# The Map of Math

## MSC Classification Codes

The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme formulated by the American Mathematical Society. I am using it here to classify all the mathematics on this website.

## Order, lattices, ordered algebraic structures

• 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
• Galois correspondences, closure operators
• None of the above, but in this section
• Lattices
• Structure theory
• Ideals, congruence relations
• Representation theory
• Varieties of lattices
• Complete lattices, completions
• Free lattices, projective lattices, word problems
• Topological lattices, order topologies
• Continuous lattices and posets, applications
• None of the above, but in this section
• Modular lattices, complemented lattices
• Modular lattices, Desarguesian lattices
• Semimodular lattices, geometric lattices
• Complemented lattices, orthocomplemented lattices and posets
• 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
• Frames, locales
• Post algebras
• De Morgan algebras, Lukasiewicz algebras
• MV-algebras
• Lattices and duality
• Fuzzy lattices (soft algebras) and related topics
• None of the above, but in this section
• Boolean algebras (Boolean rings)
• Structure theory
• Chain conditions, complete algebras
• Stone space and related constructions
• Ring-theoretic properties
• Boolean algebras with additional operations (diagonalizable algebras, etc.)
• Boolean functions
• None of the above, but in this section
• Ordered structures
• Ordered semigroups and monoids
• Quantales
• Noether lattices
• Ordered groups
• Ordered abelian groups, Riesz groups, ordered linear spaces
• Ordered rings, algebras, modules
• Topological lattices, order topologies
• BCK-algebras, BCI-algebras
• None of the above, but in this section
• 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
• Relational systems, laws of composition
• Structure theory
• Subalgebras, congruence relations
• Automorphisms, endomorphisms
• Operations, polynomials, primal algebras
• Equational compactness
• Word problems
• 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
• Equational logic, Malcev (Maltsev) conditions
• Congruence modularity, congruence distributivity
• Lattices of varieties
• Free algebras
• Products, amalgamated products, and other kinds of limits and colimits
• Subdirect products and subdirect irreducibility
• Injectives, projectives
• None of the above, but in this section
• Other classes of algebras
• Categories of algebras
• Axiomatic model classes
• Quasivarieties
• 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
• 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
• Other representations
• None of the above, but in this section
• Sequences and sets
• Density, gaps, topology
• Arithmetic progressions
• Representation functions
• Recurrences
• 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
• 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
• Matrices, determinants
• None of the above, but in this section
• Diophantine equations
• Linear 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
• Diophantine inequalities
• Congruences in many variables
• Representation problems
• $p$-adic and power series fields
• None of the above, but in this section
• Forms and linear algebraic groups
• Quadratic forms over general fields
• Quadratic forms over local rings and fields
• Forms over real fields
• Quadratic forms over global rings and fields
• 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
• $K$-theory of quadratic and Hermitian forms
• Galois cohomology of linear algebraic groups
• Forms of degree higher than two
• Algebraic theory of quadratic forms; Witt groups and rings
• $p$-adic theory
• None of the above, but in this section
• Discontinuous groups and automorphic forms
• Modular and automorphic functions
• Structure of modular groups and generalizations; arithmetic groups
• Modular forms, one variable
• Automorphic forms, one variable
• Dedekind eta function, Dedekind sums
• Relationship to Lie algebras and finite simple groups
• Relations with algebraic geometry and topology
• Hecke-Petersson operators, differential operators (one variable)
• Theta series; Weil representation
• Fourier coefficients of automorphic forms
• Modular correspondences, etc.
• Congruences for modular and $p$-adic modular forms
• Forms of half-integer weight; nonholomorphic modular forms
• Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
• Siegel modular groups and their modular and automorphic forms
• Jacobi forms
• Modular forms associated to Drinfel'd modules
• Other groups and their modular and automorphic forms (several variables)
• Hecke-Petersson operators, differential operators (several variables)
• Dirichlet series and functional equations in connection with modular forms
• Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
• 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
• None of the above, but in this section
• Arithmetic algebraic geometry (Diophantine geometry)
• Elliptic curves over global fields
• Elliptic curves over local fields
• Drinfeld modules; higher-dimensional motives, etc.
• Abelian varieties of dimension $\gtr 1$
• Complex multiplication and moduli of abelian varieties
• Elliptic and modular units
• Arithmetic aspects of modular and Shimura varieties
• Curves over finite and local fields
• Varieties over finite and local fields
• Curves of arbitrary genus or genus $\ne 1$ over global fields
• Varieties over global fields
• $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
• Geometric class field theory
• Heights
• Polylogarithms and relations with $K$-theory
• None of the above, but in this section
• Geometry of numbers
• Lattices and convex bodies
• Nonconvex bodies
• Lattice packing and covering
• 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
• Diophantine approximation, transcendental number theory
• 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
• Small fractional parts of polynomials and generalizations
• Approximation in non-Archimedean valuations
• Approximation to algebraic numbers
• Continued fractions and generalizations
• Distribution modulo one
• 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
• Probabilistic theory: distribution modulo $1$; metric theory of algorithms
• General theory of distribution modulo $1$
• Normal numbers, radix expansions, etc.
• Special sequences
• Well-distributed sequences and other variations
• Irregularities of distribution, discrepancy
• Continuous, $p$-adic and abstract analogues
• Pseudo-random numbers; Monte Carlo methods
• Metric theory of continued fractions
• Metric theory of other algorithms and expansions; measure and Hausdorff dimension
• Diophantine approximation
• Arithmetic functions
• Harmonic analysis and almost periodicity
• None of the above, but in this section
• Exponential sums and character sums
• 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 and $L$-functions: analytic theory
• $\zeta (s)$ 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
• Tauberian theorems
• None of the above, but in this section
• Multiplicative number theory
• Distribution of primes
• Primes in progressions
• Distribution of integers with specified multiplicative constraints
• Turán theory
• 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
• 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
• Inverse problems of additive number theory
• Elementary theory of partitions
• Analytic theory of partitions
• Partitions; congruences and congruential restrictions
• None of the above, but in this section
• Algebraic number theory: global fields
• Algebraic numbers; rings of algebraic integers
• PV-numbers and generalizations; other special algebraic numbers
• Polynomials (irreducibility, etc.)
• 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
• Galois cohomology
• Class field theory
• Langlands-Weil conjectures, nonabelian class field theory
• Zeta functions and $L$-functions of number fields
• Distribution of prime ideals
• Density theorems
• Other analytic theory
• Quaternion and other division algebras: arithmetic, zeta functions
• Other algebras and orders, and their zeta and $L$-functions
• Arithmetic theory of algebraic function fields
• Cyclotomic function fields (class groups, Bernoulli objects, etc.)
• Class groups and Picard groups of orders
• $K$-theory of global fields
• Totally real and totally positive fields
• 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
• Class field theory; $p$-adic formal groups
• Langlands-Weil conjectures, nonabelian class field theory
• Zeta functions and $L$-functions
• Algebras and orders, and their zeta functions
• $K$-theory of local fields
• 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
• Finite fields and commutative rings (number-theoretic aspects)
• Polynomials
• Cyclotomy
• Exponential sums
• Other character sums and Gauss sums
• Structure theory
• Arithmetic theory of polynomial rings over finite fields
• Finite upper half-planes
• Algebraic coding theory; cryptography
• None of the above, but in this section
• Connections with logic
• Decidability
• Ultraproducts
• Model theory
• Nonstandard arithmetic
• None of the above, but in this section
• Computational number theory
• Factorization
• Primality
• Algorithms; complexity
• 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

## 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)
• Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
• None of the above, but in this section
• General field theory
• Polynomials (irreducibility, etc.)
• Special polynomials
• Equations
• Skew fields, division rings
• 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
• Cohomological dimension
• None of the above, but in this section
• Differential and difference algebra
• Differential algebra
• Difference algebra
• Abstract differential equations
• $p$-adic differential equations
• 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
• Non-Archimedean valued fields
• Krasner-Tate algebras
• None of the above, but in this section
• Generalizations of fields
• Near-fields
• Semifields
• None of the above, but in this section
• Connections with logic
• Decidability
• Ultraproducts
• Model theory
• Nonstandard arithmetic
• None of the above, but in this section
• Computational aspects of field theory and polynomials

## Commutative rings and algebras

• 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
• Divisibility
• Ideals; multiplicative ideal theory
• Valuations and their generalizations
• 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
• Actions of groups on commutative rings; invariant theory
• None of the above, but in this section
• Ring extensions and related topics
• Extension theory
• Galois theory
• Morphisms
• Integral dependence
• Integral closure of rings and ideals; integrally closed rings, related rings (Japanese, etc.)
• Going up; going down; going between
• Polynomials over commutative rings
• Quotients and localization
• Completion
• Étale and flat extensions; Henselization; Artin approximation
• None of the above, but in this section
• Theory of modules and ideals
• Structure, classification theorems
• Projective and free modules and ideals
• Injective and flat modules and ideals
• Torsion modules and ideals
• Other special types
• Cohen-Macaulay modules
• Dimension theory, depth, related rings (catenary, etc.)
• Class groups
• Linkage, complete intersections and determinantal ideals
• None of the above, but in this section
• Homological methods
• Syzygies and resolutions
• (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
• Homological dimension
• Homological functors on modules (Tor, Ext, etc.)
• Deformations and infinitesimal methods
• Grothendieck groups, $K$-theory
• Homological conjectures (intersection theorems)
• Complexes
• Torsion theory
• Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
• Local cohomology
• 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, Prüfer and Krull rings and their generalizations
• Euclidean rings and generalizations
• Principal ideal rings
• Factorial rings, unique factorization domains
• Polynomial rings and ideals; rings of integer-valued polynomials
• Formal power series rings
• Valuation rings
• Excellent rings
• Seminormal rings
• Rings with straightening laws, Hodge algebras
• Face and Stanley-Reisner rings; simplicial complexes
• None of the above, but in this section
• Integral domains
• Local rings and semilocal rings
• Regular local rings
• Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
• Multiplicity theory and related topics
• None of the above, but in this section
• Topological rings and modules
• Power series rings
• Analytical algebras and rings
• Complete rings, completion
• Henselian rings
• Global topological rings
• Ordered rings
• Real algebra
• None of the above, but in this section
• Witt vectors and related rings
• Applications of logic to commutative algebra
• Finite commutative rings
• Structure
• Polynomials
• None of the above, but in this section
• Differential algebra
• Modules of differentials
• Rings of differential operators and their modules
• Derivations
• None of the above, but in this section
• Computational aspects of commutative algebra
• Polynomials, factorization
• Polynomial ideals, Gröbner bases
• 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
• Varieties and morphisms
• Schemes and morphisms
• Generalizations (algebraic spaces, stacks)
• Noncommutative algebraic geometry
• Elementary questions
• None of the above, but in this section
• Local theory
• Singularities
• Deformations of singularities
• Infinitesimal methods
• Local deformation theory, Artin approximation, etc.
• Local cohomology
• Formal neighborhoods
• Local structure of morphisms: étale, flat, etc.
• None of the above, but in this section
• Cycles and subschemes
• Parametrization (Chow and Hilbert schemes)
• Chow groups and rings
• Intersection theory, characteristic classes, intersection multiplicities
• Divisors, linear systems, invertible sheaves
• Pencils, nets, webs
• Picard groups
• Algebraic cycles
• Transcendental methods, Hodge theory, Hodge conjecture
• Torelli problem
• Applications of methods of algebraic $K$-theory
• Riemann-Roch theorems
• None of the above, but in this section
• Families, fibrations
• Structure of families (Picard-Lefschetz, monodromy, etc.)
• Fibrations, degenerations
• Variation of Hodge structures
• Arithmetic ground fields (finite, local, global)
• Formal methods; deformations
• Algebraic moduli problems, moduli of vector bundles
• 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
• Birational geometry
• Rational and birational maps
• Birational automorphisms, Cremona group and generalizations
• Rationality questions
• Global theory and resolution of singularities
• Coverings
• Ramification problems
• Embeddings
• Minimal model program (Mori theory, extremal rays)
• None of the above, but in this section
• (Co)homology theory
• Vector bundles, sheaves, related constructions
• Differentials and other special sheaves
• Vanishing theorems
• Étale and other Grothendieck topologies and cohomologies
• Brauer groups of schemes
• Classical real and complex cohomology
• $p$-adic cohomology, crystalline cohomology
• Homotopy theory; fundamental groups
• de Rham cohomology
• 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
• Arithmetic problems. Diophantine geometry
• Rational points
• Zeta-functions and related questions(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
• Arithmetic varieties and schemes; Arakelov theory; heights
• Applications to coding theory and cryptography
• None of the above, but in this section
• Curves
• Algebraic functions; function fields
• Families, moduli (algebraic)
• Families, moduli (analytic)
• Singularities, local rings
• Arithmetic ground fields
• Coverings, fundamental group
• Automorphisms
• Jacobians, Prym varieties
• Theta functions; Schottky problem
• Special curves and curves of low genus
• Plane and space curves
• Special divisors (gonality, Brill-Noether theory)
• Elliptic curves
• Riemann surfaces; Weierstrass points; gap sequences
• Vector bundles on curves and their moduli
• Relationships with integrable systems
• Relationships with physics
• None of the above, but in this section
• Surfaces and higher-dimensional varieties
• Families, moduli, classification: algebraic theory
• Moduli, classification: analytic theory; relations with modular forms
• Singularities
• Arithmetic ground fields
• Special surfaces
• 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
• 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
• Subvarieties
• Arithmetic ground fields
• Analytic theory; abelian integrals and differentials
• Complex multiplication
• Theta functions
• Picard schemes, higher Jacobians
• None of the above, but in this section
• Algebraic groups
• Formal groups, $p$-divisible groups
• Group varieties
• Group schemes
• Affine algebraic groups, hyperalgebra constructions
• Geometric invariant theory
• Group actions on varieties or schemes (quotients)
• Classical groups (geometric aspects)
• Other algebraic groups (geometric aspects)
• None of the above, but in this section
• Special varieties
• Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
• Low codimension problems
• Complete intersections
• Determinantal varieties
• Grassmannians, Schubert varieties, flag manifolds
• Homogeneous spaces and generalizations
• Rational and unirational varieties
• Toric varieties, Newton polyhedra
• Supervarieties
• None of the above, but in this section
• Projective and enumerative geometry
• Projective techniques
• Enumerative problems (combinatorial problems)
• Classical problems, Schubert calculus
• Configurations of linear subspaces
• Varieties of low degree
• Gromov-Witten invariants, quantum cohomology
• None of the above, but in this section
• Real algebraic and real analytic geometry
• Real algebraic sets
• Semialgebraic sets and related spaces
• Real analytic and semianalytic sets
• Nash functions and manifolds
• Topology of real algebraic varieties
• None of the above, but in this section
• Computational aspects in algebraic geometry
• Curves
• Surfaces, hypersurfaces
• Higher-dimensional varieties
• Effectivity
• None of the above, but in this section
• Affine geometry
• Classification of affine varieties
• Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
• Jacobian problem
• Group actions on affine varieties
• Affine fibrations
• None of the above, but in this section

## Linear and multilinear algebra; matrix 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.
• Vector spaces, linear dependence, rank
• Linear transformations, semilinear transformations
• Linear equations
• Matrix inversion, generalized inverses
• Conditioning of matrices
• Determinants, permanents, other special matrix functions
• Eigenvalues, singular values, and eigenvectors
• Canonical forms, reductions, classification
• Matrix pencils
• Factorization of matrices
• Matrix equations and identities
• Commutativity
• Inverse problems
• Algebraic systems of matrices
• Matrices over special rings (quaternions, finite fields, etc.)
• Matrices of integers
• Linear inequalities
• Inequalities involving eigenvalues and eigenvectors
• Miscellaneous inequalities involving matrices
• Positive matrices and their generalizations; cones of matrices
• Stochastic matrices
• Random matrices
• Matrices over function rings in one or more variables
• Other types of matrices (Hermitian, skew-Hermitian, etc.)
• Norms of matrices, numerical range, applications of functional analysis to matrix theory
• Quadratic and bilinear forms, inner products
• Clifford algebras, spinors
• Multilinear algebra, tensor products
• Vector and tensor algebra, theory of invariants
• Exterior algebra, Grassmann algebras
• Other algebras built from modules
• Applications of matrix theory to physics
• Miscellaneous topics

## Associative rings and algebras

• 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 and miscellaneous
• Category-theoretic methods and results (except as in 16D90)
• Applications of logic
• None of the above, but in this section
• Modules, bimodules and ideals
• General module theory
• Bimodules
• Ideals
• Infinite-dimensional simple rings (except as in 16Kxx)
• Free, projective, and flat modules and ideals
• Injective modules, self-injective rings
• Simple and semisimple modules, primitive rings and ideals
• Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation
• Other classes of modules and ideals
• Module categories; module theory in a category-theoretic context; Morita equivalence and duality
• None of the above, but in this section
• Homological methods
• Syzygies, resolutions, complexes
• Homological dimension
• Grothendieck groups, $K$-theory, etc.
• Homological functors on modules (Tor, Ext, etc.)
• (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
• Differential graded algebras and applications
• von Neumann regular rings and generalizations
• Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
• Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
• None of the above, but in this section
• Representation theory of rings and algebras
• Representations of Artinian rings
• Representations of quivers and partially ordered sets
• Representations of orders, lattices, algebras over commutative rings
• Cohen-Macaulay modules
• Representation type (finite, tame, wild, etc.)
• Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
• None of the above, but in this section
• Orders and arithmetic, separable algebras, Azumaya algebras
• Division rings and semisimple Artin rings
• Finite-dimensional
• Infinite-dimensional and general
• Brauer groups
• None of the above, but in this section
• Local rings and generalizations
• Noncommutative local and semilocal rings, perfect rings
• Quasi-Frobenius rings
• None of the above, but in this section
• Nil and nilpotent radicals, sets, ideals, rings
• Prime and semiprime rings
• None of the above, but in this section
• Chain conditions, growth conditions, and other forms of finiteness
• Finite rings and finite-dimensional algebras
• Artinian rings and modules
• Noetherian rings and modules
• Localization and Noetherian rings
• Chain conditions on annihilators and summands: Goldie-type conditions, Krull dimension
• Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence
• Growth rate, Gelfand-Kirillov dimension
• None of the above, but in this section
• Rings with polynomial identity
• $T$-ideals, identities, varieties of rings and algebras
• Semiprime p.i. rings, rings embeddable in matrices over commutative rings
• Trace rings and invariant theory
• Identities other than those of matrices over commutative rings
• Other kinds of identities (generalized polynomial, rational, involution)
• None of the above, but in this section
• Rings and algebras arising under various constructions
• Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
• Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
• Centralizing and normalizing extensions
• Universal enveloping algebras of Lie algebras
• Rings of differential operators
• Group rings, Laurent polynomial rings
• Twisted and skew group rings, crossed products
• Ordinary and skew polynomial rings and semigroup rings
• Rings arising from non-commutative algebraic geometry
• Smash products of general Hopf actions
• Endomorphism rings; matrix rings
• Rings of functions, subdirect products, sheaves of rings
• Extensions of rings by ideals
• Deformations of rings
• Maximal ring of quotients, torsion theories, radicals on module categories
• None of the above, but in this section
• Conditions on elements
• Integral domains
• Ore rings, multiplicative sets, Ore localization
• Divisibility, noncommutative UFDs
• Units, groups of units
• Center, normalizer (invariant elements)
• Generalizations of commutativity
• None of the above, but in this section
• Rings and algebras with additional structure
• Rings with involution; Lie, Jordan and other nonassociative structures
• Automorphisms and endomorphisms
• Actions of groups and semigroups; invariant theory
• Derivations, actions of Lie algebras
• Coalgebras, bialgebras, Hopf algebras; rings, modules, etc. on which these act
• Ring-theoretic aspects of quantum groups
• Super'' (or skew'') structure
• Valuations, completions, formal power series and related constructions
• Filtered rings; filtrational and graded techniques
• Topological and ordered rings and modules
• None of the above, but in this section
• Generalizations
• Near-rings
• Semirings
• None of the above, but in this section
• Computational aspects of associative rings

