CZ:Mathematics Workgroup

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Mathematics Workgroup

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The topics below are part of the division of mathematical knowledge to subdisciplines; they come from the 2000 Math. subject classification of the AMS ^[1] and ZMATH ^[2], with some minor edits. Each of these top level topics is a big subject, and it is not necessarily our first priority to write a long article on these entries. Our object with outlining this list is to establish a framework. We recommend that when you write a new article, you should try to find a proper node for it. Do not hesitate to put a link for an article you want to work on soon. Feel free to do the same to request a particular important topic to be covered.

Remarks:

We kept the original MSC numbering in places.
No, of course its not the whole MSC tree - not even close to it. We should eventually put as much of it as appropriate.
In some places we really expect some other workgroups (usually the physicists) to do the work alongside - we state where.

Caveats:

Do not copy articles from Wikipedia without carefully reading them, verifying both scope and focus. Besides, why not consider writing your article from scratch? Often this permits a better logical structure and more coherence. See CZ:How to convert Wikipedia articles to Citizendium articles
Keep in mind three audiences when writing an article: general readers, math students and professionals.

Most wanted math entries

From the encyclopedic point of view, the "high priority" articles are probably listed below at the top level of the tree (or in the table of contents). These are, however, relatively wide syntheses. You may want to start with a bit smaller tasks. A collection of important entries to write (of both types) can be found on the Core Articles page (still to be reviewed and reorganized).

Most popular Wikipedia math entries are: Pi, Mathematics , prime number, computer, trapezoid. These come from the list of 1000 most viewed pages in March 2007.^[3] This varies heavily over the time.

On Planet Math, the most popular (as of March 2007) were:^[4] circle, proof of Markov's inequality, Cauchy-Schwarz inequality, matrix inverse, Banach fixed point theorem, metric space, invariant subspace, function, manifold, eigenvalue, quartic formula, Jensen's inequality, cross product, real number, differential equation, gradient, natural number, Jacobian matrix, Gram-Schmidt orthogonalization, rational number.

The classification

00-XX General

(for calculus see 26-XX Real functions below)

elementary mathematics (pre-university level)

Elementary functions

trigonometric function

point line plane solid stubs

elementary algebra

for suggestions of useful articles to write see also here (pre-calculus or geometry sections)

01-XX History and biography

Euclid

Euler

03-XX Mathematical logic and foundations

Set theory

05-XX Combinatorics

06-XX Order, lattices, ordered algebraic structures

[See also 18B35]

08-XX General algebraic systems

11-XX Number theory

11Axx Elementary number theory {For analogues in number fields, see 11R04}
11Bxx Sequences and sets
11Cxx Polynomials and matrices
11Dxx Diophantine equations [See also 11Gxx, 14Gxx]
11Exx Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63}
11Fxx Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45}
11Gxx Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14-xx, 14Gxx, 14Kxx]
11Hxx Geometry of numbers {For applications in coding theory, see 94B75}
11Jxx Diophantine approximation, transcendental number theory [See also 11K60]
11Kxx Probabilistic theory: distribution modulo 1; metric theory of algorithms
11Lxx Exponential sums and character sums {For finite fields, see 11Txx}
11Mxx Zeta and $L$ -functions: analytic theory
11Nxx Multiplicative number theory
11Pxx Additive number theory; partitions
11Rxx Algebraic number theory: global fields {For complex multiplication, see 11G15}
11Sxx Algebraic number theory: local and $p$ -adic fields
11Txx Finite fields and commutative rings (number-theoretic aspects)
11Uxx Connections with logic
11Yxx Computational number theory [See also 11-04]

12-XX Field theory and polynomials

12Dxx Real and complex fields

real numbers

complex numbers

12Exx General field theory
12Fxx Field extensions
12Gxx Homological methods (field theory)
12Hxx Differential and difference algebra
12Jxx Topological fields
12Kxx Generalizations of fields
12Lxx Connections with logic
12Yxx
- 12Y05 Computational aspects of field theory and polynomials

