CZ:Mathematics Workgroup: Difference between revisions

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imported>David Lehavi
(added more pieces of classification tree)
imported>Aleksander Stos
(→‎The classification: wiki formating: top level goes to table of contents; some minor cleanup (BTW, what happend to 20-XX)?))
Line 17: Line 17:
* Keep in mind three audiences when writing an article: general readers, math students and professionals.
* Keep in mind three audiences when writing an article: general readers, math students and professionals.


=== The classification ===
== The classification ==


* 00-XX General and elementary mathematics (for calculus see 26-XX Real functions below)
==== 00-XX General ====
** high school mathematics
''(for calculus see 26-XX Real functions below)''
*01-XX History and biography  
* elementary mathematics
**[[Euclid]]
:: [[trigonometric function]] <!-- just an example!-->
**[[Euler]]
 
*03-XX Mathematical logic and foundations
==== 01-XX History and biography ====
*05-XX Combinatorics  
::[[Euclid]] <!-- just an example! feel free to change-->
*06-XX Order, lattices, ordered algebraic structures [See also 18B35]
::[[Euler]]
*08-XX General algebraic systems
==== 03-XX Mathematical logic and foundations ====
*11-XX [[Number theory]]
====05-XX Combinatorics ====
**11Axx Elementary number theory {For analogues in number fields, see 11R04}  
====06-XX Order, lattices, ordered algebraic structures====
**11Bxx Sequences and sets  
''[See also 18B35]''
**11Cxx Polynomials and matrices  
====08-XX General algebraic systems====
**11Dxx Diophantine equations [See also 11Gxx, 14Gxx]  
====11-XX [[Number theory]]====
**11Exx Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63}  
*11Axx Elementary number theory {For analogues in number fields, see 11R04}  
**11Fxx Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45}  
*11Bxx Sequences and sets  
**11Gxx Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14-xx, 14Gxx, 14Kxx]  
*11Cxx Polynomials and matrices  
**11Hxx Geometry of numbers {For applications in coding theory, see 94B75}  
*11Dxx Diophantine equations [See also 11Gxx, 14Gxx]  
**11Jxx Diophantine approximation, transcendental number theory [See also 11K60]  
*11Exx Forms and linear algebraic groups [See also 19Gxx] {For quadratic forms in linear algebra, see 15A63}  
**11Kxx Probabilistic theory: distribution modulo <math>1</math>; metric theory of algorithms  
*11Fxx Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45}  
**11Lxx [[Exponential sums]] and character sums {For finite fields, see 11Txx}  
*11Gxx Arithmetic algebraic geometry (Diophantine geometry) [See also 11Dxx, 14-xx, 14Gxx, 14Kxx]  
**11Mxx Zeta and <math>L</math>-functions: analytic theory  
*11Hxx Geometry of numbers {For applications in coding theory, see 94B75}  
**11Nxx Multiplicative number theory  
*11Jxx Diophantine approximation, transcendental number theory [See also 11K60]  
**11Pxx Additive number theory; [[partitions]]  
*11Kxx Probabilistic theory: distribution modulo <math>1</math>; metric theory of algorithms  
**11Rxx Algebraic number theory: [[global fields]] {For complex multiplication, see 11G15}  
*11Lxx [[Exponential sums]] and character sums {For finite fields, see 11Txx}  
**11Sxx Algebraic number theory: [[local and <math>p</math>-adic fields]]  
*11Mxx Zeta and <math>L</math>-functions: analytic theory  
**11Txx [[Finite fields]] and commutative rings (number-theoretic aspects)  
*11Nxx Multiplicative number theory  
**11Uxx Connections with logic  
*11Pxx Additive number theory; [[partitions]]  
**11Yxx Computational number theory [See also 11-04]  
*11Rxx Algebraic number theory: [[global fields]] {For complex multiplication, see 11G15}  
*12-XX Field theory and [[polynomials]]
*11Sxx Algebraic number theory: [[local and <math>p</math>-adic fields]]  
**12Dxx Real and complex fields
*11Txx [[Finite fields]] and commutative rings (number-theoretic aspects)  