## Nonassociative rings and algebras

• 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.
• Computational methods
• General nonassociative rings
• General theory
• Power-associative rings
• Noncommutative Jordan algebras
• Flexible algebras
• Algebras satisfying other identities
• Leibniz algebras
• Division algebras
• Automorphisms, derivations, other operators
• Ternary compositions
• Other $n$-ary compositions $(n \ge 3)$
• Free algebras
• Structure theory
• Superalgebras
• Composition algebras
• Valued algebras
• None of the above, but in this section
• Lie algebras and Lie superalgebras
• Identities, free Lie (super)algebras
• Structure theory
• Representations, algebraic theory (weights)
• Representations, analytic theory
• Simple, semisimple, reductive (super)algebras (roots)
• Exceptional (super)algebras
• Solvable, nilpotent (super)algebras
• Universal enveloping (super)algebras
• Quantum groups (quantized enveloping algebras) and related deformations
• Automorphisms, derivations, other operators
• Lie algebras of linear algebraic groups
• Modular Lie (super)algebras
• Homological methods in Lie (super)algebras
• Cohomology of Lie (super)algebras
• Lie (super)algebras associated with other structures (associative, Jordan, etc.)
• Lie bialgebras
• Poisson algebras
• Infinite-dimensional Lie (super)algebras
• Lie algebras of vector fields and related (super) algebras
• Kac-Moody (super)algebras (structure and representation theory)
• Virasoro and related algebras
• Vertex operators; vertex operator algebras and related structures
• Color Lie (super)algebras
• Applications to integrable systems
• Applications to physics
• None of the above, but in this section
• Jordan algebras (algebras, triples and pairs)
• Identities and free Jordan structures
• Structure theory
• Simple, semisimple algebras
• Idempotents, Peirce decompositions
• Associated groups, automorphisms
• Associated manifolds
• Associated geometries
• Exceptional Jordan structures
• Jordan structures associated with other structures
• Finite-dimensional structures
• Division algebras
• Jordan structures on Banach spaces and algebras
• Super structures
• Applications to physics
• None of the above, but in this section
• Other nonassociative rings and algebras
• Alternative rings
• Malcev (Maltsev) rings and algebras
• Right alternative rings
• $(\gamma, \delta)$-rings, including $(1,-1)$-rings
• Genetic algebras
• None of the above, but in this section

## Category theory; homological 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 theory of categories and functors
• Definitions, generalizations
• Graphs, diagram schemes, precategories
• Foundations, relations to logic and deductive systems
• Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
• Special properties of functors (faithful, full, etc.)
• Natural morphisms, dinatural morphisms
• Functor categories, comma categories
• Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
• Factorization of morphisms, substructures, quotient structures, congruences, amalgams
• Categories admitting limits (complete categories), functors preserving limits, completions
• Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
• None of the above, but in this section
• Special categories
• Category of sets, characterizations
• Category of relations, additive relations
• Embedding theorems, universal categories
• Categories of machines, automata, operative categories
• Topoi
• Categories of topological spaces and continuous mappings
• Preorders, orders and lattices (viewed as categories)
• Groupoids, semigroupoids, semigroups, groups (viewed as categories)
• None of the above, but in this section
• Categories and theories
• Equational categories
• Theories (e.g. algebraic theories), structure, and semantics
• Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples
• Algebras and Kleisli categories associated with monads
• Sketches and generalizations
• Accessible and locally presentable categories
• Categorical semantics of formal languages
• None of the above, but in this section
• Categories with structure
• Double categories, $2$-categories, bicategories and generalizations
• Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
• Closed categories (closed monoidal and Cartesian closed categories, etc.)
• Enriched categories (over closed or monoidal categories)
• Fibered categories
• Structured objects in a category (group objects, etc.)
• None of the above, but in this section
• Abelian categories
• Exact categories, abelian categories
• Grothendieck categories
• Embedding theorems
• Derived functors and satellites
• Derived categories, triangulated categories
• Localization of categories
• None of the above, but in this section
• Categories and geometry
• Local categories and functors
• Grothendieck topologies
• Abstract manifolds and fiber bundles
• Presheaves and sheaves
• Algebraic $K$-theory and $L$-theory
• Grothendieck groups
• None of the above, but in this section
• Homological algebra
• Projectives and injectives
• Resolutions; derived functors
• Ext and Tor, generalizations, Künneth formula
• Homological dimension
• Relative homological algebra, projective classes
• Simplicial sets, simplicial objects (in a category)
• Chain complexes
• Spectral sequences, hypercohomology
• Nonabelian homological algebra
• Homotopical algebra
• Other (co)homology theories
• None of the above, but in this section

## $K$-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.
• Grothendieck groups and $K_0$
• Stability for projective modules
• Efficient generation
• Frobenius induction, Burnside and representation rings
• $K_0$ of group rings and orders
• $K_0$ of other rings
• None of the above, but in this section
• Whitehead groups and $K_1$
• Stable range conditions
• Stability for linear groups
• $K_1$ of group rings and orders
• Congruence subgroup problems
• None of the above, but in this section
• Steinberg groups and $K_2$
• Central extensions and Schur multipliers
• Symbols, presentations and stability of $K_2$
• $K_2$ and the Brauer group
• Excision for $K_2$
• None of the above, but in this section
• Higher algebraic $K$-theory
• $Q$- and plus-constructions
• Algebraic $K$-theory of spaces
• Symmetric monoidal categories
• Karoubi-Villamayor-Gersten $K$-theory
• Negative $K$-theory, NK and Nil
• Higher symbols, Milnor $K$-theory
• Computations of higher $K$-theory of rings
• $K$-theory and homology; cyclic homology and cohomology
• None of the above, but in this section
• $K$-theory in geometry
• $K$-theory of schemes
• Algebraic cycles and motivic cohomology
• Relations with cohomology theories
• None of the above, but in this section
• $K$-theory in number theory
• Generalized class field theory
• Symbols and arithmetic
• Étale cohomology, higher regulators, zeta and $L$-functions
• None of the above, but in this section
• $K$-theory of forms
• Witt groups of rings
• $L$-theory of group rings
• Hermitian $K$-theory, relations with $K$-theory of rings
• None of the above, but in this section
• Obstructions from topology
• Finiteness and other obstructions in $K_0$
• Surgery obstructions
• Obstructions to group actions
• None of the above, but in this section
• $K$-theory and operator algebras
• $K_0$ as an ordered group, traces
• EXT and $K$-homology
• Kasparov theory ($KK$-theory)
• Index theory
• None of the above, but in this section
• Topological $K$-theory
• Riemann-Roch theorems, Chern characters
• $J$-homomorphism, Adams operations
• Connective $K$-theory, cobordism
• Equivariant $K$-theory
• Computations, geometric applications
• None of the above, but in this section
• Miscellaneous applications of $K$-theory

## Group theory and generalizations

• 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
• Axiomatics and elementary properties
• Metamathematical considerations
• Applications of logic to group theory
• None of the above, but in this section
• Permutation groups
• General theory for finite groups
• General theory for infinite groups
• Characterization theorems
• Primitive groups
• Multiply transitive finite groups
• Multiply transitive infinite groups
• Finite automorphism groups of algebraic, geometric, or combinatorial structures
• Infinite automorphism groups
• Symmetric groups
• Subgroups of symmetric groups
• Computational methods
• None of the above, but in this section
• Representation theory of groups
• Group rings of finite groups and their modules
• Group rings of infinite groups and their modules
• Hecke algebras and their representations
• Integral representations of finite groups
• $p$-adic representations of finite groups
• Integral representations of infinite groups
• Ordinary representations and characters
• Modular representations and characters
• Projective representations and multipliers
• Representations of finite symmetric groups
• Representations of infinite symmetric groups
• Representations of finite groups of Lie type
• Applications of group representations to physics
• Computational methods
• None of the above, but in this section
• Abstract finite groups
• Classification of simple and nonsolvable groups
• Simple groups: alternating groups and groups of Lie type
• Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks
• Nilpotent groups, $p$-groups
• Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
• Special subgroups (Frattini, Fitting, etc.)
• Series and lattices of subgroups
• Subnormal subgroups
• Products of subgroups
• Automorphisms
• Arithmetic and combinatorial problems
• None of the above, but in this section
• Structure and classification of infinite or finite groups
• Free nonabelian groups
• Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
• Subgroup theorems; subgroup growth
• Groups acting on trees
• Quasivarieties and varieties of groups
• Chains and lattices of subgroups, subnormal subgroups
• Limits, profinite groups
• Extensions, wreath products, and other compositions
• Local properties
• Residual properties and generalizations
• Maximal subgroups
• Simple groups
• General structure theorems
• General theorems concerning automorphisms of groups
• Groups with a $BN$-pair; buildings
• Conjugacy classes
• None of the above, but in this section
• Special aspects of infinite or finite groups
• Generators, relations, and presentations
• Cancellation theory; application of van Kampen diagrams
• Word problems, other decision problems, connections with logic and automata
• Commutator calculus
• Derived series, central series, and generalizations
• Solvable groups, supersolvable groups
• Formations of groups, Fitting classes
• Nilpotent groups
• Generalizations of solvable and nilpotent groups
• Other classes of groups defined by subgroup chains
• FC-groups and their generalizations
• Automorphism groups of groups
• Representations of groups as automorphism groups of algebraic systems
• Fundamental groups and their automorphisms
• Braid groups; Artin groups
• Other groups related to topology or analysis
• Associated Lie structures
• Engel conditions
• Periodic groups; locally finite groups
• Reflection and Coxeter groups
• Ordered groups
• Geometric group theory
• Hyperbolic groups and nonpositively curved groups
• Asymptotic properties of groups
• None of the above, but in this section
• Linear algebraic groups (classical groups)
• Representation theory
• Cohomology theory
• Linear algebraic groups over arbitrary fields
• Linear algebraic groups over the reals, the complexes, the quaternions
• Linear algebraic groups over local fields and their integers
• Linear algebraic groups over global fields and their integers
• Linear algebraic groups over adèles and other rings and schemes
• Linear algebraic groups over finite fields
• Quantum groups (quantized function algebras) and their representations
• Applications to physics
• None of the above, but in this section
• Other groups of matrices
• Unimodular groups, congruence subgroups
• Fuchsian groups and their generalizations
• Other geometric groups, including crystallographic groups
• Other matrix groups over fields
• Other matrix groups over rings
• Other matrix groups over finite fields
• None of the above, but in this section
• Connections with homological algebra and category theory
• Homological methods in group theory
• Cohomology of groups
• Category of groups
• None of the above, but in this section
• Abelian groups
• Finite abelian groups
• Torsion groups, primary groups and generalized primary groups
• Torsion-free groups, finite rank
• Torsion-free groups, infinite rank
• Mixed groups
• Direct sums, direct products, etc.
• Subgroups
• Automorphisms, homomorphisms, endomorphisms, etc.
• Extensions
• Homological and categorical methods
• Topological methods
• None of the above, but in this section
• Groupoids (i.e. small categories in which all morphisms are isomorphisms)
• Semigroups
• Free semigroups, generators and relations, word problems
• Varieties of semigroups
• General structure theory
• Ideal theory
• Commutative semigroups
• Mappings of semigroups
• Regular semigroups
• Inverse semigroups
• Orthodox semigroups
• Semigroups of transformations, etc.
• Semigroup rings, multiplicative semigroups of rings
• Representation of semigroups; actions of semigroups on sets
• Semigroups in automata theory, linguistics, etc.
• Connections of semigroups with homological algebra and category theory
• None of the above, but in this section
• Other generalizations of groups
• Sets with a single binary operation (groupoids)
• Loops, quasigroups
• Ternary systems (heaps, semiheaps, heapoids, etc.)
• $n$-ary systems $(n\ge 3)$
• Hypergroups
• Fuzzy groups
• None of the above, but in this section
• Probabilistic methods in group theory

## Topological groups, Lie groups

• 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.
• Topological and differentiable algebraic systems
• Structure of general topological groups
• Analysis on general topological groups
• Structure of topological semigroups
• Analysis on topological semigroups
• Topological groupoids (including differentiable and Lie groupoids)
• Representations of general topological groups and semigroups
• Topological semilattices, lattices and applications
• Other topological algebraic systems and their representations
• None of the above, but in this section
• Locally compact abelian groups (LCA groups)
• General properties and structure of LCA groups
• Structure of group algebras of LCA groups
• None of the above, but in this section
• Compact groups
• Locally compact groups and their algebras
• General properties and structure of locally compact groups
• Unitary representations of locally compact groups
• Other representations of locally compact groups
• Group algebras of locally compact groups
• Representations of group algebras
• $C^*$-algebras and $W$*-algebras in relation to group representations
• Induced representations
• Duality theorems
• Ergodic theory on groups
• Automorphism groups of locally compact groups
• None of the above, but in this section
• Lie groups
• Local Lie groups
• General properties and structure of complex Lie groups
• General properties and structure of real Lie groups
• General properties and structure of other Lie groups
• Nilpotent and solvable Lie groups
• Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
• Analysis on real and complex Lie groups
• Analysis on $p$-adic Lie groups
• Discrete subgroups of Lie groups
• Continuous cohomology
• Structure and representation of the Lorentz group
• Representations of Lie and linear algebraic groups over real fields: analytic methods
• Semisimple Lie groups and their representations
• Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
• Representations of Lie and linear algebraic groups over local fields
• Representations of Lie and linear algebraic groups over global fields and adèle rings
• Lie algebras of Lie groups
• Infinite-dimensional Lie groups and their Lie algebras
• Loop groups and related constructions, group-theoretic treatment
• Applications of Lie groups to physics; explicit representations
• None of the above, but in this section
• Noncompact transformation groups
• General theory of group and pseudogroup actions
• Measurable group actions
• Homogeneous spaces
• Groups as automorphisms of other structures

## Real functions

• 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.
• Functions of one variable
• Foundations: limits and generalizations, elementary topology of the line
• One-variable calculus
• Elementary functions
• Rate of growth of functions, orders of infinity, slowly varying functions
• Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
• Lipschitz (Hölder) classes
• Iteration
• Classification of real functions; Baire classification of sets and functions
• Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
• Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
• Singular functions, Cantor functions, functions with other special properties
• Fractional derivatives and integrals
• Antidifferentiation
• Denjoy and Perron integrals, other special integrals
• Integrals of Riemann, Stieltjes and Lebesgue type
• Functions of bounded variation, generalizations
• Absolutely continuous functions
• Monotonic functions, generalizations
• Convexity, generalizations
• None of the above, but in this section
• Functions of several variables
• Continuity and differentiation questions
• Implicit function theorems, Jacobians, transformations with several variables
• Calculus of vector functions
• Integration: length, area, volume
• Integral formulas (Stokes, Gauss, Green, etc.)
• Convexity, generalizations
• Absolutely continuous functions, functions of bounded variation
• Special properties of functions of several variables, Hölder conditions, etc.
• Representation and superposition of functions
• None of the above, but in this section
• Polynomials, rational functions
• Polynomials: analytic properties, etc.
• Polynomials: location of zeros
• Rational functions
• None of the above, but in this section
• Inequalities
• Inequalities for trigonometric functions and polynomials
• Inequalities involving other types of functions
• Inequalities involving derivatives and differential and integral operators
• Inequalities for sums, series and integrals
• Other analytical inequalities
• None of the above, but in this section
• Miscellaneous topics
• Real-analytic functions
• $C^\infty$-functions, quasi-analytic functions
• Calculus of functions on infinite-dimensional spaces
• Calculus of functions taking values in infinite-dimensional spaces
• Set-valued functions
• Non-Archimedean analysis
• Nonstandard analysis
• Constructive real analysis
• Fuzzy real analysis
• Means
• None of the above, but in this section

## Measure and integration

• 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.
• Classical measure theory
• Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
• Real- or complex-valued set functions
• Contents, measures, outer measures, capacities
• Abstract differentiation theory, differentiation of set functions
• Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
• Integration with respect to measures and other set functions
• Spaces of measures, convergence of measures
• Measures and integrals in product spaces
• Integration and disintegration of measures
• Lifting theory
• Measures on Boolean rings, measure algebras
• Length, area, volume, other geometric measure theory
• Hausdorff and packing measures
• Fractals
• None of the above, but in this section
• Set functions, measures and integrals with values in abstract spaces
• Vector-valued set functions, measures and integrals
• Group- or semigroup-valued set functions, measures and integrals
• Set functions, measures and integrals with values in ordered spaces
• Set-valued set functions and measures; integration of set-valued functions; measurable selections
• None of the above, but in this section
• Set functions and measures on spaces with additional structure
• Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
• Set functions and measures on topological groups, Haar measures, invariant measures
• Set functions and measures on topological spaces (regularity of measures, etc.)
• Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
• None of the above, but in this section
• Measure-theoretic ergodic theory
• Measure-preserving transformations
• One-parameter continuous families of measure-preserving transformations
• General groups of measure-preserving transformations
• Entropy and other invariants
• None of the above, but in this section
• Miscellaneous topics in measure theory
• Nonstandard measure theory
• Fuzzy measure theory
• Other connections with logic and set theory
• None of the above, but in this section