13-XX Commutative rings and algebras

14-XX Algebraic geometry

14Axx Foundations
- 14A10 Varieties and morphisms
- 14A15 Schemes and morphisms
- 14A20 Generalizations (algebraic spaces, stacks)
- 14A22 Noncommutative algebraic geometry
14Bxx Local theory
- 14B05 Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
- 14B07 Deformations of singularities [See also 14D15, 32S30]
- 14B10 Infinitesimal methods [See also 13D10]
- 14B12 Local deformation theory, Artin approximation, etc. [See also 13B40, 13D10]
- 14B15 Local cohomology [See also 13D45, 32C36]
- 14B20 Formal neighborhoods
- 14B25 Local structure of morphisms: étale morphism, flat morphism, etc. [See also 13B40]
14Cxx Cycles and subschemes
- 14C05 Parametrization (Chow schemes and Hilbert schemes)
- 14C15 Chow groups and rings
- 14C17 Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
- 14C20 Divisors, linear systems, invertible sheaves
- 14C21 Pencils, nets, webs [See also 53A60]
- 14C22 Picard groups
- 14C25 Algebraic cycles
- 14C30 Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture
- 14C34 Torelli problem [See also 32G20]
- 14C35 Applications of methods of algebraic K-theory [See also 19Exx]
- 14C40 Riemann-Roch theorems [See also 19E20, 19L10]
14Dxx Families, fibrations
- 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
- 14D06 Fibrations, degenerations
- 14D07 Variation of Hodge structures [See also 32G20]
- 14D10 Arithmetic ground fields (finite, local, global)
- 14D15 Formal methods; deformations [See also 13D10, 14B07, 32Gxx]
- 14D20 Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13}
- 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
- 14D22 Fine moduli spaces and coarse moduli spaces
14Exx Birational geometry
- 14E05 Rational and birational maps
- 14E07 Birational automorphisms, Cremona group and generalizations
- 14E08 Rationality questions
- 14E15 Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
- 14E20 Coverings [See also 14H30]
- 14E22 Ramification problems [See also 11S15]
- 14E25 Embeddings
- 14E30 Minimal model program (Mori theory, extremal rays)
14Fxx (Co)homology theory [See also 13Dxx]
- 14F05 Vector bundles, sheaves, related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
- 14F10 sheaf of Differentials and other special sheaves [See also 13Nxx, 32C38]
- 14F17 Vanishing theorems [See also 32L20]
- 14F20 Étale topology Etale cohomology and other Grothendieck topologies and Grothendieck cohomologies
- 14F22 Brauer groups of schemes [See also 12G05, 16K50]
- 14F25 Classical real and complex cohomology
- 14F30 p-adic cohomology, crystalline cohomology
- 14F35 Homotopy theory; fundamental groups [See also 14H30]
- 14F40 de Rham cohomology [See also 14C30, 32C35, 32L10]
- 14F42 Motivic cohomology
- 14F43 Other algebro-geometric (co)homologies (e.g., intersection cohomology, equivariant cohomology, Lawson, Deligne (co)homologies)
- 14F45 Topological properties
14Gxx Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx]
14Hxx Curves
- 14H05 Algebraic functions; function fields [See also 11R58]
- 14H10,14H15 moduli [See also 30F10, 32Gxx]
- 14H20 Singularities, local rings [See also 13Hxx, 14B05]
- 14H25 Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
- 14H30 Coverings, fundamental group [See also 14E20, 14F35]
- 14H37 Automorphisms
- 14H40 Jacobians, Prym varieties [See also 32G20]
- 14H42 Theta functions; Schottky problem [See also 14K25, 32G20]
- 14H45 Special curves and curves of low genus

hyperelliptic curve

- 14H50 Plane and space curves
- 14H51 Special divisors (gonality, Brill-Noether theory)
- 14H52 Elliptic curves [See also 11G05, 11G07, 14Kxx]
- 14H55 Riemann surfaces; Weierstrass points; gap sequences [See also 30Fxx]
- 14H60 Vector bundles on curves and their moduli [See also 14D20, 14F05]
14Jxx Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx}
- 14J10 Families, moduli, classification: algebraic theory
- 14J15 Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
- 14J17 Singularities of surfaces [See also 14B05, 14E15]
- 14J20 Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
- 14J25 Special surfaces {For Hilbert modular surfaces, see 14G35}
- 14J26 Rational surfaces and ruled surfaces
- 14J27 Elliptic surfaces
- 14J28 K3 surfaces and Enriques surfaces