*** [[real numbers]]
*11Uxx Connections with logic  
*** [[complex numbers]]
*11Yxx Computational number theory [See also 11-04]  
**12Exx General field theory
====12-XX Field theory and [[polynomials]] ====
**12Fxx Field extensions
*12Dxx Real and complex fields
**12Gxx Homological methods (field theory)
:: [[real numbers]]
**12Hxx Differential and difference algebra
:: [[complex numbers]]
**12Jxx Topological fields
*12Exx General field theory
**12Kxx Generalizations of fields
*12Fxx Field extensions
**12Lxx Connections with logic
*12Gxx Homological methods (field theory)
**12Y05 Computational aspects of field theory and polynomials*13-XX Commutative rings and algebras
*12Hxx Differential and difference algebra
*13-XX Commutative rings and algebras
*12Jxx Topological fields
*14-XX [[Algebraic geometry]]
*12Kxx Generalizations of fields
**14Axx Foundations  
*12Lxx Connections with logic
**14Bxx Local theory  
*12Yxx
**14Cxx Cycles and subschemes  
**12Y05 Computational aspects of field theory and polynomials
**14Dxx Families, fibrations  
====13-XX Commutative rings and algebras====
**14Exx [[Birational geometry]]  
====14-XX [[Algebraic geometry]]====
**14Fxx (Co)homology theory [See also 13Dxx]  
*14Axx Foundations  
**14Gxx Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx]  
*14Bxx Local theory  
**14Hxx [[algebraic curve|Curves]]
*14Cxx Cycles and subschemes  
***14H05 Algebraic functions; [[function fields]] [See also 11R58]
*14Dxx Families, fibrations  
***14H10,14H15 [[moduli]]  [See also 30F10, 32Gxx]
*14Exx [[Birational geometry]]  
***14H20 Singularities, local rings [See also 13Hxx, 14B05]
*14Fxx (Co)homology theory [See also 13Dxx]  
***14H25 Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
*14Gxx Arithmetic problems. Diophantine geometry [See also 11Dxx, 11Gxx]  
***14H30 Coverings, fundamental group [See also 14E20, 14F35]  
*14Hxx [[algebraic curve|Curves]]
***14H37 Automorphisms
**14H05 Algebraic functions; [[function fields]] [See also 11R58]
***14H40 [[Jacobians]], [[Prym varieties]] [See also 32G20]
**14H10,14H15 [[moduli]]  [See also 30F10, 32Gxx]
***14H42 [[Theta functions]]; [[Schottky problem]] [See also 14K25, 32G20]
**14H20 Singularities, local rings [See also 13Hxx, 14B05]
***14H45 Special curves and curves of low genus
**14H25 Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]
**** [[hyperelliptic curve]]
**14H30 Coverings, fundamental group [See also 14E20, 14F35]  
***14H50 Plane and space curves
**14H37 Automorphisms
***14H51 Special divisors (gonality, Brill-Noether theory)
**14H40 [[Jacobians]], [[Prym varieties]] [See also 32G20]
***14H52 [[Elliptic curves]] [See also 11G05, 11G07, 14Kxx]
**14H42 [[Theta functions]]; [[Schottky problem]] [See also 14K25, 32G20]
***14H55 [[Riemann surfaces]]; [[Weierstrass points]]; [[gap sequences]] [See also 30Fxx]
**14H45 Special curves and curves of low genus
***14H60 Vector bundles on curves and their moduli [See also 14D20, 14F05]
:: [[hyperelliptic curve]]
**14Jxx Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx}  
**14H50 Plane and space curves
***14J10 Families, moduli, classification: algebraic theory
**14H51 Special divisors (gonality, Brill-Noether theory)
***14J15 Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
**14H52 [[Elliptic curves]] [See also 11G05, 11G07, 14Kxx]
***14J17 [[Singularities of surfaces]] [See also 14B05, 14E15]
**14H55 [[Riemann surfaces]]; [[Weierstrass points]]; [[gap sequences]] [See also 30Fxx]
***14J20 Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
**14H60 Vector bundles on curves and their moduli [See also 14D20, 14F05]
***14J25 Special surfaces {For Hilbert modular surfaces, see 14G35}
*14Jxx Surfaces and higher-dimensional varieties {For analytic theory, see 32Jxx}  