## Functions of a complex variable

• 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 properties
• Monogenic properties of complex functions (including polygenic and areolar monogenic functions)
• Inequalities in the complex domain
• None of the above, but in this section
• Series expansions
• Power series (including lacunary series)
• Random power series
• Boundary behavior of power series, over-convergence
• Analytic continuation
• Dirichlet series and other series expansions, exponential series
• Completeness problems, closure of a system of functions
• Continued fractions
• None of the above, but in this section
• Geometric function theory
• Polynomials
• Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
• Conformal mappings of special domains
• Covering theorems in conformal mapping theory
• Numerical methods in conformal mapping theory
• General theory of conformal mappings
• Kernel functions and applications
• Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
• Coefficient problems for univalent and multivalent functions
• General theory of univalent and multivalent functions
• Quasiconformal mappings in the plane
• Quasiconformal mappings in $<B>R</B>^n$, other generalizations
• Extremal problems for conformal and quasiconformal mappings, variational methods
• Extremal problems for conformal and quasiconformal mappings, other methods
• Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
• Capacity and harmonic measure in the complex plane
• None of the above, but in this section
• Entire and meromorphic functions, and related topics
• Functional equations in the complex domain, iteration and composition of analytic functions
• Representations of entire functions by series and integrals
• Special classes of entire functions and growth estimates
• Entire functions, general theory
• Meromorphic functions, general theory
• Distribution of values, Nevanlinna theory
• Cluster sets, prime ends, boundary behavior
• Bloch functions, normal functions, normal families
• Blaschke products, bounded mean oscillation, bounded characteristic, bounded functions, functions with positive real part
• ${H]^p$-classes
• Quasi-analytic and other classes of functions
• None of the above, but in this section
• Miscellaneous topics of analysis in the complex domain
• Moment problems, interpolation problems
• Approximation in the complex domain
• Asymptotic representations in the complex domain
• Integration, integrals of Cauchy type, integral representations of analytic functions
• Boundary value problems
• None of the above, but in this section
• Riemann surfaces
• Compact Riemann surfaces and uniformization
• Harmonic functions on Riemann surfaces
• Classification theory of Riemann surfaces
• Ideal boundary theory
• Differentials on Riemann surfaces
• Fuchsian groups and automorphic functions
• Kleinian groups
• Conformal metrics (hyperbolic, Poincaré, distance functions)
• Klein surfaces
• Teichmüller theory
• None of the above, but in this section
• Generalized function theory
• Non-Archimedean function theory; nonstandard function theory
• Finely holomorphic functions and topological function theory
• Generalizations of Bers or Vekua type (pseudoanalytic, $p$-analytic, etc.)
• Discrete analytic functions
• Other generalizations of analytic functions (including abstract-valued functions)
• Functions of hypercomplex variables and generalized variables
• None of the above, but in this section
• Spaces and algebras of analytic functions

## Potential 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.
• Two-dimensional theory
• Harmonic, subharmonic, superharmonic functions
• Integral representations, integral operators, integral equations methods
• Potentials and capacity, harmonic measure, extremal length
• Boundary behavior (theorems of Fatou type, etc.)
• Boundary value and inverse problems
• Biharmonic, polyharmonic functions and equations, Poisson's equation
• Connections with differential equations
• None of the above, but in this section
• Higher-dimensional theory
• Harmonic, subharmonic, superharmonic functions
• Integral representations, integral operators, integral equations methods
• Potentials and capacities, extremal length
• Boundary value and inverse problems
• Boundary behavior
• Biharmonic and polyharmonic equations and functions
• Connections with differential equations
• None of the above, but in this section
• Other generalizations
• Harmonic, subharmonic, superharmonic functions
• Pluriharmonic and plurisubharmonic functions
• Potential theory on Riemannian manifolds
• Potentials and capacities
• Discrete potential theory and numerical methods
• Dirichlet spaces
• Martin boundary theory
• Fine potential theory
• Other generalizations (nonlinear potential theory, etc.)
• None of the above, but in this section

## Axiomatic potential theory

• Several complex variables and analytic spaces
• 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.
• Holomorphic functions of several complex variables
• Power series, series of functions
• Special domains (Reinhardt, Hartogs, circular, tube)
• Holomorphic functions
• Multifunctions
• Entire functions
• Special families of functions
• Bloch functions, normal functions
• Normal families of functions, mappings
• Meromorphic functions
• Nevanlinna theory (local); growth estimates; other inequalities
• Integral representations; canonical kernels (Szegó, Bergman, etc.)
• Integral representations, constructed kernels (e.g. Cauchy, Fantappiè-type kernels)
• Local theory of residues
• Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30)
• ${H]^p$-spaces, Nevanlinna spaces
• Bergman spaces
• Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
• Algebras of holomorphic functions
• Boundary behavior of holomorphic functions
• Hyperfunctions
• Harmonic analysis of several complex variables
• Singular integrals
• Zero sets of holomorphic functions
• Banach algebra techniques
• Functional analysis techniques
• None of the above, but in this section
• Local analytic geometry
• Analytic algebras and generalizations, preparation theorems
• Germs of analytic sets, local parametrization
• Analytic subsets of affine space
• Semi-analytic sets and subanalytic sets
• Triangulation and related questions
• None of the above, but in this section
• Analytic spaces
• Real-analytic manifolds, real-analytic spaces
• Real-analytic sets, complex Nash functions
• Embedding of real analytic manifolds
• Complex supergeometry
• Complex spaces
• Topology of analytic spaces
• Normal analytic spaces
• Embedding of analytic spaces
• Analytic subsets and submanifolds
• Integration on analytic sets and spaces, currents
• Analytic sheaves and cohomology groups
• Local cohomology of analytic spaces
• Duality theorems
• Sheaves of differential operators and their modules, $D$-modules
• The Levi problem in complex spaces; generalizations
• Applications to physics
• None of the above, but in this section
• Analytic continuation
• Domains of holomorphy
• Envelopes of holomorphy
• Continuation of analytic objects
• Removable singularities
• Riemann domains
• None of the above, but in this section
• Holomorphic convexity
• Holomorphically convex complex spaces, reduction theory
• Stein spaces, Stein manifolds
• Polynomial convexity
• Holomorphic and polynomial approximation, Runge pairs, interpolation
• Global boundary behavior of holomorphic functions
• The Levi problem
• None of the above, but in this section
• Geometric convexity
• $q$-convexity, $q$-concavity
• Other notions of convexity
• Finite-type conditions
• Topological consequences of geometric convexity
• Analytical consequences of geometric convexity (vanishing theorems, etc.)
• Invariant metrics and pseudodistances
• None of the above, but in this section
• Deformations of analytic structures
• Deformations of complex structures
• Deformations of special (e.g. CR) structures
• Deformations of fiber bundles
• Deformations of submanifolds and subspaces
• Analytic moduli problems
• Moduli of Riemann surfaces, Teichmüller theory
• Period matrices, variation of Hodge structure; degenerations
• Moduli and deformations for ordinary differential equations (e.g. Khnizhnik-Zamolodchikov equation)
• Applications to physics
• None of the above, but in this section
• Holomorphic mappings and correspondences
• Holomorphic mappings, (holomorphic) embeddings and related questions
• Meromorphic mappings
• Boundary uniqueness of mappings
• Picard-type theorems and generalizations
• Value distribution theory in higher dimensions
• Proper mappings, finiteness theorems
• Boundary regularity of mappings
• Iteration problems
• None of the above, but in this section
• Compact analytic spaces
• Compactification of analytic spaces
• Algebraic dependence theorems
• Compact surfaces
• Compact $3$-folds
• Compact $n$-folds
• Transcendental methods of algebraic geometry
• Compact Kähler manifolds: generalizations, classification
• Applications to physics
• None of the above, but in this section
• Generalizations of analytic spaces (should also be assigned at least one other classification number from Section 32 describing the type of problem)
• Banach analytic spaces
• Formal and graded complex spaces
• Differentiable functions on analytic spaces, differentiable spaces
• None of the above, but in this section
• Holomorphic fiber spaces
• Holomorphic bundles and generalizations
• Sheaves and cohomology of sections of holomorphic vector bundles, general results
• Bundle convexity
• Vanishing theorems
• Twistor theory, double fibrations
• Applications to physics
• None of the above, but in this section
• Complex spaces with a group of automorphisms
• Complex Lie groups, automorphism groups acting on complex spaces
• Homogeneous complex manifolds
• Almost homogeneous manifolds and spaces
• Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras
• Automorphism groups of ${\bf C]^n$ and affine manifolds
• Complex vector fields
• None of the above, but in this section
• Automorphic functions
• General theory of automorphic functions of several complex variables
• Automorphic forms
• Automorphic functions in symmetric domains
• None of the above, but in this section
• Non-Archimedean complex analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)
• Complex manifolds
• Negative curvature manifolds
• Positive curvature manifolds
• Kähler manifolds
• Kähler-Einstein manifolds
• Calabi-Yau theory
• Stein manifolds
• Uniformization
• Complex manifolds as subdomains of Euclidean space
• Embedding theorems
• Hyperbolic and Kobayashi hyperbolic manifolds
• Topological aspects of complex manifolds
• Classification theorems
• Almost complex manifolds
• Pseudoholomorphic curves
• None of the above, but in this section
• Singularities
• Local singularities
• Invariants of analytic local rings
• Equisingularity (topological and analytic)
• Global theory of singularities; cohomological properties
• Relations with arrangements of hyperplanes
• Surface and hypersurface singularities
• Deformations of singularities; vanishing cycles
• Mixed Hodge theory of singular varieties
• Monodromy; relations with differential equations and $D$-modules
• Modifications; resolution of singularities
• Topological aspects: Lefschetz theorems, topological classification, invariants
• Milnor fibration; relations with knot theory
• Stratifications; constructible sheaves; intersection cohomology
• Singularities of holomorphic vector fields and foliations
• Other operations on singularities
• None of the above, but in this section
• Pseudoconvex domains
• Domains of holomorphy
• Strongly pseudoconvex domains
• Worm domains
• Finite type domains
• Geometric and analytic invariants on weakly pseudoconvex boundaries
• Exhaustion functions
• Peak functions
• None of the above, but in this section
• Pluripotential theory
• Plurisubharmonic functions and generalizations
• Plurisubharmonic exhaustion functions
• General pluripotential theory
• Capacity theory and generalizations
• Lelong numbers
• Removable sets
• Pluricomplex Green functions
• Currents
• None of the above, but in this section
• CR manifolds
• CR structures, CR operators, and generalizations
• CR functions
• CR manifolds as boundaries of domains
• Analysis on CR manifolds
• Extension of functions and other analytic objects from CR manifolds
• Embeddings of CR manifolds
• Finite type conditions on CR manifolds
• Real submanifolds in complex manifolds
• None of the above, but in this section
• Differential operators in several variables
• $\overline\partial$ and $\overline\partial$-Neumann operators
• $\overline\partial_b$ and $\overline\partial_b$-Neumann operators
• Complex Monge-Ampère operators
• Pseudodifferential operators in several complex variables
• Heat kernels in several complex variables
• Other partial differential equations of complex analysis
• None of the above, but in this section

## Special functions (33-XX deals with the properties of functions as functions)

• 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 classical functions
• Exponential and trigonometric functions
• Gamma, beta and polygamma functions
• Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
• Higher logarithm functions
• None of the above, but in this section
• Hypergeometric functions
• Classical hypergeometric functions, $_2F_1$
• Bessel and Airy functions, cylinder functions, $_0F_1$
• Confluent hypergeometric functions, Whittaker functions, $_1F_1$
• Generalized hypergeometric series, $_pF_q$
• Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
• Other special orthogonal polynomials and functions
• Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
• Orthogonal polynomials and functions associated with root systems
• Spherical harmonics
• Hypergeometric integrals and functions defined by them ($E$, $G$ and ${H]$ functions)
• Appell, Horn and Lauricella functions
• Hypergeometric functions associated with root systems
• Other hypergeometric functions and integrals in several variables
• Elliptic integrals as hypergeometric functions
• Connections with groups and algebras, and related topics
• Applications
• None of the above, but in this section
• Basic hypergeometric functions
• $q$-gamma functions, $q$-beta functions and integrals
• Basic hypergeometric functions in one variable, ${]_r\phi_s$
• Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
• Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable
• Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
• Basic hypergeometric integrals and functions defined by them
• Bibasic functions and multiple bases
• Basic hypergeometric functions associated with root systems
• Other basic hypergeometric functions and integrals in several variables
• Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics
• Applications
• None of the above, but in this section
• Other special functions
• Elliptic functions and integrals
• Lamé, Mathieu, and spheroidal wave functions
• Mittag-Leffler functions and generalizations
• Other wave functions
• Painlevé-type functions
• Other functions defined by series and integrals
• Other functions coming from differential, difference and integral equations
• Special functions in characteristic $p$ (gamma functions, etc.)
• None of the above, but in this section
• Computational aspects
• Numerical approximation
• Symbolic computation (Gosper and Zeilberger algorithms, etc.)
• None of the above, but in this section
• Ordinary differential equations
• 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 theory
• Explicit solutions and reductions
• Implicit equations, differential-algebraic equations
• Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
• Analytical theory: series, transformations, transforms, operational calculus, etc.
• Geometric methods in differential equations
• Linear equations and systems, general
• Nonlinear equations and systems, general
• Differential equations of infinite order
• Discontinuous equations
• Differential equations with impulses
• Differential inequalities
• Theoretical approximation of solutions
• Inverse problems
• Differential inclusions
• None of the above, but in this section
• Boundary value problems
• Linear boundary value problems
• Linear boundary value problems with nonlinear dependence on the spectral parameter
• Multi-parameter boundary value problems
• Boundary value problems with an indefinite weight
• Multipoint boundary value problems
• Nonlinear boundary value problems
• Singular nonlinear boundary value problems
• Positive solutions of nonlinear boundary value problems
• Weyl theory and its generalizations
• Sturm-Liouville theory
• Green functions
• Special equations (Mathieu, Hill, Bessel, etc.)
• Boundary value problems with impulses
• Boundary value problems on infinite intervals
• Boundary value problems on graphs and networks
• Applications
• None of the above, but in this section
• Qualitative theory
• Location of integral curves, singular points, limit cycles
• Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
• Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.)
• Oscillation theory, zeros, disconjugacy and comparison theory
• Growth, boundedness, comparison of solutions
• Monotone systems
• Symmetries, invariants
• Nonlinear oscillations, coupled oscillators
• Transformation and reduction of equations and systems, normal forms
• Bifurcation
• Periodic solutions
• Relaxation oscillations
• Almost periodic solutions
• Complex behavior, chaotic systems
• Averaging method
• Manifolds of solutions
• Homoclinic and heteroclinic solutions
• Equations and systems on manifolds
• Equivalence, asymptotic equivalence
• Method of integral manifolds
• Hysteresis
• Applications
• None of the above, but in this section
• Stability theory
• Asymptotic properties
• Characteristic and Lyapunov exponents
• Dichotomy, trichotomy
• Perturbations
• Singular perturbations
• Lyapunov stability
• Global stability
• Structural stability and analogous concepts
• Stability of manifolds of solutions
• Ultimate boundedness
• Attractors
• None of the above, but in this section
• Asymptotic theory
• Asymptotic expansions
• Perturbations, asymptotics
• Multiple scale methods
• Singular perturbations, general theory
• Methods of nonstandard analysis
• Singular perturbations, turning point theory, WKB methods
• None of the above, but in this section
• Equations and systems with randomness
• Differential equations in abstract spaces
• Linear equations
• Nonlinear equations
• Evolution inclusions
• None of the above, but in this section
• Control problems
• Functional-differential and differential-difference equations
• General theory
• Linear functional-differential equations
• Theoretical approximation of solutions
• Boundary value problems
• Oscillation theory
• Growth, boundedness, comparison of solutions
• Periodic solutions
• Almost periodic solutions
• Transformation and reduction of equations and systems, normal forms
• Bifurcation theory
• Invariant manifolds
• Stability theory
• Complex (chaotic) behavior of solutions
• Asymptotic theory
• Singular perturbations
• Numerical approximation of solutions
• Inverse problems
• Equations in abstract spaces
• Control problems
• Neutral equations
• Equations with impulses
• Stochastic delay equations
• Applications
• None of the above, but in this section
• Ordinary differential operators
• General spectral theory
• Eigenfunction expansions, completeness of eigenfunctions
• Estimation of eigenvalues, upper and lower bounds
• Numerical approximation of eigenvalues and of other parts of the spectrum
• Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
• Scattering theory
• Nonlinear ordinary differential operators
• Particular operators (Dirac, one-dimensional Schrödinger, etc.)
• None of the above, but in this section
• Differential equations in the complex domain
• Entire and meromorphic solutions
• Oscillation, growth of solutions
• Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical)
• Nonanalytic aspects
• Formal solutions, transform techniques
• Asymptotics, summation methods
• Singularities, monodromy, local behavior of solutions, normal forms
• Resurgence phenomena
• Stokes phenomena and connection problems (linear and nonlinear)
• Differential equations on complex manifolds
• Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.)
• Painlevé and other special equations; classification, hierarchies; isomonodromic deformations
• Singular perturbation problems in the complex domain (complex WKB, turning points, steepest descent)
• None of the above, but in this section

