, but in this section Factorization Primality Algorithms; complexity [See also 68Q25] Analytic computations Algebraic number theory computations Computer solution of Diophantine equations Calculation of integer sequences Evaluation of constants Continued fraction calculations Values of arithmetic functions; tables None of the above, but in this section    Field theory and polynomials Polynomials: factorization Polynomials: location of zeros (algebraic theorems) {For the analytic theory, see 26C10, 30C15} Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx] None of the above, but in this section Polynomials (irreducibility, etc.) Special polynomials Equations Skew fields, division rings [See also 11R52, 11R54, 11S45, 16Kxx] Finite fields (field-theoretic aspects) Hilbertian fields; Hilbert's irreducibility theorem Field arithmetic None of the above, but in this section Algebraic extensions Separable extensions, Galois theory Inverse Galois theory Inseparable extensions Transcendental extensions None of the above, but in this section Galois cohomology [See also 14F22, 16H05, 16K50] Cohomological dimension None of the above, but in this section Differential algebra [See also 13Nxx] Difference algebra [See also 39Axx] Abstract differential equations [See also 34Mxx] $p$-adic differential equations [See also 11S80, 14G20] None of the above, but in this section Normed fields Valued fields Formally $p$-adic fields Ordered fields Topological semifields General valuation theory [See also 13A18] Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10] Krasner-Tate algebras [See mainly 32P05; see also 46S10, 47S10] None of the above, but in this section Near-fields [See also 16Y30] Semifields [See also 16Y60] None of the above, but in this section Decidability [See also 03B25] Ultraproducts [See also 03C20] Model theory [See also 03C60] Nonstandard arithmetic [See also 03H15] None of the above, but in this section    Commutative rings and algebras Graded rings [See also 16W50] Divisibility Radical theory Ideals; multiplicative ideal theory Valuations and their generalizations [See also 12J20] Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure [See also 13B22] Actions of groups on commutative rings; invariant theory [See also 14L24] None of the above, but in this section Extension theory Galois theory Morphisms Integral dependence Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) Going up; going down; going between Polynomials over commutative rings [See also 11C08, 13F20, 13M10] Quotients and localization Completion [See also 13J10] Étale and flat extensions; Henselization; Artin approximation [See also 13J15, 14B12, 14B25] None of the above, but in this section Structure, classification theorems Projective and free modules and ideals [See also 19A13] Injective and flat modules and ideals Torsion modules and ideals Other special types Cohen-Macaulay modules [See also 13H10] Dimension theory, depth, related rings (catenary, etc.) Class groups [See also 11R29] Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] None of the above, but in this section Noetherian rings and modules Artinian rings and modules, finite-dimensional algebras Rings and modules of finite generation or presentation; number of generators None of the above, but in this section Dedekind, Prüfer and Krull rings and their generalizations Euclidean rings and generalizations Principal ideal rings Factorial rings, unique factorization domains [See also 14M05] Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] Formal power series rings [See also 13J05] Valuation rings [See also 13A18] Excellent rings Seminormal rings Rings with straightening laws, Hodge algebras Face and Stanley-Reisner rings; simplicial complexes [See also 55U10] None of the above, but in this section Regular local rings Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] Multiplicity theory and related topics [See also 14C17] None of the above, but in this section Power series rings [See also 13F25] Analytical algebras and rings [See also 32B05] Complete rings, completion [See also 13B35] Henselian rings [See also 13B40] Global topological rings Ordered rings [See also 06F25] Real algebra [See also 12D15, 14Pxx] None of the above, but in this section Structure Polynomials None of the above, but in this section Modules of differentials Rings of differential operators and their modules [See also 16S32, 32C38] Derivations None of the above, but in this section Polynomials, factorization [See also 12Y05] Polynomial ideals, Gröbner bases [See also 13F20] None of the above, but in this section    Algebraic geometry Relevant commutative algebra [See also 13-XX] Varieties and morphisms Schemes and morphisms Generalizations (algebraic spaces, stacks) Noncommutative algebraic geometry Elementary questions None of the above, but in this section Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] Deformations of singularities [See also 14D15, 32S30] Infinitesimal methods [See also 13D10] Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10] Local cohomology [See also 13D45, 32C36] Formal neighborhoods Local structure of morphisms: étale, flat, etc. [See also 13B40] None of the above, but in this section Parametrization (Chow and Hilbert schemes) Chow groups and rings Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] Divisors, linear systems, invertible sheaves Pencils, nets, webs [See also 53A60] Picard groups Algebraic cycles Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture Torelli problem [See also 32G20] Applications of methods of algebraic $K$-theory [See also 19Exx] Riemann-Roch theorems [See also 19E20, 19L10] None of the above, but in this section Structure of families (Picard-Lefschetz, monodromy, etc.) Fibrations, degenerations Variation of Hodge structures [See also 32G20] Arithmetic ground fields (finite, local, global) Formal methods; deformations [See also 13D10, 14B07, 32Gxx] Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Fine and coarse moduli spaces None of the above, but in this section Rational and birational maps Birational automorphisms, Cremona group and generalizations Rationality questions Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] Coverings [See also 14H30] Ramification problems [See also 11S15] Embeddings Minimal model program (Mori theory, extremal rays) None of the above, but in this section Vector bundles, sheaves, related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] Differentials and other special sheaves [See also 13Nxx, 32C38] Vanishing theorems [See also 32L20] Étale and other Grothendieck topologies and cohomologies Brauer groups of schemes [See also 12G05, 16K50] Classical real and complex cohomology $p$-adic cohomology, crystalline cohomology Homotopy theory; fundamental groups [See also 14H30] de Rham cohomology [See also 14C30, 32C35, 32L10] Motivic cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Topological properties None of the above, but in this section Rational points Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture) Finite ground fields Local ground fields Rigid analytic geometry Global ground fields Other nonalgebraically closed ground fields Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Modular and Shimura varieties [See also 11F41, 11F46, 11G18] Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50] Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40] None of the above, but in this section Algebraic functions; function fields [See also 11R58] Families, moduli (algebraic) Families, moduli (analytic) [See also 30F10, 32Gxx] Singularities, local rings [See also 13Hxx, 14B05] Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx] Coverings, fundamental group [See also 14E20, 14F35] Automorphisms Jacobians, Prym varieties [See also 32G20] Theta functions; Schottky problem [See also 14K25, 32G20] Special curves and curves of low genus Plane and space curves Special divisors (gonality, Brill-Noether theory) Elliptic curves [See also 11G05, 11G07, 14Kxx] Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx] Vector bundles on curves and their moduli [See also 14D20, 14F05] Relationships with integrable systems Relationships with physics None of the above, but in this section Families, moduli, classification: algebraic theory Moduli, classification: analytic theory; relations with modular forms [See also 32G13] Singularities [See also 14B05, 14E15] Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] Special surfaces {For Hilbert modular surfaces, see 14G35} Rational and ruled surfaces Elliptic surfaces $K3$ surfaces and Enriques surfaces Surfaces of general type $3$-folds Calabi-Yau manifolds, mirror symmetry $4$-folds $n$-folds ($n>4$) Fano varieties Automorphisms of surfaces and higher-dimensional varieties Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]