***14J26 [[Rational surfaces]] and [[ruled surfaces]]
**14J10 Families, moduli, classification: algebraic theory
***14J27 [[Elliptic surfaces]]
**14J15 Moduli, classification: analytic theory; relations with modular forms [See also 32G13]
***14J28 [[K3 surfaces]] and [[Enriques surfaces]]
**14J17 [[Singularities of surfaces]] [See also 14B05, 14E15]
**** [[Kummer surfaces]]
**14J20 Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx]
***14J29 [[Surfaces of general type]]
**14J25 Special surfaces {For Hilbert modular surfaces, see 14G35}
***14J30 [[3-folds]]
**14J26 [[Rational surfaces]] and [[ruled surfaces]]
***14J32 [[Calabi-Yau manifolds]], [[mirror symmetry]]
**14J27 [[Elliptic surfaces]]
***14J35 [[4-folds]]
**14J28 [[K3 surfaces]] and [[Enriques surfaces]]
***14J40 [[n-folds]] (n>4)
:: [[Kummer surfaces]]
***14J45 [[Fano varieties]]
**14J29 [[Surfaces of general type]]
***14J50 Automorphisms of surfaces and higher-dimensional varieties
**14J30 [[3-folds]]
***14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
**14J32 [[Calabi-Yau manifolds]], [[mirror symmetry]]
***14J70 Hypersurfaces
**14J35 [[4-folds]]
***14J80 Topology of surfaces ([[Donaldson polynomials]], [[Seiberg-Witten invariants]])
**14J40 [[n-folds]] (n>4)
**14Kxx [[Abelian variety|Abelian varieties]] and schemes  
**14J45 [[Fano varieties]]
**14Lxx Algebraic groups {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45}  
**14J50 Automorphisms of surfaces and higher-dimensional varieties
**14Mxx Special varieties  
**14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
**14Nxx Projective and enumerative geometry [See also 51-xx]  
**14J70 Hypersurfaces
**14Pxx [[Real algebraic]] and real analytic geometry  
**14J80 Topology of surfaces ([[Donaldson polynomials]], [[Seiberg-Witten invariants]])
**14Qxx Computational algebraic geometry [See also 12Y05, 13Pxx, 68W30]  
*14Kxx [[Abelian variety|Abelian varieties]] and schemes  
**14Rxx Affine geometry
*14Lxx Algebraic groups {For linear algebraic groups, see 20Gxx; for Lie algebras, see 17B45}  
*15-XX Linear and multilinear algebra; matrix theory
*14Mxx Special varieties  
*16-XX Associative rings and algebras  
*14Nxx Projective and enumerative geometry [See also 51-xx]  
*17-XX Nonassociative rings and algebras
*14Pxx [[Real algebraic]] and real analytic geometry  
*18-XX [[Category theory]]; [[homological algebra]]  
*14Qxx Computational algebraic geometry [See also 12Y05, 13Pxx, 68W30]  
**18Axx General theory of categories and functors
*14Rxx Affine geometry
**18Bxx Special categories
====15-XX Linear and multilinear algebra; matrix theory ====
**18Cxx Categories and theories
====16-XX Associative rings and algebras ====
**18Dxx Categories with structure
====17-XX Nonassociative rings and algebras====
**18Exx [[Abelian categories]]
====18-XX [[Category theory]]; [[homological algebra]] ====
**18Fxx Categories and geometry
*18Axx General theory of categories and functors
**18Gxx Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx]*19-XX <math>K</math>-theory
*18Bxx Special categories
***18G05 [[Projective objects]] and [[injective objects]] [See also 13C10, 13C11, 16D40, 16D50]
*18Cxx Categories and theories
***18G10 [[Resolutions]]; [[derived functors]] [See also 13D02, 16E05, 18E25]
*18Dxx Categories with structure
***18G15 [[Ext]] and [[Tor]], generalizations, [[Künneth formula]] [See also 55U25]
*18Exx [[Abelian categories]]
***18G20 [[Homological dimension]] [See also 13D05, 16E10]
*18Fxx Categories and geometry
***18G25 Relative homological algebra, projective classes
*18Gxx Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx]