## Partial differential equations

• 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 theory
• General existence and uniqueness theorems
• Local existence and uniqueness theorems
• Fundamental solutions
• Cauchy-Kovalevskaya theorems
• Variational methods
• Parametrices
• Wave front sets
• Analytic methods, singularities
• Propagation of singularities
• Transform methods (e.g. integral transforms)
• Other special methods
• Microlocal methods; methods of sheaf theory and homological algebra in PDE
• Geometric theory, characteristics, transformations
• Theoretical approximation to solutions
• None of the above, but in this section
• Qualitative properties of solutions
• General behavior of solutions of PDE (comparison theorems; oscillation, zeros and growth of solutions; mean value theorems)
• Periodic solutions
• Almost periodic solutions
• Perturbations
• Singular perturbations
• Homogenization; partial differential equations in media with periodic structure
• Dependence of solutions of PDE on initial and boundary data, parameters
• Bifurcation
• Critical exponents
• Resonances
• Stability, boundedness
• PDE in connection with control problems
• Critical points
• Asymptotic behavior of solutions
• Attractors
• Inertial manifolds
• A priori estimates
• Maximum principles
• Continuation and prolongation of solutions of PDE
• Smoothness and regularity of solutions of PDE
• None of the above, but in this section
• Representations of solutions
• Solutions in closed form
• Series solutions, expansion theorems
• Integral representations of solutions of PDE
• Asymptotic expansions
• None of the above, but in this section
• Generalized solutions of partial differential equations
• Existence of generalized solutions
• Regularity of generalized solutions
• None of the above, but in this section
• Equations and systems with constant coefficients
• Fundamental solutions
• Convexity properties
• Initial value problems
• General theory
• None of the above, but in this section
• General first-order equations and systems
• General theory of linear first-order PDE
• Initial value problems for linear first-order PDE, linear evolution equations
• Boundary value problems for linear first-order PDE
• General theory of nonlinear first-order PDE
• Initial value problems for nonlinear first-order PDE, nonlinear evolution equations
• Boundary value problems for nonlinear first-order PDE
• None of the above, but in this section
• General higher-order equations and systems
• General theory of linear higher-order PDE
• Initial value problems for linear higher-order PDE, linear evolution equations
• Boundary value problems for linear higher-order PDE
• General theory of nonlinear higher-order PDE
• Initial value problems for nonlinear higher-order PDE, nonlinear evolution equations
• Boundary value problems for nonlinear higher-order PDE
• None of the above, but in this section
• Close-to-elliptic equations
• Hypoelliptic equations
• Subelliptic equations
• Quasi-elliptic equations
• None of the above, but in this section
• Partial differential equations of elliptic type
• Laplace equation, reduced wave equation (Helmholtz), Poisson equation
• Schrödinger operator
• General theory of second-order, elliptic equations
• Variational methods for second-order, elliptic equations
• Boundary value problems for second-order, elliptic equations
• General theory of higher-order, elliptic equations
• Variational methods for higher-order, elliptic equations
• Boundary value problems for higher-order, elliptic equations
• General theory of elliptic systems of PDE
• Variational methods for elliptic systems
• Boundary value problems for elliptic systems
• Nonlinear PDE of elliptic type
• Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
• Boundary values of solutions to elliptic PDE
• Elliptic partial differential equations of degenerate type
• Unilateral problems and variational inequalities for elliptic PDE
• None of the above, but in this section
• Parabolic equations and systems
• Heat equation
• General theory of second-order, parabolic equations
• Initial value problems for second-order, parabolic equations
• Boundary value problems for second-order, parabolic equations
• General theory of higher-order, parabolic equations
• Initial value problems for higher-order, parabolic equations
• Boundary value problems for higher-order, parabolic equations
• General theory of parabolic systems of PDE
• Initial value problems for parabolic systems
• Boundary value problems for parabolic systems
• Nonlinear PDE of parabolic type
• Reaction-diffusion equations
• Nonlinear boundary value problems for linear parabolic PDE; boundary value problems for nonlinear parabolic PDE
• Parabolic partial differential equations of degenerate type
• Ultraparabolic, pseudoparabolic PDE, etc.
• Unilateral problems and variational inequalities for parabolic PDE
• Abstract parabolic evolution equations
• None of the above, but in this section
• Partial differential equations of hyperbolic type
• Wave equation
• General theory of second-order, hyperbolic equations
• Initial value problems for second-order, hyperbolic equations
• Boundary value problems for second-order, hyperbolic equations
• General theory of higher-order, hyperbolic equations
• Initial value problems for higher-order, hyperbolic equations
• Boundary value problems for higher-order, hyperbolic equations
• General theory of hyperbolic systems of first-order PDE
• Initial value problems for hyperbolic systems of first-order PDE
• Boundary value problems for hyperbolic systems of first-order PDE
• Hyperbolic systems of higher-order PDE
• Nonlinear first-order PDE of hyperbolic type
• Conservation laws
• Shocks and singularities
• Nonlinear second-order PDE of hyperbolic type
• Nonlinear hyperbolic PDE of higher ($\gtr 2$) order
• Hyperbolic PDE of degenerate type
• Pseudohyperbolic equations
• Unilateral problems; variational inequalities for hyperbolic PDE
• Abstract hyperbolic evolution equations
• None of the above, but in this section
• Partial differential equations of special type (mixed, composite, etc.)
• PDE of mixed type
• PDE of composite type
• None of the above, but in this section
• Overdetermined systems
• Overdetermined systems with constant coefficients
• Overdetermined systems with variable coefficients (general)
• $\overline\partial$-Neumann problem and generalizations; formal complexes
• None of the above, but in this section
• Spectral theory and eigenvalue problems for partial differential operators
• General spectral theory of PDE
• Completeness of eigenfunctions, eigenfunction expansions for PDO
• Estimation of eigenvalues, upper and lower bounds
• Asymptotic distribution of eigenvalues and eigenfunctions for PDO
• Scattering theory for PDE
• Nonlinear eigenvalue problems, nonlinear spectral theory for PDO
• None of the above, but in this section
• Equations of mathematical physics and other areas of application
• Euler-Poisson-Darboux equation and generalizations
• Riemann-Hilbert problems
• Stokes and Navier-Stokes equations
• Other equations arising in fluid mechanics
• Equations from quantum mechanics
• Solitons
• KdV-like equations (Korteweg-de Vries, Burgers, sine-Gordon, sinh-Gordon, etc.)
• NLS-like (nonlinear Schrödinger) equations
• Other completely integrable equations
• Equations of electromagnetic theory and optics
• Other equations from mechanics
• PDE in relativity
• Applications of PDE in areas other than physics
• None of the above, but in this section
• Miscellaneous topics involving partial differential equations
• PDE with discontinuous coefficients or data
• Partial functional-differential or differential-difference equations, with or without deviating arguments
• Impulsive partial differential equations
• Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables)
• Partial operator-differential equations (i.e. PDE on finite-dimensional spaces for abstract space valued functions)
• Improperly posed problems for PDE
• Inverse problems (undetermined coefficients, etc.) for PDE
• Free boundary problems for PDE
• Partial differential inequalities
• Partial differential equations of infinite order
• Partial differential equations with randomness
• PDE with multivalued right-hand sides
• None of the above, but in this section
• Pseudodifferential operators and other generalizations of partial differential operators
• General theory of PsDO
• Initial value problems for PsDO
• Boundary value problems for PsDO
• Fourier integral operators
• Topological aspects: intersection cohomology, stratified sets, etc.
• None of the above, but in this section

## Dynamical systems and ergodic 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.
• Ergodic theory
• Measure-preserving transformations
• One-parameter continuous families of measure-preserving transformations
• General groups of measure-preserving transformations
• Homogeneous flows
• Orbit equivalence, cocycles, ergodic equivalence relations
• Ergodicity, mixing, rates of mixing
• Ergodic theorems, spectral theory, Markov operators
• Entropy and other invariants, isomorphism, classification
• Nonsingular (and infinite-measure preserving) transformations
• Relations with number theory and harmonic analysis
• Relations with probability theory and stochastic processes
• Relations with the theory of $C^*$-algebras
• Dynamical systems in statistical mechanics
• None of the above, but in this section
• Topological dynamics
• Transformations and group actions with special properties (minimality, distality, proximality, etc.)
• Symbolic dynamics
• Cellular automata
• Notions of recurrence
• Lyapunov functions and stability; attractors, repellers
• Index theory, Morse-Conley indices
• Gradient-like and recurrent behavior; isolated (locally-maximal) invariant sets
• Topological entropy
• Continua theory in dynamics
• Multi-dimensional shifts of finite type, tiling dynamics
• Nonautonomous dynamical systems
• None of the above, but in this section
• Smooth dynamical systems: general theory
• Smooth mappings and diffeomorphisms
• Vector fields, flows, ordinary differential equations
• Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
• Generic properties, structural stability
• Fixed points, periodic points, fixed-point index theory
• Periodic orbits of vector fields and flows
• Homoclinic and heteroclinic orbits
• Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
• Orbit growth
• Smooth ergodic theory, invariant measures
• Dimension theory of dynamical systems
• Approximate trajectories (pseudotrajectories, shadowing, etc.)
• Periodic and quasiperiodic flows and diffeomorphisms
• Nonautonomous smooth dynamical systems
• Monotone flows
• Attractors and repellers, topological structure
• Stability theory
• Symmetries, equivariant dynamical systems
• Dynamics of group actions other than <B>Z</B> and <B>R</B>, and foliations
• None of the above, but in this section
• Dynamical systems with hyperbolic behavior
• Hyperbolic orbits and sets
• Invariant manifold theory
• Morse-Smale systems
• Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
• Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
• Partially hyperbolic systems and dominated splittings
• Thermodynamic formalism, variational principles, equilibrium states
• Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
• Strange attractors, chaotic dynamics
• Hyperbolic systems with singularities (billiards, etc.)
• None of the above, but in this section
• Low-dimensional dynamical systems
• Maps of the interval (piecewise continuous, continuous, smooth)
• Maps of the circle
• Combinatorial dynamics (types of periodic orbits)
• Universality, renormalization
• Maps of trees and graphs
• Homeomorphisms and diffeomorphisms of planes and surfaces
• Flows on surfaces
• Twist maps
• Rotation numbers and vectors
• None of the above, but in this section
• Complex dynamical systems
• Relations and correspondences
• Polynomials; rational maps; entire and meromorphic functions
• Expanding maps; hyperbolicity; structural stability
• Combinatorics and topology
• Renormalization
• Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems
• Conformal densities and Hausdorff dimension
• Geometric limits
• Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
• Small divisors, rotation domains and linearization; Fatou and Julia sets
• Holomorphic foliations and vector fields
• None of the above, but in this section
• Local and nonlocal bifurcation theory
• Normal forms
• Bifurcations of singular points
• Bifurcations of limit cycles and periodic orbits
• Hyperbolic singular points with homoclinic trajectories
• Bifurcations connected with nontransversal intersection
• Infinite nonwandering sets arising in bifurcations
• Attractors and their bifurcations
• Symmetries, equivariant bifurcation theory
• None of the above, but in this section
• Random dynamical systems
• Foundations, general theory of cocycles, algebraic ergodic theory
• Generation, random and stochastic difference and differential equations
• Multiplicative ergodic theory, Lyapunov exponents
• Bifurcation theory
• None of the above, but in this section
• Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
• General theory, relations with symplectic geometry and topology
• Symplectic mappings, fixed points
• Symmetries, invariants, invariant manifolds, momentum maps, reduction
• Bifurcation problems
• Stability problems
• Obstructions to integrability (nonintegrability criteria)
• Completely integrable systems, topological structure of phase space, integration methods
• Perturbations, normal forms, small divisors, KAM theory, Arnold diffusion
• Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
• Action-minimizing orbits and measures
• Contact systems
• Nonholonomic dynamical systems
• None of the above, but in this section
• Infinite-dimensional Hamiltonian systems
• Hamiltonian structures, symmetries, variational principles, conservation laws
• Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)
• Integration of completely integrable systems by inverse spectral and scattering methods
• Relations with algebraic geometry, complex analysis, special functions
• Relations with differential geometry
• Relations with infinite-dimensional Lie algebras and other algebraic structures
• Lie-Bäcklund and other transformations
• Soliton theory, asymptotic behavior of solutions
• Stability problems
• Bifurcation problems
• Perturbations, KAM for infinite-dimensional systems
• Lattice dynamics
• Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
• None of the above, but in this section
• Infinite-dimensional dissipative dynamical systems
• General theory, nonlinear semigroups, evolution equations
• Normal forms, center manifold theory, bifurcation theory
• Stability problems
• Symmetries
• Inertial manifolds and other invariant attracting sets
• Attractors and their dimensions, Lyapunov exponents
• Invariant measures
• Hyperbolicity; Lyapunov functions
• Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems
• Infinite-dimensional random dynamical systems; stochastic equations
• Lattice dynamics
• Special approximation methods (nonlinear Galerkin, etc.)
• None of the above, but in this section
• Approximation methods and numerical treatment of dynamical systems
• Simulation
• Time series analysis
• Symplectic integrators
• Computational methods for bifurcation problems
• Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy)
• None of the above, but in this section
• Applications
• Dynamical systems in classical and celestial mechanics
• Dynamical systems in fluid mechanics, oceanography and meteorology
• Dynamical systems in solid mechanics
• Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
• Dynamical systems in biology
• Dynamical systems in numerical analysis
• Dynamical systems in control
• Dynamical systems in optimization and economics
• None of the above, but in this section

## Difference and functional equations

• General 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.
• Difference equations
• General
• Stability and asymptotics of difference equations; oscillatory and periodic solutions, etc.
• Discrete version of topics in analysis
• Difference equations, scaling ($q$-differences)
• Multiplicative and other generalized difference equations, e.g. of Lyness type
• Difference operators
• None of the above, but in this section
• Functional equations and inequalities
• General
• Iteration theory, iterative and composite equations
• Equations for real functions
• Equations for complex functions
• Matrix and operator equations
• Equations for functions with more general domains and/or ranges
• Orthogonal additivity and other conditional equations
• Functional inequalities, including subadditivity, convexity, etc.
• Systems of functional equations and inequalities
• Stability, separation, extension, and related topics
• None of the above, but in this section

## Sequences, series, summability

• 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.
• Convergence and divergence of infinite limiting processes
• Convergence and divergence of series and sequences
• Convergence and divergence of integrals
• Convergence and divergence of continued fractions
• Convergence and divergence of infinite products
• Approximation to limiting values (summation of series, etc.)
• Convergence and divergence of series and sequences of functions
• None of the above, but in this section
• Multiple sequences and series {(should also be assigned at least one other classification number in this section)]
• General summability methods
• Matrix methods
• Integral methods
• Function-theoretic methods (including power series methods and semicontinuous methods)
• None of the above, but in this section
• Direct theorems on summability
• General theorems
• Structure of summability fields
• Tauberian constants and oscillation limits
• Convergence factors and summability factors
• Summability and bounded fields of methods
• Inclusion and equivalence theorems
• None of the above, but in this section
• Inversion theorems
• Tauberian theorems, general
• Growth estimates
• Lacunary inversion theorems
• Tauberian constants
• None of the above, but in this section
• Absolute and strong summability
• Special methods of summability
• Cesàro, Euler, Nörlund and Hausdorff methods
• Abel, Borel and power series methods
• None of the above, but in this section
• Functional analytic methods in summability
• Summability in abstract structures

## Approximations and expansions

• 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.
• Interpolation
• Approximation by polynomials
• Spline approximation
• Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski\u\i-type inequalities)
• Approximation by rational functions
• Rate of convergence, degree of approximation
• Inverse theorems
• Simultaneous approximation
• Approximation with constraints
• Approximation by other special function classes
• Approximation by operators (in particular, by integral operators)
• Approximation by positive operators
• Saturation
• Best constants
• Approximation by arbitrary linear expressions
• Approximation by arbitrary nonlinear expressions; widths and entropy
• Best approximation, Chebyshev systems
• Uniqueness of best approximation
• Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
• Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
• Multidimensional problems (should also be assigned at least one other classification number in this section)
• Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
• Remainders in approximation formulas
• Miscellaneous topics

## Fourier analysis

• 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.
• Fourier analysis in one variable
• Trigonometric polynomials, inequalities, extremal problems
• Trigonometric approximation
• Trigonometric interpolation
• Fourier coefficients, Fourier series of functions with special properties, special Fourier series
• Convergence and absolute convergence of Fourier and trigonometric series
• Summability and absolute summability of Fourier and trigonometric series
• Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
• Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
• Multipliers
• Conjugate functions, conjugate series, singular integrals
• Lacunary series of trigonometric and other functions; Riesz products
• Probabilistic methods
• Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
• Completeness of sets of functions
• Trigonometric moment problems
• Classical almost periodic functions, mean periodic functions
• Positive definite functions
• Convolution, factorization
• None of the above, but in this section
• Fourier analysis in several variables
• Fourier series and coefficients
• Summability
• Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
• Multipliers
• Singular integrals (Calderón-Zygmund, etc.)
• Maximal functions, Littlewood-Paley theory
• $H^p$-spaces
• Function spaces arising in harmonic analysis
• None of the above, but in this section
• Nontrigonometric Fourier analysis
• Orthogonal functions and polynomials, general theory
• Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
• Series of general orthogonal functions, generalized Fourier expansions, nonorthogonal expansions
• Rearrangements and other transformations of Fourier and other orthogonal series
• Uniqueness and localization for orthogonal series
• Completeness of sets of functions
• Wavelets
• None of the above, but in this section

## Abstract harmonic analysis

• 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.
• Measures on groups and semigroups, etc.
• Means on groups, semigroups, etc.; amenable groups
• Measure algebras on groups, semigroups, etc.
• $L^p$-spaces and other function spaces on groups, semigroups, etc.
• Analysis on ordered groups, ${H]^p$-theory
• $L^1$-algebras on groups, semigroups, etc.
• Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
• Fourier and Fourier-Stieltjes transforms on locally compact abelian groups
• Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
• Other transforms and operators of Fourier type
• Positive definite functions on groups, semigroups, etc.
• Character groups and dual objects
• Spectral synthesis on groups, semigroups, etc.
• Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
• Convergence of Fourier series and of inverse transforms
• Summability methods on groups, semigroups, etc.
• Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
• Hypergroups
• Representations of groups, semigroups, etc.
• Analysis on specific locally compact abelian groups
• Analysis on specific compact groups
• Analysis on general compact groups
• Analysis on other specific Lie groups
• Analysis on homogeneous spaces
• Spherical functions
• Categorical methods
• Miscellaneous topics

## Integral transforms, operational calculus

• 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 transforms
• Laplace transform
• Special transforms (Legendre, Hilbert, etc.)
• Transforms of special functions
• Multiple transforms
• Convolution
• Calculus of Mikusi\'nski and other operational calculi
• Classical operational calculus
• Discrete operational calculus
• Moment problems
• Miscellaneous topics

## Integral equations

• 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.
• Linear integral equations
• Fredholm integral equations
• Eigenvalue problems
• Volterra integral equations
• Singular integral equations
• Integral equations with kernels of Cauchy type
• Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
• None of the above, but in this section
• Systems of linear integral equations
• Systems of nonsingular linear integral equations
• Dual, triple, etc., integral and series equations
• Systems of singular linear integral equations
• None of the above, but in this section
• Nonlinear integral equations
• Singular nonlinear integral equations
• Other nonlinear integral equations
• Systems of nonlinear integral equations
• Miscellaneous special kernels
• Integro-ordinary differential equations
• Integro-partial differential equations
• Theoretical approximation of solutions
• Qualitative behavior
• Asymptotics
• Stability theory
• Periodic solutions
• Positive solutions
• None of the above, but in this section
• Abstract integral equations, integral equations in abstract spaces
• Integral operators
• Inverse problems
• Random integral equations