Kummer surfaces

- 14J29 Surfaces of general type
- 14J30 3-folds
- 14J32 Calabi-Yau manifolds, mirror symmetry
- 14J35 4-folds
- 14J40 n-folds (n > 4)
- 14J45 Fano varieties
- 14J50 Automorphisms of surfaces and higher-dimensional varieties
- 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
- 14J70 Hypersurfaces
- 14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
14Kxx Abelian varieties and schemes
14Lxx Algebraic groups {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45}
14Mxx Special varieties
14Nxx Projective and enumerative geometry [See also 51-xx]
14Pxx Real algebraic and real analytic geometry
14Qxx Computational algebraic geometry [See also 12Y05, 13Pxx, 68W30]
14Rxx Affine geometry

15-XX Linear and multilinear algebra; matrix theory

16-XX Associative rings and algebras

17-XX Nonassociative rings and algebras

18-XX Category theory; homological algebra

18Axx General theory of categories and functors
18Bxx Special categories
18Cxx Categories and theories
18Dxx Categories with structure
18Exx Abelian categories
18Fxx Categories and geometry
18Gxx Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx]
- 18G05 Projective objects and injective objects [See also 13C10, 13C11, 16D40, 16D50]
- 18G10 Resolutions; derived functors [See also 13D02, 16E05, 18E25]
- 18G15 Ext and Tor, generalizations, Künneth formula [See also 55U25]
- 18G20 Homological dimension [See also 13D05, 16E10]
- 18G25 Relative homological algebra, projective classes
- 18G30 Simplicial sets, simplicial objects (in a category) [See also 55U10]
- 18G35 Chain complexes [See also 18E30, 55U15]
- 18G40 Spectral sequences, hypercohomology [See also 55Txx]
- 18G50 Nonabelian homological algebra
- 18G55 Homotopical algebra
- 18G60 Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]*20-XX Group theory and generalizations

19-XX K-theory

20-XX Group theory and generalizations

20Axx Foundations
20Bxx Permutation groups
20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)]
20Dxx Abstract finite groups
20Exx Structure and classification of infinite or finite groups
20Fxx Special aspects of infinite or finite groups
20Gxx Linear algebraic groups (classical groups) {For arithmetic theory, see 11E57, 11H56; for geometric theory, see 14Lxx, 22Exx; for other methods in representation theory, see 15A30, 22E45, 22E46, 22E47, 22E50, 22E55}
- 20Hxx Other groups of matrices [See also 15A30]
20Jxx Connections with homological algebra and category theory
20Kxx Abelian groups
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms) {For sets with a single binary operation, see 20N02; for topological groupoids, see 22A22, 58H05}
20Mxx Semigroups
20Nxx Other generalizations of groups
20P05 Probabilistic methods in group theory [See also 60Bxx]*22-XX Topological groups, Lie groups

22-XX Topological groups, Lie groups

For some useful suggestions see: Lie groups topics

26-XX Real functions

Calculus

Mean value theorem

for some more suggestions of useful articles to write see also list of mathematical topics

28-XX Measure and integration

28Axx Classical measure theory
- 28A05 Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
- 28A10 Real- or complex-valued set functions
- 28A12 Contents, measures, outer measures, capacities

Lebesgue measure

- 28A15 Abstract differentiation theory, differentiation of set functions [See also 26A24]
- 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
- 28A25 Integration with respect to measures and other set functions