***18G30 [[Simplicial sets]], [[simplicial objects]] (in a category) [See also 55U10]
**18G05 [[Projective objects]] and [[injective objects]] [See also 13C10, 13C11, 16D40, 16D50]
***18G35 [[Chain complexes]] [See also 18E30, 55U15]
**18G10 [[Resolutions]]; [[derived functors]] [See also 13D02, 16E05, 18E25]
***18G40 [[Spectral sequences]], [[hypercohomology]] [See also 55Txx]
**18G15 [[Ext]] and [[Tor]], generalizations, [[Künneth formula]] [See also 55U25]
***18G50 [[Nonabelian homological algebra]]
**18G20 [[Homological dimension]] [See also 13D05, 16E10]
***18G55 [[Homotopical algebra]]
**18G25 Relative homological algebra, projective classes
***18G60 Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]*20-XX [[Group theory]] and generalizations
**18G30 [[Simplicial sets]], [[simplicial objects]] (in a category) [See also 55U10]
**20Axx Foundations
**18G35 [[Chain complexes]] [See also 18E30, 55U15]
**20Bxx [[Permutation groups]]
**18G40 [[Spectral sequences]], [[hypercohomology]] [See also 55Txx]
**20Cxx Representation theory of groups [See also 19A22 (for representation rings and Burnside rings)]
**18G50 [[Nonabelian homological algebra]]
**20Dxx Abstract finite groups
**18G55 [[Homotopical algebra]]
**20Exx Structure and classification of infinite or finite groups
**18G60 Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]*20-XX [[Group theory]] and generalizations
**20Fxx Special aspects of infinite or finite groups
====19-XX ''K''-theory====
**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}
==== 20-XX (name?) ====
*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]
**20Hxx Other groups of matrices [See also 15A30]
**20Jxx Connections with homological algebra and category theory
*20Jxx Connections with homological algebra and category theory
**20Kxx [[Abelian groups]]
*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}
*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]]
*20Mxx [[Semigroups]]
**20Nxx Other generalizations of groups
*20Nxx Other generalizations of groups
**20P05 Probabilistic methods in group theory [See also 60Bxx]*22-XX [[Topological groups]],  [[Lie groups]]
*20P05 Probabilistic methods in group theory [See also 60Bxx]*22-XX [[Topological groups]],  [[Lie groups]]
*22-XX [[Topological groups]],  [[Lie groups]]
====22-XX [[Topological groups]],  [[Lie groups]] ====
*26-XX Real functions   
====26-XX Real functions  ====
** [[Calculus]]
* [[Calculus]]
*** [[Mean Value Theorem]]
:: [[Mean Value Theorem]]
*28-XX Measure and integration  
====28-XX Measure and integration ====
** 28Axx Classical measure theory
* 28Axx Classical measure theory
***28A05 Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
**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
** 28A10 Real- or complex-valued set functions
*** 28A12 Contents, measures, outer measures, capacities
** 28A12 Contents, measures, outer measures, capacities
**** [[Lebesgue measure]]
:: [[Lebesgue measure]]
*** 28A15 Abstract differentiation theory, differentiation of set functions [See also 26A24]
** 28A15 Abstract differentiation theory, differentiation of set functions [See also 26A24]
*** 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
** 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
*** 28A25 Integration with respect to measures and other set functions
** 28A25 Integration with respect to measures and other set functions
**** [[Lebesgue integral]]
:: [[Lebesgue integral]]
**** [[Bounded convergence theorem]]
:: [[Bounded convergence theorem]]
**** [[Monotone convergence theorem]]
:: [[Monotone convergence theorem]]
**** [[Fatou's lemma]]
:: [[Fatou's lemma]]
*** 28A33 Spaces of measures, convergence of measures [See also 46E27, 60Bxx]
** 28A33 Spaces of measures, convergence of measures [See also 46E27, 60Bxx]
*** 28A35 Measures and integrals in product spaces
** 28A35 Measures and integrals in product spaces
*** 28A50 Integration and disintegration of measures
** 28A50 Integration and disintegration of measures
*** 28A51 Lifting theory [See also 46G15]
** 28A51 Lifting theory [See also 46G15]
*** 28A60 Measures on Boolean rings, measure algebras [See also 54H10]
** 28A60 Measures on Boolean rings, measure algebras [See also 54H10]
*** 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
** 28A75 Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
*** 28A78 Hausdorff and [[packing measure]]s
** 28A78 Hausdorff and [[packing measure]]s
**** [[Hausdorff measure]]
:: [[Hausdorff measure]]
**** [[Hausdorff dimension]]
:: [[Hausdorff dimension]]
**** [[Packing dimension]]
:: [[Packing dimension]]