## Functional analysis

• 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.
• Topological linear spaces and related structures
• General theory of locally convex spaces
• Locally convex Fréchet spaces and (DF)-spaces
• Barrelled spaces, bornological spaces
• Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
• Spaces defined by inductive or projective limits (LB, LF, etc.)
• Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
• Bornologies and related structures; Mackey convergence, etc.
• Other topological'' linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than ${\bf R]$, etc.)
• Duality theory
• Theorems of Hahn-Banach type; extension and lifting of functionals and operators
• Reflexivity and semi-reflexivity
• Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)
• Spaces of linear operators; topological tensor products; approximation properties
• Summability and bases
• Ordered topological linear spaces, vector lattices
• Sequence spaces (including Köthe sequence spaces)
• Compactness in topological linear spaces; angelic spaces, etc.
• Convex sets in topological linear spaces; Choquet theory
• Graded Fréchet spaces and tame operators
• Topological invariants ((DN), ($\Omega$), etc.)
• Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
• Modular spaces
• None of the above, but in this section
• Normed linear spaces and Banach spaces; Banach lattices
• Isomorphic theory (including renorming) of Banach spaces
• Isometric theory of Banach spaces
• Local theory of Banach spaces
• Ultraproduct techniques in Banach space theory
• Probabilistic methods in Banach space theory
• Duality and reflexivity
• Summability and bases
• Geometry and structure of normed linear spaces
• Radon-Nikodym, Krein-Milman and related properties
• Classical Banach spaces in the general theory
• Nonseparable Banach spaces
• Spaces of operators; tensor products; approximation properties
• Ordered normed spaces
• Banach lattices
• Banach sequence spaces
• Compactness in Banach (or normed) spaces
• Interpolation between normed linear spaces
• None of the above, but in this section
• Inner product spaces and their generalizations, Hilbert spaces
• Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
• Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.)
• Characterizations of Hilbert spaces
• Spaces with indefinite inner product (Krein spaces, Pontryagin spaces, etc.)
• Generalizations of inner products (semi-inner products, partial inner products, etc.)
• None of the above, but in this section
• Linear function spaces and their duals
• Lattices of continuous, differentiable or analytic functions
• Topological linear spaces of continuous, differentiable or analytic functions
• Banach spaces of continuous, differentiable or analytic functions
• Hilbert spaces of continuous, differentiable or analytic functions
• Hilbert spaces with reproducing kernels (= proper functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
• Rings and algebras of continuous, differentiable or analytic functions
• Spaces of measures
• Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
• Sobolev (and similar kinds of) spaces of functions of discrete variables
• Spaces of vector- and operator-valued functions
• Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
• None of the above, but in this section
• Distributions, generalized functions, distribution spaces
• Topological linear spaces of test functions, distributions and ultradistributions
• Operations with distributions
• Integral transforms in distribution spaces
• Hyperfunctions, analytic functionals
• Distributions and ultradistributions as boundary values of analytic functions
• Distributions on infinite-dimensional spaces
• Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
• None of the above, but in this section
• Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)
• Derivatives
• Vector-valued measures and integration
• Measures and integration on abstract linear spaces
• Functional analytic lifting theory
• Infinite-dimensional holomorphy
• (Spaces of) multilinear mappings, polynomials
• None of the above, but in this section
• Topological algebras, normed rings and algebras, Banach algebras
• General theory of topological algebras
• Ideals and subalgebras
• Representations of topological algebras
• Structure, classification of topological algebras
• Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
• Functional calculus in topological algebras
• Topological algebras of operators
• Automatic continuity
• Nonassociative topological algebras
• None of the above, but in this section
• Commutative Banach algebras and commutative topological algebras
• General theory of commutative topological algebras
• Banach algebras of continuous functions, function algebras
• Banach algebras of differentiable or analytic functions, ${H]^p$-spaces
• Ideals, maximal ideals, boundaries
• Representations of commutative topological algebras
• Subalgebras
• Structure, classification of commutative topological algebras
• None of the above, but in this section
• Topological (rings and) algebras with an involution
• General theory of topological algebras with involution
• Representations of topological algebras with involution
• Hilbert algebras
• Nonselfadjoint (sub)algebras in algebras with involution
• Nonassociative topological algebras with an involution
• None of the above, but in this section
• Selfadjoint operator algebras ($C^*$-algebras, von Neumann ($W$*-) algebras, etc.)
• General theory of $C^*$-algebras
• Tensor products of $C^*$-algebras
• Operator spaces and completely bounded maps
• $C^*$-modules
• Free products of $C^*$-algebras
• General theory of von Neumann algebras
• States
• Classifications of $C^*$-algebras, factors
• Subfactors and their classification
• Automorphisms
• Decomposition theory for $C^*$-algebras
• Noncommutative measure and integration
• Noncommutative function spaces
• Noncommutative probability and statistics
• Free probability and free operator algebras
• Noncommutative dynamical systems
• Derivations, dissipations and positive semigroups in $C^*$-algebras
• Applications of selfadjoint operator algebras to physics
• Quantizations, deformations
• $K$-theory and operator algebras (including cyclic theory)
• Noncommutative topology
• Noncommutative differential geometry
• Other noncommutative'' mathematics based on $C^*$-algebra theory
• None of the above, but in this section
• Methods of category theory in functional analysis
• Tensor products
• Ultraproducts
• Projective and injective objects
• Categories, functors
• Homological methods (exact sequences, right inverses, lifting, etc.)
• Methods of algebraic topology (cohomology, sheaf and bundle theory, etc.)
• Abstract interpolation of topological vector spaces
• Inductive and projective limits
• None of the above, but in this section
• Miscellaneous applications of functional analysis
• Applications in optimization, convex analysis, mathematical programming, economics
• Applications to differential and integral equations
• Applications in probability theory and statistics
• Applications in numerical analysis
• Applications in quantum physics
• Applications in statistical physics
• Applications in biology and other sciences
• None of the above, but in this section
• Other (nonclassical) types of functional analysis
• Functional analysis over fields other than <B>R</B> or <B>C</B> or the quaternions; non-Archimedean functional analysis
• Nonstandard functional analysis
• Constructive functional analysis
• Fuzzy functional analysis
• Functional analysis in probabilistic metric linear spaces
• Functional analysis on superspaces (supermanifolds) or graded spaces
• None of the above, but in this section
• Nonlinear functional analysis
• Infinite-dimensional manifolds
• Manifolds of mappings
• Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds
• Continuous and differentiable maps
• Holomorphic maps
• Distributions and generalized functions on nonlinear spaces
• None of the above, but in this section

## Operator 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.
• General theory of linear operators
• General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
• Linear relations (multivalued linear operators)
• Forms (bilinear, sesquilinear, multilinear)
• Spectrum, resolvent
• Local spectral properties
• Several-variable operator theory (spectral, Fredholm, etc.)
• Invariant subspaces
• Cyclic and hypercyclic vectors
• Dilations, extensions, compressions
• Spectral sets
• Norms (inequalities, more than one norm, etc.)
• Ergodic theory
• Scattering theory
• Canonical models for contractions and nonselfadjoint operators
• Chains (nests) of projections or of invariant subspaces, integrals along chains, etc.
• Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
• Equations and inequalities involving linear operators, with vector unknowns
• Ill-posed problems, regularization
• (Semi-) Fredholm operators; index theories
• Perturbation theory
• Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
• Operator methods in interpolation, moment and extension problems
• Operator approximation theory
• Functional calculus
• Equations involving linear operators, with operator unknowns
• Operator inequalities
• Operator means, shorted operators, etc.
• Structure theory
• Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal operators
• Representation theory
• Factorization theory (including Wiener-Hopf and spectral factorizations)
• (Generalized) eigenfunction expansions; rigged Hilbert spaces
• Eigenvalue problems
• Tensor products of operators
• None of the above, but in this section
• Special classes of linear operators
• Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
• Operators defined by compactness properties
• Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.)
• Hermitian and normal operators (spectral measures, functional calculus, etc.)
• Subnormal operators, hyponormal operators, etc.
• Symmetric and selfadjoint operators (unbounded)
• Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
• Composition operators
• Kernel operators
• Toeplitz operators, Hankel operators, Wiener-Hopf operators
• Jacobi (tridiagonal) operators (matrices) and generalizations
• Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
• Operators on function spaces (general)
• Difference operators
• Spectral operators, decomposable operators, well-bounded operators, etc.
• Accretive operators, dissipative operators, etc.
• Commutators, derivations, elementary operators, etc.
• Operators on Banach algebras
• Transformers (= operators on spaces of operators)
• Operators on spaces with an indefinite metric
• Operators on ordered spaces
• Positive operators and order-bounded operators
• Random operators
• None of the above, but in this section
• Individual linear operators as elements of algebraic systems
• Operators in algebras
• Operators in $^*$-algebras
• Operators in $C^*$- or von Neumann algebras
• None of the above, but in this section
• Groups and semigroups of linear operators, their generalizations and applications
• Groups and semigroups of linear operators
• One-parameter semigroups and linear evolution equations
• Markov semigroups and applications to diffusion processes
• Schrödinger and Feynman-Kac semigroups
• Operator sine and cosine functions and higher-order Cauchy problems
• $C$-semigroups
• Integrated semigroups
• None of the above, but in this section
• Ordinary differential operators
• Partial differential operators
• Integral, integro-differential, and pseudodifferential operators
• Integral operators
• Integro-differential operators
• Pseudodifferential operators
• None of the above, but in this section
• Nonlinear operators and their properties
• Set-valued operators
• Monotone operators (with respect to duality)
• Accretive operators, dissipative operators, etc.
• Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
• Nonexpansive mappings, and their generalizations (ultimately compact mappings, measures of noncompactness and condensing mappings, $A$-proper mappings, $K$-set contractions, etc.)
• Fixed-point theorems
• Degree theory
• Perturbations of nonlinear operators
• Semigroups of nonlinear operators
• Particular nonlinear operators (superposition, Hammerstein, Nemytskii, Uryson, etc.)
• Random operators
• Potential operators
• Multilinear and polynomial operators
• None of the above, but in this section
• Equations and inequalities involving nonlinear operators
• Equations involving nonlinear operators (general)
• Nonlinear ill-posed problems
• Abstract inverse mapping and implicit function theorems
• Nonlinear eigenvalue problems
• Abstract bifurcation theory
• Variational and other types of inequalities involving nonlinear operators (general)
• Methods for solving nonlinear operator equations (general)
• Variational methods
• Nonlinear evolution equations
• Equations with hysteresis operators
• None of the above, but in this section
• Linear spaces and algebras of operators
• Linear spaces of operators
• Convex sets and cones of operators
• Algebras of operators on Banach spaces and other topological linear spaces
• Operator algebras with symbol structure
• Operator ideals
• Operator spaces (= matricially normed spaces)
• Abstract operator algebras on Hilbert spaces
• Nest algebras, CSL algebras
• Limit algebras, subalgebras of $C^*$-algebras
• Dual algebras; weakly closed singly generated operator algebras
• Dual spaces of operator algebras
• Representations of (nonselfadjoint) operator algebras
• Algebras of unbounded operators; partial algebras of operators
• Crossed product algebras (analytic crossed products)
• Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
• Applications of operator algebras to physics
• None of the above, but in this section
• Miscellaneous applications of operator theory
• Applications in optimization, convex analysis, mathematical programming, economics
• Applications to differential and integral equations
• Applications in probability theory and statistics
• Applications in numerical analysis
• Applications in quantum physics
• Applications in statistical physics
• Applications in biology and other sciences
• Applications in systems theory, circuits, etc.
• None of the above, but in this section
• Other (nonclassical) types of operator theory
• Operator theory over fields other than <B>R</B>, <B>C</B> or the quaternions; non-Archimedean operator theory
• Nonstandard operator theory
• Constructive operator theory
• Fuzzy operator theory
• Operator theory in probabilistic metric linear spaces
• None of the above, but in this section

## Calculus of variations and optimal control; optimization

• 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.
• Existence theories
• Free problems in one independent variable
• Free problems in two or more independent variables
• Optimal control problems involving ordinary differential equations
• Optimal control problems involving partial differential equations
• Optimal control problems involving integral equations
• Optimal control problems involving differential inclusions
• Optimal control problems involving equations with retarded arguments
• Problems in abstract spaces
• Optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
• Minimax problems
• Variational methods including variational inequalities
• Methods involving semicontinuity and convergence; relaxation
• Fréchet and Gateaux differentiability
• Nonsmooth analysis
• Set-valued and variational analysis
• Problems involving randomness
• None of the above, but in this section
• Necessary conditions and sufficient conditions for optimality
• Free problems in one independent variable
• Free problems in two or more independent variables
• Problems involving ordinary differential equations
• Problems involving partial differential equations
• Problems involving integral equations
• Problems involving differential inclusions
• Problems involving equations with retarded arguments
• Problems in abstract spaces
• Optimal solutions belonging to restricted classes
• Minimax problems
• Sensitivity, stability, well-posedness
• Problems involving randomness
• None of the above, but in this section
• Hamilton-Jacobi theories, including dynamic programming
• Dynamic programming method
• Viscosity solutions
• None of the above, but in this section
• Methods of successive approximations
• Methods based on necessary conditions
• Methods of Newton-Raphson, Galerkin and Ritz types
• Methods of relaxation type
• Discrete approximations
• Decomposition methods
• Methods involving duality
• Other methods, not based on necessary conditions (penalty function, etc.)
• Methods of nonlinear programming type
• None of the above, but in this section
• Miscellaneous topics
• Linear optimal control problems
• Duality theory
• Periodic optimization
• Impulsive optimal control problems
• Problems with incomplete information
• Optimal feedback synthesis
• Inverse problems
• Regularity of solutions
• Differential games
• Pursuit and evasion games
• Applications of optimal control and differential games
• None of the above, but in this section
• Manifolds
• Minimal surfaces
• Optimization of shapes other than minimal surfaces
• Sensitivity analysis
• Geometric measure and integration theory, integral and normal currents
• Variational problems in a geometric measure-theoretic setting
• None of the above, but in this section
• Variational methods for eigenvalues of operators
• Variational principles of physics

## 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.
• Linear incidence geometry
• General theory and projective geometries
• Homomorphism, automorphism and dualities
• Structures with parallelism
• Configuration theorems
• Algebraization
• Desarguesian and Pappian geometries
• Non-Desarguesian affine and projective planes
• Incidence structures imbeddable into projective geometries
• Polar geometry, symplectic spaces, orthogonal spaces
• None of the above, but in this section
• Nonlinear incidence geometry
• General theory
• Möbius geometries
• Laguerre geometries
• Minkowski geometries
• Lie geometries
• None of the above, but in this section
• Ring geometry (Hjelmslev, Barbilian, etc.)
• Geometric closure systems
• Abstract (Maeda) geometries
• Abstract geometries with exchange axiom
• Abstract geometries with parallelism
• Combinatorial geometries
• Lattices of subspaces
• Continuous geometries and related topics
• None of the above, but in this section
• Finite geometry and special incidence structures
• General block designs
• Steiner systems
• Finite partial geometries (general), nets, partial spreads
• Affine and projective planes
• Combinatorial structures in finite projective spaces
• Blocking sets, ovals, $k$-arcs
• Linear codes and caps in Galois spaces
• Buildings and the geometry of diagrams
• Other finite nonlinear geometries
• Other finite linear geometries
• Other finite incidence structures
• None of the above, but in this section
• Metric geometry
• Absolute planes
• Absolute spaces
• Reflection groups, reflection geometries
• Congruence and orthogonality
• Orthogonal and unitary groups
• None of the above, but in this section
• Ordered geometries (ordered incidence structures, etc.)
• Topological geometry
• General theory
• Topological linear incidence structures
• Topological nonlinear incidence structures
• Topological geometries on manifolds
• Geometries with differentiable structure
• Geometries with algebraic manifold structure
• None of the above, but in this section
• Incidence groups
• General theory
• Projective incidence groups
• Kinematic spaces
• Representation by near-fields and near-algebras
• None of the above, but in this section
• Distance geometry
• General theory
• Synthetic differential geometry
• None of the above, but in this section
• Geometric order structures
• Geometry of orders of nondifferentiable curves
• Directly differentiable curves
• $n$-vertex theorems via direct methods
• Geometry of orders of surfaces
• None of the above, but in this section
• Real and complex geometry
• Elementary problems in Euclidean geometries
• Euclidean geometries (general) and generalizations
• Elementary problems in hyperbolic and elliptic geometries
• Hyperbolic and elliptic geometries (general) and generalizations
• Geometric constructions
• Inequalities and extremum problems
• Polyhedra and polytopes; regular figures, division of spaces
• Length, area and volume
• Line geometries and their generalizations
• Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
• None of the above, but in this section
• Analytic and descriptive geometry
• Descriptive geometry
• Affine analytic geometry
• Projective analytic geometry
• Euclidean analytic geometry
• Analytic geometry with other transformation groups
• Geometry of classical groups
• Questions of classical algebraic geometry
• None of the above, but in this section
• Geometry and physics (should also be assigned at least one other classification number from Sections 70--86)

## Convex and discrete 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.
• General convexity
• Axiomatic and generalized convexity
• Convex sets without dimension restrictions
• Convex sets in topological vector spaces
• Convex sets in $2$ dimensions (including convex curves)
• Convex sets in $3$ dimensions (including convex surfaces)
• Convex sets in $n$ dimensions (including convex hypersurfaces)
• Finite-dimensional Banach spaces (including special norms, zonoids, etc.)
• Random convex sets and integral geometry
• Approximation by convex sets
• Variants of convex sets (star-shaped, ($m, n$)-convex, etc.)
• Helly-type theorems and geometric transversal theory
• Other problems of combinatorial convexity
• Length, area, volume
• Mixed volumes and related topics
• Inequalities and extremum problems
• Convex functions and convex programs
• Spherical and hyperbolic convexity
• None of the above, but in this section
• Polytopes and polyhedra
• Combinatorial properties (number of faces, shortest paths, etc.)
• Three-dimensional polytopes
• $n$-dimensional polytopes
• Special polytopes (linear programming, centrally symmetric, etc.)
• Symmetry properties of polytopes
• Lattice polytopes (including relations with commutative algebra and algebraic geometry)
• Shellability
• Gale and other diagrams
• Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
• Dissections and valuations (Hilbert's third problem, etc.)
• Computational aspects related to convexity
• Isoperimetric problems for polytopes
• Polyhedral manifolds
• None of the above, but in this section
• Discrete geometry
• Lattices and convex bodies in $2$ dimensions
• Lattices and convex bodies in $n$ dimensions
• Erdös problems and related topics of discrete geometry
• Packing and covering in $2$ dimensions
• Packing and covering in $n$ dimensions
• Tilings in $2$ dimensions
• Tilings in $n$ dimensions
• Quasicrystals, aperiodic tilings
• Rigidity and flexibility of structures
• Circle packings and discrete conformal geometry
• Planar arrangements of lines and pseudolines
• Arrangements of points, flats, hyperplanes
• Oriented matroids
• Combinatorial complexity of geometric structures
• None of the above, but in this section

## Differential 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.
• Classical differential geometry
• Curves in Euclidean space
• Surfaces in Euclidean space
• Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
• Minimal surfaces, surfaces with prescribed mean curvature
• Affine differential geometry
• Kinematics
• Projective differential geometry
• Differential line geometry
• Conformal differential geometry
• Non-Euclidean differential geometry
• Other special differential geometries
• Vector and tensor analysis
• Differential invariants (local theory), geometric objects
• Geometry of webs
• None of the above, but in this section
• Local differential geometry
• Linear and affine connections
• Projective connections
• Other connections
• Local Riemannian geometry
• Methods of Riemannian geometry
• Local submanifolds
• Lorentz metrics, indefinite metrics
• Hermitian and Kählerian structures
• Finsler spaces and generalizations (areal metrics)
• Applications to physics
• None of the above, but in this section
• Global differential geometry
• Connections, general theory
• Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)
• $G$-structures
• Foliations (differential geometric aspects)
• General geometric structures on manifolds (almost complex, almost product structures, etc.)
• Sub-Riemannian geometry
• Global Riemannian geometry, including pinching
• Methods of Riemannian geometry, including PDE methods; curvature restrictions
• Geodesics
• Global topological methods (à la Gromov)
• Rigidity results
• Special Riemannian manifolds (Einstein, Sasakian, etc.)
• Hyper-Kähler and quaternionic Kähler geometry, special'' geometry
• Spin and Spin$^c$ geometry
• Twistor methods
• Issues of holonomy
• Homogeneous manifolds
• Symmetric spaces
• Calibrations and calibrated geometries
• Global submanifolds
• Immersions (minimal, prescribed curvature, tight, etc.)
• Differential geometric aspects of harmonic maps
• Geometric evolution equations (mean curvature flow)
• Global surface theory (convex surfaces à la A. D. Aleksandrov)
• Lorentz manifolds, manifolds with indefinite metrics
• Hermitian and Kählerian manifolds
• Other complex differential geometry
• Finsler spaces and generalizations (areal metrics)
• Integral geometry; differential forms, currents, etc.
• Direct methods ($G$-spaces of Busemann, etc.)
• Geometric orders, order geometry
• Applications to physics
• None of the above, but in this section
• Symplectic geometry, contact geometry
• Symplectic manifolds, general
• Contact manifolds, general
• Lagrangian submanifolds; Maslov index
• Almost contact and almost symplectic manifolds
• Poisson manifolds
• Momentum maps; symplectic reduction
• Canonical transformations
• Geodesic flows
• Symplectic structures of moduli spaces
• Global theory of symplectic and contact manifolds
• Floer homology and cohomology, symplectic aspects
• Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
• Geometric quantization
• Deformation quantization, star products
• None of the above, but in this section
• Applications to physics

