CZ:Mathematics Workgroup

Template:Naturalsciences

Work plan white paper

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 Citizendium Pilot:How to convert Wikipedia articles to Citizendium articles
• Keep in mind three audiences when writing an article: general readers, math students and professionals.

The classification

00-XX General

(for calculus see 26-XX Real functions below)

• elementary mathematics (pre-university level)

trigonometric function

point line plane solid stubs

elementary algebra

for suggestions of useful articles to write see also list of calculus topics (pre-calculus section)

01-XX History and biography

Euclid

Euler

06-XX Order, lattices, ordered algebraic structures

[See also 18B35]

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

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]
    • Riemann-Roch theorem
    • Clifford's theorem
    • Grothendieck Riemann-Roch theorem
• 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

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

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 useful suggestions see: List of Lie group topics

26-XX Real functions

• Calculus

Mean Value Theorem

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

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

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}

35-XX Partial differential equations

schrodinger equation

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

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

68-XX Computer science

(do we leave it for the computers Workgorup ?)

70-XX Mechanics of particles and systems

(do we leave it for physics Workgroup??)

74-XX Mechanics of deformable solids

(do we leave it for physics Workgroup??)

76-XX Fluid mechanics

(do we leave it for physics Workgroup??)

78-XX Optics, electromagnetic theory

{For quantum optics, see 81V80} (do we leave it for physics Workgroup??)

80-XX Classical thermodynamics, heat transfer

(do we leave it for physics Workgroup??)

82-XX Statistical mechanics, structure of matter

(do we leave it for physics Workgroup??)

83-XX Relativity and gravitational theory

(do we leave it for physics Workgroup??)

85-XX Astronomy and astrophysics

(do we leave it for physics Workgroup??)

86-XX Geophysics

(do we leave it for physics Workgroup??)

91-XX Game theory, economics, social and behavioral sciences

(do we leave it for economy Workgroup??)

92-XX Biology and other natural sciences

(do we leave it for biology Workgroup??)

93-XX Systems theory; control

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