# 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.### General Math

## History and biography

## Mathematical logic and foundations

## Combinatorics

## 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 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
- Radix representation; digital problems
- Other representations
- None of the above, but in this section
- Sequences and sets
- Density, gaps, topology
- Additive bases
- 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
- Quadratic and bilinear equations
- Cubic and quartic equations
- Higher degree equations; Fermat's equation
- Counting solutions of Diophantine equations
- Multiplicative and norm form equations
- Thue-Mahler equations
- Exponential equations
- Rational numbers as sums of fractions
- Equations in many variables
- 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 binary quadratic forms
- General ternary and quaternary quadratic forms; forms of more than two variables
- Sums of squares and representations by other particular quadratic forms
- Bilinear and Hermitian forms
- Class numbers of quadratic and Hermitian forms
- Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
- Classical groups
- $K$-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
- Quadratic spaces; Clifford algebras
- $p$-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
- Additive number theory; partitions
- 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.)
- Quadratic extensions
- Cubic and quartic extensions
- Cyclotomic extensions
- Other abelian and metabelian extensions
- Other number fields
- Iwasawa theory
- Units and factorization
- Class numbers, class groups, discriminants
- Galois theory
- Integral representations related to algebraic numbers; Galois module structure of rings of integers
- 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
- Adèle rings and groups
- 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
- Graded rings
- Divisibility
- Radical theory
- 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)
- Linkage
- 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
- Adjunction problems
- 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
- Radicals and radical properties of rings
- Jacobson radical, quasimultiplication
- Nil and nilpotent radicals, sets, ideals, rings
- Prime and semiprime rings
- General radicals and 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
- Quadratic and Koszul algebras
- 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
- Graded rings and modules
- ``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 $(n \ge 3)$
- Quadratic algebras (but not quadratic Jordan algebras)
- Free algebras
- Structure theory
- Radical theory
- Superalgebras
- Composition algebras
- Valued algebras
- None of the above, but in this section
- 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
- Graded Lie (super)algebras
- 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
- Radicals
- 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
- Lie-admissible algebras
- 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)
- Strong functors, strong adjunctions
- Fibered categories
- Structured objects in a category (group objects, etc.)
- Operads
- None of the above, but in this section
- Abelian categories
- Preadditive, additive categories
- Exact categories, abelian categories
- Grothendieck categories
- Embedding theorems
- Derived functors and satellites
- Derived categories, triangulated categories
- Localization of categories
- Torsion theories, radicals
- 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
- Stability for quadratic modules
- 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 $K_0$
- Whitehead (and related) torsion
- 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
- Representations of sporadic groups
- 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
- Simple groups: sporadic groups
- 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
- Radical 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.
- Paradifferential operators
- 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
- Difference equations, additive
- Stability and asymptotics of difference equations; oscillatory and periodic solutions, etc.
- Discrete version of topics in analysis
- Difference equations, scaling ($q$-differences)
- 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
- Padé approximation
- 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
- Approximate quadratures
- 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
- Radon 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
- Radical Banach 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
- Nonassociative selfadjoint operator algebras
- $K$-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
- Numerical range, numerical radius
- 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)
- Nonassociative nonselfadjoint operator algebras
- Other nonselfadjoint operator algebras
- 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
- Linear-quadratic 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
- Translation planes and spreads
- 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
- Generalized quadrangles, generalized polygons
- 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
- Spreads and packing problems
- 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
- Spanier-Whitehead duality
- Eckmann-Hilton duality
- Loop spaces
- Suspensions
- Stable homotopy theory, spectra
- Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
- ${H]$-spaces and duals
- Infinite loop spaces
- Loop space machines, operads
- 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
- Whitehead products and generalizations
- Homotopy groups of wedges, joins, and simple spaces
- Hopf invariants
- Operations in homotopy groups
- Homotopy groups of spheres
- Stable homotopy of spheres
- $J$-morphism
- $v_n$-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
- Adams 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 ($4 < n < \infty$)
- 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
- Supermanifolds and graded manifolds
- 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
- Admissibility
- 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
- Robustness and adaptive procedures
- 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
- Quadrature and cubature formulas
- 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
- Learning and adaptive systems
- 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
- Adiabatic invariants
- 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
- Equilibrium (steady-state) problems
- 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
- Waves and radiation
- 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
- Radiative transfer
- 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
- Marketing, advertising
- 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
- Quadratic 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
- Adaptive control
- 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
- Adaptive or robust stabilization
- 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

08dec16 | admin |