## General topology

• 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.
• Generalities
• Topological spaces and generalizations (closure spaces, etc.)
• Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
• Syntopogeneous structures
• Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)
• Cardinality properties (cardinal functions and inequalities, discrete subsets)
• Consistency and independence results
• Fuzzy topology
• None of the above, but in this section
• Basic constructions
• Subspaces
• Product spaces
• Quotient spaces, decompositions
• Adjunction spaces and similar constructions
• Hyperspaces
• Categorical methods
• Spectra
• Presheaves and sheaves
• None of the above, but in this section
• Maps and general types of spaces defined by maps
• Continuous maps
• Weak and generalized continuity
• Special maps on topological spaces (open, closed, perfect, etc.)
• Retraction
• Extension of maps
• Embedding
• Real-valued functions
• Function spaces
• Algebraic properties of function spaces
• $C$- and $C^*$-embedding
• Special sets defined by functions
• Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
• Shape theory
• Set-valued maps
• Selections
• Entropy
• None of the above, but in this section
• Fairly general properties
• Connected and locally connected spaces (general aspects)
• Lower separation axioms ($T_0$--$T_3$, etc.)
• Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
• Noncompact covering properties (paracompact, Lindelöf, etc.)
• $P$-minimal'' and $P$-closed'' spaces
• Compactness
• Extensions of spaces (compactifications, supercompactifications, completions, etc.)
• Remainders
• Local compactness, $\sigma$-compactness
• $k$-spaces
• Sequential spaces
• Realcompactness and realcompactification
• Separability
• Base properties
• Special constructions of spaces (spaces of ultrafilters, etc.)
• None of the above, but in this section
• Spaces with richer structures
• Proximity structures and generalizations
• Uniform structures and generalizations
• Nearness spaces
• $p$-spaces, $M$-spaces, $\sigma$-spaces, etc.
• Stratifiable spaces, cosmic spaces, etc.
• Semimetric spaces
• Moore spaces
• Metric spaces, metrizability
• Special maps on metric spaces
• Compact (locally compact) metric spaces
• Complete metric spaces
• Baire category, Baire spaces
• Bitopologies
• Probabilistic metric spaces
• None of the above, but in this section
• Special properties
• Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
• Continua and generalizations
• Higher-dimensional local connectedness
• Dimension theory
• Spaces of dimension $\leq 1$; curves, dendrites
• Unicoherence, multicoherence
• Topological characterizations of particular spaces
• None of the above, but in this section
• Peculiar spaces
• Extremally disconnected spaces, $F$-spaces, etc.
• $P$-spaces
• Scattered spaces
• Pathological spaces
• Counterexamples
• None of the above, but in this section
• Connections with other structures, applications
• Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• Topological representations of algebraic systems
• Topological groups
• Topological lattices, etc.
• Topological fields, rings, etc.
• Transformation groups and semigroups
• Topological dynamics
• Fixed-point and coincidence theorems
• None of the above, but in this section
• Nonstandard topology
• Algebraic topology
• 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.

## Classical topics

• Duality
• Dimension theory
• Absolute neighborhood retracts
• Fixed points and coincidences
• Degree, winding number
• Ljusternik-Schnirelman (Lyusternik-Shnirelman) category of a space
• Finite groups of transformations (including Smith theory)
• None of the above, but in this section
• Homology and cohomology theories
• Cech types
• Steenrod-Sitnikov homologies
• Singular theory
• $K$-theory
• Generalized (extraordinary) homology and cohomology theories
• Bordism and cobordism theories, formal group laws
• Homology with local coefficients, equivariant cohomology
• Sheaf cohomology
• Intersection homology and cohomology
• Elliptic cohomology
• Other homology theories
• Axioms for homology theory and uniqueness theorems
• Products and intersections
• Equivariant homology and cohomology
• None of the above, but in this section
• Homotopy theory
• Homotopy extension properties, cofibrations
• Homotopy equivalences
• Classification of homotopy type
• Eilenberg-Mac Lane spaces
• Eckmann-Hilton duality
• Loop spaces
• Suspensions
• Stable homotopy theory, spectra
• Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
• ${H]$-spaces and duals
• Infinite loop spaces
• Shape theory
• Proper homotopy theory
• Localization and completion
• Rational homotopy theory
• Homotopy functors
• Equivariant homotopy theory
• Relations between equivariant and nonequivariant homotopy theory
• None of the above, but in this section
• Homotopy groups
• Homotopy groups, general; sets of homotopy classes
• Shape groups
• Stable homotopy groups
• Homotopy groups of wedges, joins, and simple spaces
• Hopf invariants
• Operations in homotopy groups
• Homotopy groups of spheres
• Stable homotopy of spheres
• $J$-morphism
• $v_n$-periodicity
• Homotopy groups of special spaces
• Cohomotopy groups
• Homotopy groups of special types
• Equivariant homotopy groups
• None of the above, but in this section
• Fiber spaces and bundles
• Fiber spaces
• Fiber bundles
• Transfer
• Classification
• Spectral sequences and homology of fiber spaces
• Sphere bundles and vector bundles
• Classifying spaces of groups and ${H]$-spaces
• Maps between classifying spaces
• Homology of classifying spaces, characteristic classes
• Homology and homotopy of $B{\rm O]$ and $B{\rm U]$; Bott periodicity
• Stable classes of vector space bundles, $K$-theory
• Fiberings with singularities
• Microbundles and block bundles
• Generalizations of fiber spaces and bundles
• Fibrewise topology
• Discriminantal varieties, configuration spaces
• Equivariant fiber spaces and bundles
• None of the above, but in this section
• Operations and obstructions
• Primary cohomology operations
• Steenrod algebra
• Dyer-Lashof operations
• Symmetric products, cyclic products
• Secondary and higher cohomology operations
• $K$-theory operations and generalized cohomology operations
• Massey products
• Obstruction theory
• Extension and compression of mappings
• Classification of mappings
• Sectioning fiber spaces and bundles
• Postnikov systems, $k$-invariants
• Equivariant operations and obstructions
• None of the above, but in this section
• Spectral sequences
• General
• Serre spectral sequences
• Eilenberg-Moore spectral sequences
• Generalized cohomology
• None of the above, but in this section
• Applied homological algebra and category theory
• Abstract complexes
• Simplicial sets and complexes
• Chain complexes
• Universal coefficient theorems, Bockstein operator
• Homology of a product, Künneth formula
• Duality
• Abstract and axiomatic homotopy theory
• Topological categories, foundations of homotopy theory
• None of the above, but in this section

## Manifolds and cell complexes

• 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.
• Low-dimensional topology
• Fundamental group, presentations, free differential calculus
• Topological methods in group theory
• Covering spaces
• Special coverings, e.g. branched
• Relations with graph theory
• Two-dimensional complexes
• Knots and links in $S^3$
• Invariants of knots and 3-manifolds
• Wild knots and surfaces, etc., wild embeddings
• Dehn's lemma, sphere theorem, loop theorem, asphericity
• Characterizations of $E^3$ and $S^3$ (Poincaré conjecture)
• Geometric structures on low-dimensional manifolds
• Group actions in low dimensions
• None of the above, but in this section
• Topological manifolds
• Topology of $E^2$, $2$-manifolds
• Topology of general $3$-manifolds
• Topology of $E^3$ and $S^3$
• Topology of $E^4$, $4$-manifolds
• Topology of $E^n$, $n$-manifolds ($4 &lt; n &lt; \infty$)
• Geometric structures on manifolds
• Topology of topological vector spaces
• Topology of infinite-dimensional manifolds
• Shapes
• Engulfing
• Embeddings and immersions
• Isotopy and pseudo-isotopy
• Neighborhoods of submanifolds
• Flatness and tameness
• $S^{n-1]\subset E^n$, Schoenflies problem
• Microbundles and block bundles
• Cellularity
• Algebraic topology of manifolds
• Cobordism and concordance
• General position and transversality
• Stratifications
• None of the above, but in this section
• Generalized manifolds
• Local properties of generalized manifolds
• Poincaré duality spaces
• None of the above, but in this section
• PL-topology
• General topology of complexes
• Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
• Wall finiteness obstruction for CW-complexes
• Triangulating manifolds
• Cobordism
• Comparison of PL-structures: classification, Hauptvermutung
• Engulfing
• Embeddings and immersions
• Isotopy
• Regular neighborhoods
• Knots and links (in high dimensions)
• Microbundles and block bundles
• Approximations
• Cobordism and concordance
• General position and transversality
• Equivariant PL-topology
• None of the above, but in this section
• Differential topology
• Triangulating
• Smoothing
• Smooth approximations
• Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
• Symplectic and contact topology
• Algebraic topology on manifolds
• Characteristic classes and numbers
• Topology of vector bundles and fiber bundles
• Vector fields, frame fields
• Controllability of vector fields on $C^\infty$ and real-analytic manifolds
• Foliations; geometric theory
• Classifying spaces for foliations; Gelfand-Fuks cohomology
• Differentiable mappings
• Embeddings
• Immersions
• Singularities of differentiable mappings
• Diffeomorphisms
• Isotopy
• Differentiable structures
• Topological quantum field theories
• Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants
• Floer homology
• Homotopy spheres, Poincaré conjecture
• Surgery and handlebodies
• Surgery obstructions, Wall groups
• Critical points and critical submanifolds
• O- and SO-cobordism
• Complex cobordism (U- and SU-cobordism)
• $h$- and $s$-cobordism
• Equivariant cobordism
• Other types of cobordism
• Equivariant algebraic topology of manifolds
• Realizing cycles by submanifolds
• None of the above, but in this section
• Topological transformation groups
• Topological properties of groups of homeomorphisms or diffeomorphisms
• Compact groups of homeomorphisms
• Compact Lie groups of differentiable transformations
• Finite transformation groups
• Noncompact Lie groups of transformations
• Groups acting on specific manifolds
• Discontinuous groups of transformations
• None of the above, but in this section
• Homology and homotopy of topological groups and related structures
• Hopf algebras
• Homology and cohomology of Lie groups
• Homology and cohomology of homogeneous spaces of Lie groups
• Homotopy groups of topological groups and homogeneous spaces
• Homology and cohomology of ${H]$-spaces
• Bar and cobar constructions
• Applications of Eilenberg-Moore spectral sequences
• None of the above, but in this section

## Global analysis, analysis on manifolds

• 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 theory of differentiable manifolds
• Topos-theoretic approach to differentiable manifolds
• Differentiable manifolds, foundations
• Real-analytic and Nash manifolds
• Differential forms
• de Rham theory
• Hodge theory
• Exterior differential systems (Cartan theory)
• Pfaffian systems
• Jets
• Currents
• Vector distributions (subbundles of the tangent bundles)
• Natural bundles
• Stratified sets
• Differential spaces
• None of the above, but in this section
• Infinite-dimensional manifolds
• Homotopy and topological questions
• Differentiability questions
• Questions of holomorphy
• Fredholm structures
• Riemannian, Finsler and other geometric structures
• Group structures and generalizations on infinite-dimensional manifolds
• Geometry of quantum groups
• Noncommutative geometry (à la Connes)
• None of the above, but in this section
• Calculus on manifolds; nonlinear operators
• Real-valued functions
• Set valued and function-space valued mappings
• Continuity properties of mappings
• Holomorphic maps
• Implicit function theorems; global Newton methods
• Differentiation theory (Gateaux, Fréchet, etc.)
• Differentiable maps
• Fixed point theorems on manifolds
• Integration on manifolds; measures on manifolds
• Spectral theory; eigenvalue problems
• Analysis on supermanifolds or graded manifolds
• None of the above, but in this section
• Spaces and manifolds of mappings (including nonlinear versions of 46Exx)
• Groups of diffeomorphisms and homeomorphisms as manifolds
• Groups and semigroups of nonlinear operators
• Spaces of imbeddings and immersions
• Manifolds of mappings
• Manifolds of metrics (esp. Riemannian)
• Group actions and symmetry properties
• Measures (Gaussian, cylindrical, etc.) on manifolds of maps
• Equations in function spaces; evolution equations
• Moduli problems for differential geometric structures
• Moduli problems for topological structures
• Applications (in quantum mechanics (Feynman path integrals), relativity, fluid dynamics, etc.)
• None of the above, but in this section
• Variational problems in infinite-dimensional spaces
• Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirelman) theory, etc.)
• Abstract bifurcation theory
• Group-invariant bifurcation theory
• Applications to the theory of geodesics (problems in one independent variable)
• Critical metrics
• Applications to minimal surfaces (problems in two independent variables)
• Application to extremal problems in several variables; Yang-Mills functionals, etc.
• Pareto optimality, etc., applications to economics
• Harmonic maps, etc.
• Applications to control theory
• Variational principles
• Variational inequalities (global problems)
• Group actions
• Applications
• None of the above, but in this section
• Pseudogroups, differentiable groupoids and general structures on manifolds
• Pseudogroups and differentiable groupoids
• Cohomology of classifying spaces for pseudogroup structures (Spencer, Gelfand-Fuks, etc.)
• Deformations of structures
• None of the above, but in this section
• Partial differential equations on manifolds; differential operators
• Elliptic equations on manifolds, general theory
• Differential complexes; elliptic complexes
• Relations with hyperfunctions
• Index theory and related fixed point theorems
• Exotic index theories
• Elliptic genera
• Eta-invariants, Chern-Simons invariants
• Spectral flows
• Boundary value problems on manifolds
• Heat and other parabolic equation methods
• Perturbations; asymptotics
• Pseudodifferential and Fourier integral operators on manifolds
• Noncommutative global analysis, noncommutative residues
• Hyperbolic equations
• Propagation of singularities; initial value problems
• Spectral problems; spectral geometry; scattering theory
• Determinants and determinant bundles, analytic torsion
• Isospectrality
• Bifurcation
• Relations with special manifold structures (Riemannian, Finsler, etc.)
• Diffusion processes and stochastic analysis on manifolds
• Invariance and symmetry properties
• Correspondences and other transformation methods (e.g. Lie-Bäcklund)
• Applications
• None of the above, but in this section
• Theory of singularities and catastrophe theory
• Critical points of functions and mappings
• Monodromy
• Topological properties of mappings
• Algebraic and analytic properties of mappings
• Stability
• Global theory
• Catastrophe theory
• Classification; finite determinacy of map germs
• Singularities of vector fields, topological aspects
• Normal forms
• Asymptotic behavior
• Deformation of singularities
• Topological invariants
• Symmetries, equivariance
• None of the above, but in this section
• Applications to physics

## Probability theory and stochastic processes

• 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.
• Computational methods (not classified at a more specific level)
• Foundations of probability theory
• Axioms; other general questions
• Probabilistic measure theory
• None of the above, but in this section
• Probability theory on algebraic and topological structures
• Probability measures on topological spaces
• Convergence of probability measures
• Probability theory on linear topological spaces
• Limit theorems for vector-valued random variables (infinite-dimensional case)
• Probability measures on groups, Fourier transforms, factorization
• None of the above, but in this section
• Combinatorial probability
• Geometric probability, stochastic geometry, random sets
• Distribution theory
• Distributions: general theory
• Infinitely divisible distributions; stable distributions
• Characteristic functions; other transforms
• Inequalities; stochastic orderings
• None of the above, but in this section
• Limit theorems
• Central limit and other weak theorems
• Large deviations
• Strong theorems
• Functional limit theorems; invariance principles
• Zero-one laws
• $L^p$-limit theorems
• None of the above, but in this section
• Stochastic processes
• Foundations of stochastic processes
• General theory of processes
• Exchangeability
• Stationary processes
• General second-order processes
• Gaussian processes
• Sample path properties
• Self-similar processes
• Generalized stochastic processes
• Prediction theory
• Continuity and singularity of induced measures
• Applications (signal detection, filtering, etc.)
• Stopping times; optimal stopping problems; gambling theory
• Martingales with discrete parameter
• Martingales with continuous parameter
• Martingales and classical analysis
• Generalizations of martingales
• Sums of independent random variables; random walks
• Processes with independent increments
• Stable processes
• Point processes
• Random measures
• Random fields
• Extreme value theory; extremal processes
• None of the above, but in this section
• Stochastic analysis
• Stochastic integrals
• Stochastic calculus of variations and the Malliavin calculus
• Stochastic ordinary differential equations
• Stochastic partial differential equations
• Stochastic integral equations
• Random operators and equations
• Applications of stochastic analysis (to PDE, etc.)
• Computational methods for stochastic equations
• White noise theory
• None of the above, but in this section
• Markov processes
• Markov processes with discrete parameter
• Markov chains with discrete parameter
• Applications of discrete Markov processes (social mobility, learning theory, industrial processes, etc.)
• Computational methods in Markov chains
• Markov processes with continuous parameter
• Markov chains with continuous parameter
• Transition functions, generators and resolvents
• Right processes
• Probabilistic potential theory
• Boundary theory
• Local time and additive functionals
• Multiplicative functionals
• Diffusion processes
• Brownian motion
• Applications of diffusion theory (population genetics, absorption problems, etc.)
• Jump processes
• Branching processes (Galton-Watson, birth-and-death, etc.)
• Applications of branching processes
• None of the above, but in this section
• Special processes
• Renewal theory
• Applications (reliability, demand theory, etc.)
• Markov renewal processes, semi-Markov processes
• Applications of Markov renewal processes (reliability, queueing networks, etc.)
• Queueing theory
• Applications (congestion, allocation, storage, traffic, etc.)
• Interacting random processes; statistical mechanics type models; percolation theory
• Processes in random environments
• Other physical applications of random processes
• None of the above, but in this section

