CZ:Mathematics Workgroup

Template:Naturalsciences

Work plan white paper

The topics below are part of the division of mathematical knowledge to subdiciplines; 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 otherwork group (usually the physicsits) to do the work for us - we state where.

Caveats:

• Do not copy articles from wikipedia without carefully reading them, verifying both scope and focus. 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 and elementary mathematics (for calculus see 26-XX Real functions below)
  • high school mathematics
• 01-XX History and biography
  • Euclid
  • Euler
• 03-XX Mathematical logic and foundations
• 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
  • 12Y05 Computational aspects of field theory and polynomials*13-XX Commutative rings and algebras
• 13-XX Commutative rings and algebras
• 14-XX Algebraic geometry
  • 14Axx Foundations
  • 14Bxx Local theory
  • 14Cxx Cycles and subschemes
  • 14Dxx Families, fibrations
  • 14Exx Birational geometry
  • 14Fxx (Co)homology theory [See also 13Dxx]
  • 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]*19-XX $K$-theory
    • 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
  • 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
• 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
• 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
• 35-XX Partial differential equations
• 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
• 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
• 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
• 62-XX Statistics
• 65-XX Numerical analysis
• 68-XX Computer science (we leave it for the computers Workgorup ?)
• 70-XX Mechanics of particles and systems (we leave it for physics Workgroup??)
• 74-XX Mechanics of deformable solids (we leave it for physics Workgroup??)
• 76-XX Fluid mechanics (we leave it for physics Workgroup??)
• 78-XX Optics, electromagnetic theory {For quantum optics, see 81V80} (we leave it for physics Workgroup??)
• 80-XX Classical thermodynamics, heat transfer (we leave it for physics Workgroup??)
• 81-XX Quantum theory
• 82-XX Statistical mechanics, structure of matter (we leave it for physics Workgroup??)
• 83-XX Relativity and gravitational theory (we leave it for physics Workgroup??)
• 85-XX Astronomy and astrophysics (we leave it for physics Workgroup??)
• 86-XX Geophysics (we leave it for physics Workgroup??)
• 90-XX Operations research, mathematical programming
• 91-XX Game theory, economics, social and behavioral sciences (we leave it for economy Workgroup??)
• 92-XX Biology and other natural sciences (we leave it for biology Workgroup??)
• 93-XX Systems theory; control

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