Lebesgue integral

Bounded convergence theorem

Monotone convergence theorem

Fatou's lemma

- 28A33 Spaces of measures, convergence of measures [See also 46E27, 60Bxx]
- 28A35 Measures and integrals in product spaces
- 28A50 Integration and disintegration of measures
- 28A51 Lifting theory [See also 46G15]
- 28A60 Measures on Boolean rings, measure algebras [See also 54H10]
- 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
- 28A78 Hausdorff and packing measures

Hausdorff measure

Hausdorff dimension

Packing dimension

- 28A80 Fractals [See also 37Fxx]
- 28A99 None of the above, but in this section
28Bxx Set functions, measures and integrals with values in abstract spaces
28Exx Miscellaneous topics in measure theory

30-XX Functions of a complex variable

31-XX Potential theory

31Axx Two-dimensional theory
- 31A05 Harmonic, subharmonic, superharmonic functions
- 31A10 Integral representations, integral operators, integral equations methods
- 31A15 Potentials and capacity, harmonic measure, extremal length [See also 30C85]
- 31A20 Boundary behavior (theorems of Fatou type, etc.)
- 31A25 Boundary value and inverse problems
- 31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation
- 31A35 Connections with differential equations
- 31A99 None of the above, but in this section
31Bxx Higher-dimensional theory
- 31B05 Harmonic, subharmonic, superharmonic functions
- 31B10 Integral representations, integral operators, integral equations methods
- 31B15 Potentials and capacities, extremal length
- 31B20 Boundary value and inverse problems
- 31B25 Boundary behavior
- 31B30 Biharmonic and polyharmonic equations and functions
- 31B35 Connections with differential equations
- 31B99 None of the above, but in this section
31Cxx Other generalizations
31D05 Axiomatic potential theory

32-XX Several complex variables and analytic spaces

33-XX Special functions

(33-XX deals with the properties of functions as functions) {For orthogonal functions, see 42Cxx; for aspects of combinatorics see 05Axx; for number-theoretic aspects see 11-XX; for representation theory see 22Exx}

34-XX Ordinary differential equations

see the current article on differential equations

35-XX Partial differential equations

Schrödinger equation

37-XX Dynamical systems and ergodic theory

39-XX Difference and functional equations

40-XX Sequences, series, summability

40Axx Convergence and divergence of infinite limiting processes
40B05 Multiple sequences and series {(should also be assigned at least one other classification number in this section)}
40Cxx General summability methods
40Dxx Direct theorems on summability
40Exx Inversion theorems
40F05 Absolute and strong summability
40Gxx Special methods of summability
40H05 Functional analytic methods in summability
40J05 Summability in abstract structures [See also 43A55, 46A35, 46B15]

41-XX Approximations and expansions

42-XX Fourier analysis

42-04 Explicit machine computation and programs (not the theory of computation or programming)

Fourier series

42Axx Fourier analysis in one variable
42Bxx Fourier analysis in several variables {For automorphic theory, see mainly 11F30}
42Cxx Nontrigonometric Fourier analysis

43-XX Abstract harmonic analysis

44-XX Integral transforms, operational calculus

45-XX Integral equations

46-XX Functional analysis

47-XX Operator theory

49-XX Calculus of variations and optimal control; optimization

51-XX Geometry

for a list of possible suggestions see list of geometry topics

52-XX Convex and discrete geometry

53-XX Differential geometry

54-XX General topology

55-XX Algebraic topology

57-XX Manifolds and cell complexes

58-XX Global analysis, analysis on manifolds

60-XX Probability theory and stochastic processes

60Axx Foundations of probability theory
60Bxx Probability theory on algebraic and topological structures
- 60C05 Combinatorial probability
- 60D05 Geometric probability, stochastic geometry, random sets [See also 52A22, 53C65]
60Exx Distribution theory [See also 62Exx, 62Hxx]
60Fxx Limit theorems [See also 28Dxx, 60B12]
60Gxx Stochastic processes
60Hxx Stochastic analysis [See also 58J65]
60Jxx Markov processes
60Kxx Special processes