## Statistics

• 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.
• Data analysis
• Graphical methods
• Foundational and philosophical topics
• Sufficiency and information
• Sufficient statistics and fields
• Information-theoretic topics
• Theory of statistical experiments
• None of the above, but in this section
• Decision theory
• General considerations
• Complete class results
• Bayesian problems; characterization of Bayes procedures
• Empirical decision procedures; empirical Bayes procedures
• Minimax procedures
• Compound decision problems
• None of the above, but in this section
• Sampling theory, sample surveys
• Distribution theory
• Characterization and structure theory
• Exact distribution theory
• Approximations to distributions (nonasymptotic)
• Asymptotic distribution theory
• None of the above, but in this section
• Parametric inference
• Hypothesis testing
• Asymptotic properties of tests
• Ranking and selection
• Point estimation
• Asymptotic properties of estimators
• Bayesian inference
• Tolerance and confidence regions
• Inference under constraints
• Bootstrap, jackknife and other resampling methods
• None of the above, but in this section
• Nonparametric inference
• Estimation
• Density estimation
• Nonparametric regression
• Resampling methods
• Hypothesis testing
• Tolerance and confidence regions
• Asymptotic properties
• Order statistics; empirical distribution functions
• Statistics of extreme values; tail inference
• Robustness
• None of the above, but in this section
• Multivariate analysis
• Characterization and structure theory
• Distribution of statistics
• Directional data; spatial statistics
• Estimation
• Hypothesis testing
• Contingency tables
• Measures of association (correlation, canonical correlation, etc.)
• Factor analysis and principal components; correspondence analysis
• Classification and discrimination; cluster analysis
• Image analysis
• None of the above, but in this section
• Linear inference, regression
• General nonlinear regression
• Linear regression
• Ridge regression; shrinkage estimators
• Analysis of variance and covariance
• Generalized linear models
• Paired and multiple comparisons
• Diagnostics
• None of the above, but in this section
• Design of experiments
• Optimal designs
• Block designs
• Factorial designs
• Response surface designs
• Robust parameter designs
• None of the above, but in this section
• Sequential methods
• Sequential design
• Sequential analysis
• Sequential estimation
• Optimal stopping
• Stochastic approximation
• None of the above, but in this section
• Inference from stochastic processes
• Markov processes: hypothesis testing
• Markov processes: estimation
• Non-Markovian processes: hypothesis testing
• Non-Markovian processes: estimation
• Time series, auto-correlation, regression, etc.
• Spectral analysis
• Prediction; filtering
• Spatial processes
• Random fields; image analysis
• Neural nets and related approaches
• None of the above, but in this section
• Survival analysis and censored data
• Censored data models
• Estimation
• Testing
• Reliability and life testing
• None of the above, but in this section
• Applications
• Applications to actuarial sciences and financial mathematics
• Applications to biology and medical sciences
• Applications to environmental and related topics
• Applications to psychology
• Applications to economics
• Applications to social sciences
• Applications in engineering and industry
• Applications to physics
• None of the above, but in this section
• Statistical tables

## Numerical analysis

• 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)
• Experimental papers
• Proceedings, conferences, collections, etc.
• Tables
• Acceleration of convergence
• Extrapolation to the limit, deferred corrections
• Summation of series
• Euler-Maclaurin formula
• None of the above, but in this section
• Probabilistic methods, simulation and stochastic differential equations
• Monte Carlo methods
• Random number generation
• Models, numerical methods
• Stochastic differential and integral equations
• Stochastic particle methods
• Computational Markov chains
• Other computational problems in probability
• Computational problems in statistics
• None of the above, but in this section
• Numerical approximation and computational geometry {Primarily algorithms; for theory, see 41-XX and 68Uxx]
• Interpolation
• Splines
• Smoothing, curve fitting
• Algorithms for functional approximation
• Computer aided design (modeling of curves and surfaces)
• Computer graphics and computational geometry
• Computation of special functions, construction of tables
• Numerical differentiation
• Numerical integration
• None of the above, but in this section
• Numerical methods in complex analysis (potential theory, etc.)
• Numerical linear algebra
• Direct methods for linear systems and matrix inversion
• Iterative methods for linear systems
• Eigenvalues, eigenvectors
• Inverse eigenvalue problems
• Overdetermined systems, pseudoinverses
• Ill-posedness, regularization
• Orthogonalization
• Other matrix algorithms
• Matrix norms, conditioning, scaling
• Determinants
• Sparse matrices
• None of the above, but in this section
• Error analysis and interval analysis
• Algorithms with automatic result verification
• Interval and finite arithmetic
• General methods in interval analysis
• Roundoff error
• None of the above, but in this section
• Nonlinear algebraic or transcendental equations
• Single equations
• Systems of equations
• Eigenvalues, eigenvectors
• Global methods, including homotopy approaches
• None of the above, but in this section
• Numerical analysis in abstract spaces
• General theory
• Equations with linear operators (do not use 65Fxx)
• Equations with nonlinear operators (do not use 65Hxx)
• Improperly posed problems; regularization
• Inverse problems
• None of the above, but in this section
• Mathematical programming, optimization and variational techniques
• Mathematical programming {Algorithms; for theory see 90Cxx]
• Optimization and variational techniques
• None of the above, but in this section
• Ordinary differential equations
• Initial value problems
• Multistep, Runge-Kutta and extrapolation methods
• Numerical investigation of stability of solutions
• Improperly posed problems
• Inverse problems
• Boundary value problems
• Finite difference methods
• Eigenvalue problems
• Stability and convergence of numerical methods
• Mesh generation and refinement
• Finite elements, Rayleigh-Ritz, Galerkin and collocation methods
• Error bounds
• Methods for differential-algebraic equations
• None of the above, but in this section
• Partial differential equations, initial value and time-dependent initial-boundary value problems
• Finite difference methods
• Stability and convergence of numerical methods
• Error bounds
• Method of lines
• Method of characteristics
• Improperly posed problems
• Inverse problems
• Mesh generation and refinement
• Multigrid methods; domain decomposition
• Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
• Spectral, collocation and related methods
• None of the above, but in this section
• Partial differential equations, boundary value problems
• Finite difference methods
• Stability and convergence of numerical methods
• Error bounds
• Inverse problems
• Solution of discretized equations
• Eigenvalue problems
• Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
• Spectral, collocation and related methods
• Boundary element methods
• Method of lines
• Method of contraction of the boundary
• Mesh generation and refinement
• Multigrid methods; domain decomposition
• None of the above, but in this section
• Numerical problems in dynamical systems
• Hamiltonian systems including symplectic integrators
• Numerical chaos
• Bifurcation problems
• Nonlinear stabilities
• None of the above, but in this section
• Difference and functional equations, recurrence relations
• Integral equations, integral transforms
• Integral transforms
• Integral equations
• Improperly posed problems
• Inverse problems
• None of the above, but in this section
• Graphical methods
• Numerical methods in Fourier analysis
• Trigonometric approximation and interpolation
• Discrete and fast Fourier transforms
• Wavelets
• None of the above, but in this section
• Computer aspects of numerical algorithms
• Parallel computation
• Algorithms for specific classes of architectures
• Packaged methods
• Complexity and performance of numerical algorithms
• None of the above, but in this section
• Applications to physics

## Computer science

• 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.
• Computer system organization
• General
• Mathematical problems of computer architecture
• Network design and communication
• Network protocols
• Distributed systems
• Reliability, testing and fault tolerance
• Performance evaluation; queueing; scheduling
• None of the above, but in this section

## Software

• General
• Programming languages
• Logic programming
• Functional programming and lambda calculus
• Other programming techniques (object-oriented, sequential, concurrent, automatic, etc.)
• Compilers and interpreters
• Operating systems
• Mathematical aspects of software engineering (specification, verification, metrics, requirements, etc.)
• None of the above, but in this section
• Theory of data
• General
• Data structures
• Searching and sorting
• Database theory
• Information storage and retrieval
• Data encryption
• Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.)
• None of the above, but in this section
• Theory of computing
• General
• Models of computation (Turing machines, etc.)
• Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)
• Complexity classes (hierarchies, relations among complexity classes, etc.)
• Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
• Descriptive complexity and finite models
• Analysis of algorithms and problem complexity
• Algorithmic information theory (Kolmogorov complexity, etc.)
• Computational learning theory
• Grammars and rewriting systems
• Formal languages and automata
• Semantics
• Specification and verification (program logics, model checking, etc.)
• Abstract data types; algebraic specification
• Algebraic theory of languages and automata
• Cellular automata
• Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
• None of the above, but in this section
• Discrete mathematics in relation to computer science
• General
• Combinatorics
• Graph theory
• Combinatorics on words
• None of the above, but in this section
• Artificial intelligence
• General
• Pattern recognition, speech recognition
• Theorem proving (deduction, resolution, etc.)
• Problem solving (heuristics, search strategies, etc.)
• Logic in artificial intelligence
• Knowledge representation
• Languages and software systems (knowledge-based systems, expert systems, etc.)
• Reasoning under uncertainty
• Robotics
• Machine vision and scene understanding
• Natural language processing
• None of the above, but in this section
• Computing methodologies and applications
• General
• Computer graphics; computational geometry
• Computer-aided design
• Image processing
• Text processing; mathematical typography
• Simulation
• Information systems (hypertext navigation, interfaces, decision support, etc.)
• None of the above, but in this section

## Algorithms

• General
• Nonnumerical algorithms
• Parallel algorithms
• Distributed algorithms
• Randomized algorithms
• Approximation algorithms
• Symbolic computation and algebraic computation
• VLSI algorithms
• Analysis of algorithms
• None of the above, but in this section

## Mechanics of particles and 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Computational methods
• Axiomatics, foundations
• Kinematics
• Kinematics of a particle
• Kinematics of a rigid body
• Mechanisms, robots
• None of the above, but in this section
• Statics
• Dynamics of a rigid body and of multibody systems
• Motion of the gyroscope
• Free motion of a rigid body
• Motion of a rigid body with a fixed point
• Motion of a rigid body in contact with a solid surface
• Perturbation methods for rigid body dynamics
• Integrable cases of motion
• Higher-dimensional generalizations
• Stability problems
• Dynamics of multibody systems
• Robot dynamics and control
• None of the above, but in this section
• Dynamics of a system of particles, including celestial mechanics
• Two-body problems
• Three-body problems
• $n$-body problems
• Celestial mechanics
• Collisions in celestial mechanics, regularization
• Inverse problems
• Holonomic systems
• Nonholonomic systems
• Collision of rigid or pseudo-rigid bodies
• Problems with friction
• Infinite particle systems
• None of the above, but in this section
• General models, approaches, and methods
• Generalized coordinates; event, impulse-energy, configuration, state, or phase space
• Topological and differential-topological methods
• Differential-geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.)
• Algebraic geometry methods
• Dynamical systems methods
• Symmetries, Lie-group and Lie-algebra methods
• Functional-analytic methods
• Variational methods
• None of the above, but in this section
• Hamiltonian and Lagrangian mechanics
• Lagrange's equations
• Hamilton's equations
• Completely integrable systems and methods of integration
• Nonintegrable systems
• Nearly integrable Hamiltonian systems, KAM theory
• Perturbation theories
• Periodic and almost periodic solutions
• Stability problems
• Canonical and symplectic transformations
• Hamilton-Jacobi equations
• Hamilton's principle
• Other variational principles
• Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
• Relativistic dynamics
• Constrained dynamics, Dirac's theory of constraints
• Higher-order theories
• None of the above, but in this section
• Linear vibration theory
• Modal analysis
• Stability
• Free motions
• Forced motions
• Parametric resonances
• Systems arising from the discretization of structural vibration problems
• None of the above, but in this section
• Nonlinear dynamics
• Phase plane analysis, limit cycles
• Stability
• Free motions
• Parametric resonances
• Nonlinear resonances
• Forced motions
• Equilibria and periodic trajectories
• Quasi-periodic motions and invariant tori
• Homoclinic and heteroclinic trajectories
• Normal forms
• Bifurcations and instability
• Transition to stochasticity (chaotic behavior)
• General perturbation schemes
• Averaging of perturbations
• Systems with slow and fast motions
• Nonlinear modes
• None of the above, but in this section
• Random vibrations
• Orbital mechanics
• Variable mass, rockets
• Control of mechanical systems
• Classical field theories
• Lagrangian formalism and Hamiltonian formalism
• Symmetries and conservation laws
• Yang-Mills and other gauge theories
• More general nonquantum field theories
• None of the above, but in this section

## Mechanics of deformable solids

• 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Generalities, axiomatics, foundations of continuum mechanics of solids
• Kinematics of deformation
• Stress
• Thermodynamics
• Theory of constitutive functions
• Molecular, statistical, and kinetic theories
• Nonsimple materials
• Polar materials
• Random materials and composite materials
• Theories of fracture and damage
• Structured surfaces and interfaces, coexistent phases
• Theories of friction (tribology)
• Micromechanical theories
• Reactive materials
• None of the above, but in this section
• Elastic materials
• Classical linear elasticity
• Linear elasticity with initial stresses
• Equations linearized about a deformed state (small deformations superposed on large)
• Nonlinear elasticity
• None of the above, but in this section
• Plastic materials, materials of stress-rate and internal-variable type
• Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials)
• Small-strain, rate-dependent theories (including theories of viscoplasticity)
• Large-strain, rate-independent theories (including nonlinear plasticity)
• Large-strain, rate-dependent theories
• None of the above, but in this section
• Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
• Linear constitutive equations
• Nonlinear constitutive equations
• None of the above, but in this section
• Material properties given special treatment
• Inhomogeneity
• Anisotropy
• Crystalline structure
• Granularity
• Texture
• Composite and mixture properties
• Random structure
• Chemical structure
• None of the above, but in this section
• Coupling of solid mechanics with other effects
• Thermal effects
• Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
• Electromagnetic effects
• Mixture effects
• Chemical and reactive effects
• None of the above, but in this section
• Explicit solutions
• Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.)
• Numerical approximation of solutions
• Local existence of solutions (near a given solution)
• Global existence of solutions
• Uniqueness of solutions
• Multiplicity of solutions
• Regularity of solutions
• Bounds for solutions
• Saint-Venant's principle
• Qualitative behavior of solutions
• Bifurcation and buckling
• Energy minimization
• Stress concentrations, singularities
• Inverse problems
• None of the above, but in this section
• Dynamical problems
• Explicit solutions
• Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.)
• Numerical approximation of solutions
• Existence of solutions
• Uniqueness of solutions
• Regularity of solutions
• Singularities, blowup, stress concentrations
• Long-time behavior of solutions
• Vibrations
• Random vibrations
• Stability
• Dynamical bifurcation
• Chaotic behavior
• None of the above, but in this section
• Waves
• Linear waves
• Bulk waves
• Surface waves
• Wave scattering
• Inverse problems
• Nonlinear waves
• Solitary waves
• Shocks and related discontinuities
• None of the above, but in this section
• Thin bodies, structures
• Strings
• Rods (beams, columns, shafts, arches, rings, etc.)
• Membranes
• Plates
• Shells
• Junctions
• Thin films
• None of the above, but in this section
• Special subfields of solid mechanics
• Geophysical solid mechanics
• Soil and rock mechanics
• Biomechanical solid mechanics
• None of the above, but in this section
• Special kinds of problems
• Control, switches and devices (smart materials'')
• Friction
• Contact
• Impact
• Micromechanics
• None of the above, but in this section
• Phase transformations in solids
• Crystals
• Displacive transformations
• Analysis of microstructure
• Dynamics of phase boundaries
• Transformations involving diffusion
• Problems involving hysteresis
• None of the above, but in this section
• Optimization
• Compliance or weight optimization
• Optimization of other properties
• Topological methods
• Geometrical methods
• None of the above, but in this section
• Homogenization, determination of effective properties
• Homogenization in equilibrium problems
• Homogenization and oscillations in dynamical problems
• Effective constitutive equations
• Bounds on effective properties
• None of the above, but in this section
• Fracture and damage
• Brittle damage
• Brittle fracture
• High-velocity fracture
• Anelastic fracture and damage
• None of the above, but in this section
• Numerical methods
• Finite element methods
• Finite volume methods
• Boundary element methods
• Finite difference methods
• Spectral and related methods
• Other numerical methods
• None of the above, but in this section
• Fluid mechanics
• 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Foundations, constitutive equations, rheology
• Foundations of fluid mechanics
• Non-Newtonian fluids
• Viscoelastic fluids
• Liquid crystals
• Thin fluid films
• Superfluids (classical aspects)
• None of the above, but in this section
• Incompressible inviscid fluids
• Existence, uniqueness, and regularity theory
• Free-surface potential flows
• Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
• Water waves, gravity waves; dispersion and scattering, nonlinear interaction
• Ship waves
• Solitary waves
• Capillarity (surface tension)
• Vortex flows
• Internal waves
• Atmospheric waves
• Rossby waves
• Stratification effects in inviscid fluids
• Flow control and optimization
• None of the above, but in this section
• Incompressible viscous fluids
• Existence, uniqueness, and regularity theory
• Navier-Stokes equations
• Statistical solutions of Navier-Stokes and related equations
• Stokes and related (Oseen, etc.) flows
• Lubrication theory
• Viscous-inviscid interaction
• Boundary-layer theory, separation and reattachment, higher-order effects
• Viscous vortex flows
• Wakes and jets
• Other free-boundary flows; Hele-Shaw flows
• Waves
• Capillarity (surface tension)
• Stratification effects in viscous fluids
• Flow control and optimization
• None of the above, but in this section
• Hydrodynamic stability
• Parallel shear flows
• Convection
• Rotation
• Stability and instability of nonparallel flows
• Absolute and convective instability and stability
• Interfacial stability and instability
• Compressibility effects
• Stability and instability of geophysical and astrophysical flows
• Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
• Nonlinear effects
• None of the above, but in this section
• Turbulence
• Fundamentals
• Isotropic turbulence; homogeneous turbulence
• Transition to turbulence
• Shear flows
• Dynamical systems approach to turbulence
• Turbulent transport, mixing
• Renormalization and other field-theoretical methods
• Convective turbulence
• Turbulent boundary layers
• Stratification effects
• Compressibility effects
• Statistical turbulence modeling
• $k$-$\varepsilon$ modeling
• Direct numerical and large eddy simulation of turbulence
• Control of turbulent flows
• None of the above, but in this section
• General aerodynamics and subsonic flows
• Transonic flows
• Supersonic flows
• Hypersonic flows
• Shock waves and blast waves
• Basic methods in fluid mechanics
• Finite element methods
• Finite volume methods
• Boundary element methods
• Finite difference methods
• Spectral methods
• Vortex methods
• Other numerical methods
• Visualization algorithms
• Particle methods and lattice-gas methods
• Variational methods
• Stochastic analysis
• Complex-variables methods
• Asymptotic methods, singular perturbations
• Homogenization
• Dimensional analysis and similarity
• Symmetry analysis, Lie group and algebra methods
• None of the above, but in this section
• Compressible fluids and gas dynamics, general
• Existence, uniqueness, and regularity theory
• Gas dynamics, general
• Viscous-inviscid interaction
• Boundary-layer theory
• Flow control and optimization
• None of the above, but in this section
• Rarefied gas flows, Boltzmann equation
• Hydro- and aero-acoustics
• Diffusion and convection
• Forced convection
• Free convection
• Diffusion
• None of the above, but in this section
• Flows in porous media; filtration; seepage
• Two-phase and multiphase flows
• Liquid-gas two-phase flows, bubbly flows
• Dusty-gas two-phase flows
• Suspensions
• Granular flows
• Three or more component flows
• None of the above, but in this section
• Rotating fluids
• Reaction effects in flows
• Magnetohydrodynamics and electrohydrodynamics
• Ionized gas flow in electromagnetic fields; plasmic flow
• Quantum hydrodynamics and relativistic hydrodynamics
• Biological fluid mechanics
• Physiological flows
• Biopropulsion in water and in air
• None of the above, but in this section

