View source for CZ:Mathematics Workgroup

{{Workgroup|group=Mathematics}}

* Our mailing list is called ''cz-mathematics'': [http://mail.citizendium.org/mailman/listinfo/cz-mathematcs subscribe], [http://mail.citizendium.org/pipermail/cz-astronomy/ archive]. 
* Useful page for mathematics authors: [[CZ:Formatting mathematics]].
* Is there any [http://en.citizendium.org/wiki/Special:Newpages new hot math-related stuff?] If so, let's mark it by inserting <nowiki>[[Category:Mathematics Workgroup]]</nowiki>.

== 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 <ref>[http://www.ams.org/msc/index.html#browse]</ref> and ZMATH <ref>[http://zmath.u-strasbg.fr/math-cgi-bin/zmscen?level=0&zb=/math-cgi-bin/zmen&maxdocs=300&type=zmath&form=/ZMATH/en/quick.html]</ref>, 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==
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.<ref>[http://hemlock.knams.wikimedia.org/%7Eleon/stats/wikicharts/index.php?wiki=enwiki&ns=articles&limit=1000&mode=view]</ref> This varies heavily over the time.

On [[Planet Math]],  the most popular (as of March 2007) were:<ref>see [http://planetmath.org/?op=enlist&mode=hits Planet Math] stats page</ref>
[[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]].

Some more or less random and useful suggestions can be found [[/List of mathematical topics|here]].

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. Thus, we recommend starting with a bit smaller tasks.

__TOC__

== The classification ==

==== 00-XX General ====
''(for calculus see 26-XX Real functions below)''
* elementary mathematics (pre-university level) <!-- high school is not well known e.g. in continental europe-->
:: [[Elementary functions]]
:: [[trigonometric function]] <!-- just an example!-->
:: [[point]] [[line]] [[plane]] [[solid (geometry)|solid]] stubs
:: [[elementary algebra]]
:: for suggestions of useful articles to write see also [[/List of mathematical topics#calculus topics|here]]  (pre-calculus or geometry sections)

==== 01-XX History and biography ====
::[[Euclid]] <!-- just an example! feel free to change-->
::[[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 <math>1</math>; metric theory of algorithms 
*11Lxx [[Exponential sums]] and character sums {For finite fields, see 11Txx} 
*11Mxx Zeta and <math>L</math>-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 <math>p</math>-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 [[polynomial]]s ====
*12Dxx Real and complex fields
:: [[real number]]s
:: [[complex number]]s
*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 [[algebraic variety|Varieties]] and [[morphism]]s
**14A15 [[scheme (alegbraic geometry)|Schemes]] and morphisms
**14A20 Generalizations ([[algebraic space]]s, [[stack (algebraic geometry)|stacks]])
**14A22 [[Noncommutative algebraic geometry]]
*14Bxx Local theory 
**14B05 [[Singularity (algebraic geomtetry|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 neighborhood]]s
**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 [[Divisor]]s, [[linear system]]s, [[invertible sheaf|invertible sheaves]]
**14C21 [[Pencils, nets, webs]] [See also 53A60]
**14C22 [[Picard groups]]
**14C25 [[Algebraic cycle]]s
**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 [[algebraic curve|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 [[Algebraic surface|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 variety|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: [[/List of mathematical topics#Lie groups topics|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|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 measure]]s
:: [[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 equation]]s ====
::see the current article on [[differential equation]]s
==== 35-XX [[Partial differential equation]]s ====
::[[Schrödinger equation]]
==== 37-XX [[Dynamical system]]s and [[ergodic theory]] ====
====39-XX Difference and functional equations ====
====40-XX [[sequence|Sequences]], [[series (mathematics)|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
====62-XX [[Statistics]] ====
====65-XX [[Numerical analysis]]====
====68-XX Computer science====
''<small>(do we leave it for the computers Workgorup ?) </small> ''
====70-XX Mechanics of particles and systems====
''<small>(do we leave it for physics Workgroup??)</small>''
====74-XX Mechanics of deformable solids ====
''<small>(do we leave it for physics Workgroup??)</small>''
====76-XX Fluid mechanics====
''<small>(do we leave it for physics Workgroup??)</small>''
====78-XX Optics, electromagnetic theory ====
'' {For quantum optics, see 81V80}           (do we leave it for physics Workgroup??)''
====80-XX Classical thermodynamics, heat transfer ====
''<small>(do we leave it for physics Workgroup??)</small>''
====81-XX Quantum theory ==== 
====82-XX Statistical mechanics, structure of matter ====
''<small>(do we leave it for physics Workgroup??)</small>''
====83-XX Relativity and gravitational theory ====

====85-XX Astronomy and astrophysics ====
''<small>(do we leave it for physics Workgroup??)</small>''
====86-XX Geophysics====
''<small>(do we leave it for physics Workgroup??)</small>''
====90-XX Operations research, mathematical programming ====
====91-XX Game theory, economics, social and behavioral sciences ====
''<small>(do we leave it for economy Workgroup??)</small>''
====92-XX Biology and other natural sciences ====
''<small>(do we leave it for biology Workgroup??)</small>''
====93-XX Systems theory; control====

==References==
{{reflist}}