## Optics, electromagnetic 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• General
• Foundations
• Geometric optics
• Physical optics
• Electron optics
• Space charge waves
• Electromagnetic theory, general
• Electro- and magnetostatics
• Motion of charged particles
• Diffraction, scattering
• Inverse scattering problems
• Composite media; random media
• Antennas, wave-guides
• Technical applications
• Lasers, masers, optical bistability, nonlinear optics
• Biological applications
• Mathematically heuristic optics and electromagnetic theory (must also be assigned at least one other classification number in this section)
• Miscellaneous topics
• Basic methods
• Method of moments
• Finite element methods
• Boundary element methods
• Finite difference methods
• Other numerical methods
• Variational methods
• Asymptotic analysis
• Homogenization
• Optimization
• None of the above, but in this section

## Classical thermodynamics, heat transfer

• 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Thermodynamics and heat transfer
• Foundations
• Classical thermodynamics, including relativistic
• Thermodynamics of continua
• Heat and mass transfer, heat flow
• Stefan problems, phase changes, etc.
• Inverse problems
• Combustion
• Chemical kinetics
• Chemically reacting flows
• Chemistry (general)
• None of the above, but in this section
• Basic methods
• Finite element methods
• Boundary element methods
• Finite difference methods
• Other numerical methods
• Variational methods
• Asymptotic analysis
• Homogenization
• Optimization
• None of the above, but in this section

## Quantum 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)
• Experimental papers
• Proceedings, conferences, collections, etc.
• Computational methods
• Axiomatics, foundations, philosophy
• General and philosophical
• Logical foundations of quantum mechanics; quantum logic
• Quantum measurement theory
• Stochastic mechanics (including stochastic electrodynamics)
• Quantum computation and quantum cryptography
• None of the above, but in this section
• General mathematical topics and methods in quantum theory
• Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations
• Selfadjoint operator theory in quantum theory, including spectral analysis
• Perturbation theories for operators and differential equations
• Semiclassical techniques including WKB and Maslov methods
• Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
• Bethe-Salpeter and other integral equations
• Quantum chaos
• Supersymmetric quantum mechanics
• Differential-geometric methods, including holonomy, Berry and Hannay phases, etc.
• None of the above, but in this section
• Groups and algebras in quantum theory
• Finite-dimensional groups and algebras motivated by physics and their representations
• Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current algebras and their representations
• Relations with integrable systems
• Operator algebra methods
• Covariant wave equations
• Spinor and twistor methods
• Coherent states; squeezed states
• Symmetry breaking
• Quantum groups and related algebraic methods
• Noncommutative geometry
• None of the above, but in this section
• General quantum mechanics and problems of quantization
• Commutation relations and statistics
• Geometry and quantization, symplectic methods
• Stochastic quantization
• Quantum stochastic calculus
• Phase space methods including Wigner distributions, etc.
• Path integrals
• None of the above, but in this section
• Quantum field theory; related classical field theories
• Axiomatic quantum field theory; operator algebras
• Constructive quantum field theory
• Model quantum field theories
• Yang-Mills and other gauge theories
• Perturbative methods of renormalization
• Nonperturbative methods of renormalization
• Renormalization group methods
• Feynman diagrams
• Quantum field theory on curved space backgrounds
• Quantum field theory on lattices
• Continuum limits
• String and superstring theories; other extended objects (e.g., branes)
• Two-dimensional field theories, conformal field theories, etc.
• Topological field theories
• Anomalies
• Supersymmetric field theories
• Quantization in field theory; cohomological methods
• Noncommutative geometry methods
• Simulation and numerical modeling
• None of the above, but in this section
• Scattering theory
• $2$-body potential scattering theory
• $n$-body potential scattering theory
• Exactly and quasi-solvable systems
• $S$-matrix theory, etc.
• Dispersion theory, dispersion relations
• Inverse scattering problems
• None of the above, but in this section
• Applications to specific physical systems
• Strong interaction, including quantum chromodynamics
• Electromagnetic interaction; quantum electrodynamics
• Weak interaction
• Gravitational interaction
• Other fundamental interactions
• Unified theories
• Other elementary particle theory
• Nuclear physics
• Atomic physics
• Molecular physics
• Many-body theory; quantum Hall effect
• Quantum optics
• None of the above, but in this section

## Statistical mechanics, structure of matter

• 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)
• Experimental papers
• Proceedings, conferences, collections, etc.
• Computational methods
• Equilibrium statistical mechanics
• Foundations
• Classical equilibrium statistical mechanics (general)
• Quantum equilibrium statistical mechanics (general)
• Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
• Continuum models (systems of particles, etc.)
• Exactly solvable models; Bethe ansatz
• Interface problems; diffusion-limited aggregation
• Phase transitions (general)
• Critical phenomena
• Renormalization group methods
• Statistical thermodynamics
• Stochastic methods
• Irreversible thermodynamics, including Onsager-Machlup theory
• Kinetic theory of gases
• Random walks, random surfaces, lattice animals, etc.
• Percolation
• Disordered systems (random Ising models, random Schrödinger operators, etc.)
• Numerical methods (Monte Carlo, series resummation, etc.)
• None of the above, but in this section
• Time-dependent statistical mechanics (dynamic and nonequilibrium)
• Foundations
• Classical dynamic and nonequilibrium statistical mechanics (general)
• Quantum dynamics and nonequilibrium statistical mechanics (general)
• Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
• Dynamic continuum models (systems of particles, etc.)
• Interacting particle systems
• Exactly solvable dynamic models
• Interface problems; diffusion-limited aggregation
• Dynamic and nonequilibrium phase transitions (general)
• Dynamic critical phenomena
• Dynamic renormalization group methods
• Stochastic methods (Fokker-Planck, Langevin, etc.)
• Neural nets
• Irreversible thermodynamics, including Onsager-Machlup theory
• Kinetic theory of gases
• Dynamics of random walks, random surfaces, lattice animals, etc.
• Time-dependent percolation
• Dynamics of disordered systems (random Ising systems, etc.)
• Transport processes
• Numerical methods (Monte Carlo, series resummation, etc.)
• None of the above, but in this section
• Applications to specific types of physical systems
• Gases
• Plasmas
• Liquids
• Solids
• Crystals
• Random media, disordered materials (including liquid crystals and spin glasses)
• Metals
• Semiconductors
• Magnetic materials
• Ferroelectrics
• Superfluids
• Superconductors
• Polymers
• Nuclear reactor theory; neutron transport
• None of the above, but in this section

## Relativity and gravitational 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Computational methods
• Special relativity
• Observational and experimental questions
• General relativity
• Einstein's equations (general structure, canonical formalism, Cauchy problems)
• Equations of motion
• Exact solutions
• Classes of solutions; algebraically special solutions, metrics with symmetries
• Einstein-Maxwell equations
• Approximation procedures, weak fields
• Lattice gravity, Regge calculus and other discrete methods
• Asymptotic procedures (radiation, news functions, {\scr H]-spaces, etc.)
• Gravitational waves
• Gravitational energy and conservation laws; groups of motions
• Quantization of the gravitational field
• Methods of quantum field theory
• Electromagnetic fields
• Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
• Black holes
• Spinor and twistor methods; Newman-Penrose formalism
• Methods of noncommutative geometry
• Space-time singularities, cosmic censorship, etc.
• Analogues in lower dimensions
• None of the above, but in this section
• Relativistic gravitational theories other than Einstein's, including asymmetric field theories
• Unified, higher-dimensional and super field theories
• Geometrodynamics
• Kaluza-Klein and other higher-dimensional theories
• String and superstring theories
• Supergravity
• None of the above, but in this section
• Cosmology
• Astronomy and astrophysics
• 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Computational methods
• General
• Galactic and stellar dynamics
• Galactic and stellar structure
• Planetary atmospheres
• Hydrodynamic and hydromagnetic problems
• Statistical astronomy
• Cosmology
• Miscellaneous topics
• Geophysics
• 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)
• Experimental work
• Proceedings, conferences, collections, etc.
• Computational methods
• General
• Hydrology, hydrography, oceanography
• Meteorology and atmospheric physics
• Seismology
• Global dynamics, earthquake problems
• Potentials, prospecting
• Inverse problems
• Geo-electricity and geomagnetism
• Geodesy, mapping problems
• Geostatistics
• Glaciology
• Geological problems
• Miscellaneous topics

## Operations research, mathematical programming

• 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.
• Computational methods
• Operations research and management science
• Inventory, storage, reservoirs
• Transportation, logistics
• Network models, deterministic
• Network models, stochastic
• Communication networks
• Traffic problems
• Queues and service
• Reliability, availability, maintenance, inspection
• Production models
• Scheduling theory, deterministic
• Scheduling theory, stochastic
• Search theory
• Management decision making, including multiple objectives
• Theory of organizations, manpower planning
• Discrete location and assignment
• Continuous location
• Case-oriented studies
• None of the above, but in this section
• Mathematical programming
• Linear programming
• Large-scale problems
• Special problems of linear programming (transportation, multi-index, etc.)
• Boolean programming
• Integer programming
• Mixed integer programming
• Stochastic programming
• Semidefinite programming
• Convex programming
• Nonconvex programming
• Combinatorial optimization
• Multi-objective and goal programming
• Nonlinear programming
• Sensitivity, stability, parametric optimization
• Fractional programming
• Complementarity problems
• Semi-infinite programming
• Programming involving graphs or networks
• Dynamic programming
• Markov and semi-Markov decision processes
• Optimality conditions, duality
• Minimax problems
• Programming in abstract spaces
• Extreme-point and pivoting methods
• Interior-point methods
• Methods of reduced gradient type
• Methods of quasi-Newton type
• Methods of successive quadratic programming type
• Derivative-free methods
• Polyhedral combinatorics, branch-and-bound, branch-and-cut
• Approximation methods and heuristics
• Abstract computational complexity for mathematical programming problems
• Fuzzy programming
• Applications of mathematical programming
• None of the above, but in this section

## Game theory, economics, social and behavioral sciences

• 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.
• Computational methods
• Game theory
• 2-person games
• $n$-person games, $n>2$
• Noncooperative games
• Cooperative games
• Games with infinitely many players
• Stochastic games
• Games in extensive form
• Multistage and repeated games
• Evolutionary games
• Differential games
• Positional games (pursuit and evasion, etc.)
• Dynamic games
• Rationality, learning
• Signaling, communication
• Utility theory for games
• Decision theory for games
• Game-theoretic models
• Games involving graphs
• Games involving topology or set theory
• Combinatorial games
• Discrete-time games
• Games of timing
• Probabilistic games; gambling
• Hierarchical games
• Spaces of games
• Applications of game theory
• Experimental studies
• None of the above, but in this section
• Mathematical economics
• Fundamental topics (basic mathematics, methodology; applicable to economics in general)
• Decision theory
• Individual preferences
• Group preferences
• Voting theory
• Social choice
• Utility theory
• Public goods
• Price theory and market structure
• Market models (auctions, bargaining, bidding, selling, etc.)
• Finance, portfolios, investment
• Risk theory, insurance
• Resource and cost allocation
• Production theory, theory of the firm
• Labor market, contracts
• Consumer behavior, demand theory
• Informational economics
• Equilibrium: general theory
• Special types of equilibria
• Special types of economies
• General economic models, trade models
• Dynamic economic models, growth models
• Macro-economic models (monetary models, models of taxation)
• Multisectoral models
• Matching models
• Stochastic models
• Spatial models
• Models of real-world systems
• Environmental economics (natural resource models, harvesting, pollution, etc.)
• Statistical methods; economic indices and measures
• Economic time series analysis
• None of the above, but in this section
• Social and behavioral sciences: general topics
• Measurement theory
• One- and multidimensional scaling
• Clustering
• None of the above, but in this section
• Mathematical sociology (including anthropology)
• Models of societies, social and urban evolution
• Mathematical geography and demography
• Spatial models
• Social networks
• Manpower systems
• None of the above, but in this section
• Mathematical psychology
• Cognitive psychology
• Psychophysics and psychophysiology; perception
• Memory and learning
• Measurement and performance
• None of the above, but in this section
• Other social and behavioral sciences (mathematical treatment)
• History, political science
• Linguistics
• None of the above, but in this section

## Biology and other natural sciences

• 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.
• Computational methods
• Mathematical biology in general
• General biology and biomathematics
• Taxonomy, statistics
• General biostatistics
• Neural networks, artificial life and related topics
• None of the above, but in this section
• Physiological, cellular and medical topics
• Biophysics
• Biomechanics
• Developmental biology, pattern formation
• Cell movement (chemotaxis, etc.)
• Neural biology
• Physiology (general)
• Physiological flow
• Cell biology
• Biochemistry, molecular biology
• Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
• Medical applications (general)
• Biomedical imaging and signal processing
• Medical epidemiology
• Plant biology
• None of the above, but in this section
• Genetics and population dynamics
• Genetics
• Problems related to evolution
• Protein sequences, DNA sequences
• Population dynamics (general)
• Epidemiology
• Ecology
• Animal behavior
• None of the above, but in this section
• Chemistry
• Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
• Classical flows, reactions, etc.
• None of the above, but in this section
• Other natural sciences
• Systems theory; control
• 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

• Axiomatic system theory
• General systems
• Hierarchical systems
• Decentralized systems
• Large scale systems
• Mathematical modeling (models of systems, model-matching, etc.)
• None of the above, but in this section
• Controllability, observability, and system structure
• Attainable sets
• Controllability
• Observability
• Canonical structure
• System structure simplification
• Variable structure systems
• Realizations from input-output data
• Transformations
• Linearizations
• Minimal systems representations
• Algebraic methods
• Geometric methods (including algebro-geometric)
• Operator-theoretic methods
• Differential-geometric methods
• System identification
• Sensitivity (robustness)
• ${H]^\infty$-control
• Computational methods
• Synthesis problems
• Design techniques (robust design, computer-aided design, etc.)
• Feedback control
• Pole and zero placement problems
• Eigenvalue problems
• None of the above, but in this section
• Control systems, guided systems
• Linear systems
• Nonlinear systems
• Systems governed by ordinary differential equations
• Systems governed by partial differential equations
• Systems governed by functional-differential equations
• Systems in abstract spaces
• Systems governed by functional relations other than differential equations
• Multivariable systems
• Problems with incomplete information
• Fuzzy control
• Discrete-time systems
• Sampled-data systems
• Digital systems
• Discrete event systems
• Time-scale analysis and singular perturbations
• Perturbations
• Frequency-response methods
• Control problems involving computers (process control, etc.)
• Automated systems (robots, etc.)
• Applications
• None of the above, but in this section
• Stability
• Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.)
• Robust stability
• Popov-type stability of feedback systems
• Stabilization of systems by feedback
• Asymptotic stability
• Input-output approaches
• Scalar and vector Lyapunov functions
• None of the above, but in this section
• Stochastic systems and control
• Stochastic systems, general
• Estimation and detection
• Filtering
• System identification
• Data smoothing
• Stochastic stability
• Optimal stochastic control
• Least squares and related methods
• Other computational methods
• Stochastic learning and adaptive control
• None of the above, but in this section
• Information and communication, circuits
• 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.

## Communication, information

• Communication theory
• Image processing (compression, reconstruction, etc.)
• Application of orthogonal functions in communication
• Signal theory (characterization, reconstruction, etc.)
• Detection theory
• Modulation and demodulation
• Information theory, general
• Measures of information, entropy
• Sampling theory
• Coding theorems (Shannon theory)
• Source coding
• Rate-distortion theory
• Channel models
• Prefix, length-variable, comma-free codes
• Theory of questionnaires
• Shift register sequences and sequences over finite alphabets
• Cryptography
• Authentication and secret sharing
• None of the above, but in this section
• Theory of error-correcting codes and error-detecting codes
• Linear codes, general
• Convolutional codes
• Combined modulation schemes (including trellis codes)
• Cyclic codes
• Burst-correcting codes
• Combinatorial codes
• Geometric methods (including applications of algebraic geometry)
• Majority codes
• Decoding
• Arithmetic codes
• Synchronization error-correcting codes
• Other types of codes
• Bounds on codes
• Error probability
• Applications of the theory of convex sets and geometry of numbers (covering radius, etc.)
• None of the above, but in this section
• Circuits, networks
• Analytic circuit theory
• Switching theory, application of Boolean algebra; Boolean functions
• Fault detection; testing
• Applications of graph theory
• Applications of design theory
• None of the above, but in this section
• Fuzzy sets and logic (in connection with questions of Section 94)

## Mathematics education

• 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
• Recreational mathematics
• Sociological issues
• Standards
• Fiction and games
• Educational policy and educational systems
• Educational research and planning
• General education
• Vocational education
• Higher education
• Teacher education
• Out-of-school education. Adult and further education
• Syllabuses. Curriculum guides, official documents
• None of the above, but in this section
• Psychology of and research in mathematics education
• Affective aspects (motivation, anxiety, persistence, etc.)
• Student learning and thinking (misconceptions, cognitive development, problem solving, etc.)
• Assessment (large scale assessment, validity, reliability, etc.)
• Theoretical perspectives (learning theories, epistemology, philosophies of teaching and learning, etc.)
• Sociological aspects of learning (culture, group interactions, equity issues, etc.)
• Teachers, and research on teacher education (teacher development, etc.)
• Technological tools and other materials in teaching and learning (research on innovations, role in student learning, use of tools by teachers, etc.)
• Teaching and curriculum (innovations, teaching practices, studies of curriculum materials, effective teaching, etc. )
• None of the above, but in this section
• Education and instruction in mathematics
• Comparative studies on mathematics education
• Philosophical and theoretical contributions to mathematical education
• Goals of mathematics teaching. Curriculum development
• Teaching methods and classroom techniques. Lesson preparation. Educational principles
• Teaching problem solving and heuristic strategies
• Achievement control and rating
• Diagnosis, analysis and remediation of learning difficulties and student errors
• Teaching units, draft lessons and master lessons
• None of the above, but in this section
• Educational material and media. Educational technology
• Analysis of textbooks, development and evaluation of textbooks. Textbook use in the classroom
• Teacher manuals and planning aids
• Problem books; student competitions, examination questions
• Computer assisted instruction and programmed instruction
• Manipulative materials and their use in the classroom
• Technological tools (computers, calculators, software, etc.) and their use in the classroom
• Audiovisual media and their use in instruction
• None of the above, but in this section

http://thevikidtruth.com/5000/?mathmap