*** 28A80 Fractals [See also 37Fxx]
** 28A80 Fractals [See also 37Fxx]
*** 28A99 None of the above, but in this section
** 28A99 None of the above, but in this section
** 28Bxx Set functions, measures and integrals with values in abstract spaces
* 28Bxx Set functions, measures and integrals with values in abstract spaces
** 28Exx Miscellaneous topics in measure theory  
* 28Exx Miscellaneous topics in measure theory  
*30-XX Functions of a complex variable  
====30-XX Functions of a complex variable ====
*31-XX [[Potential theory]]  
====31-XX [[Potential theory]] ====
**31Axx Two-dimensional theory
*31Axx Two-dimensional theory
***31A05 Harmonic, subharmonic, superharmonic functions
**31A05 Harmonic, subharmonic, superharmonic functions
***31A10 Integral representations, integral operators, integral equations methods
**31A10 Integral representations, integral operators, integral equations methods
***31A15 Potentials and capacity, harmonic measure, extremal length [See also 30C85]
**31A15 Potentials and capacity, harmonic measure, extremal length [See also 30C85]
***31A20 Boundary behavior (theorems of Fatou type, etc.)
**31A20 Boundary behavior (theorems of Fatou type, etc.)
***31A25 Boundary value and inverse problems
**31A25 Boundary value and inverse problems
***31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation
**31A30 Biharmonic, polyharmonic functions and equations, Poisson's equation
***31A35 Connections with differential equations
**31A35 Connections with differential equations
***31A99 None of the above, but in this section
**31A99 None of the above, but in this section
**31Bxx Higher-dimensional theory
*31Bxx Higher-dimensional theory
***31B05 Harmonic, subharmonic, superharmonic functions
**31B05 Harmonic, subharmonic, superharmonic functions
***31B10 Integral representations, integral operators, integral equations methods
**31B10 Integral representations, integral operators, integral equations methods
***31B15 Potentials and capacities, extremal length
**31B15 Potentials and capacities, extremal length
***31B20 Boundary value and inverse problems
**31B20 Boundary value and inverse problems
***31B25 Boundary behavior
**31B25 Boundary behavior
***31B30 Biharmonic and polyharmonic equations and functions
**31B30 Biharmonic and polyharmonic equations and functions
***31B35 Connections with differential equations
**31B35 Connections with differential equations
***31B99 None of the above, but in this section
**31B99 None of the above, but in this section
**31Cxx Other generalizations
*31Cxx Other generalizations
**31D05 Axiomatic potential theory
*31D05 Axiomatic potential theory
*32-XX Several complex variables and analytic spaces  
====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}
====33-XX Special functions ====
*34-XX [[Ordinary differential equations]]
''(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]]
====34-XX [[Ordinary differential equations]] ====
*37-XX [[Dynamical systems]] and [[ergodic theory]]  
==== 35-XX [[Partial differential equations]] ====
*39-XX Difference and functional equations
::[[schrodinger equation]]
*40-XX [[Sequences]], [[series]], [[summability]]
==== 37-XX [[Dynamical systems]] and [[ergodic theory]] ====
**40Axx Convergence and divergence of infinite limiting processes
====39-XX Difference and functional equations ====
**40B05 Multiple sequences and series {(should also be assigned at least one other classification number in this section)}
====40-XX [[Sequences]], [[series]], [[summability]]====
**40Cxx General summability methods
*40Axx Convergence and divergence of infinite limiting processes
**40Dxx Direct theorems on summability
*40B05 Multiple sequences and series {(should also be assigned at least one other classification number in this section)}
**40Exx Inversion theorems
*40Cxx General summability methods
**40F05 Absolute and strong summability
*40Dxx Direct theorems on summability
**40Gxx Special methods of summability
*40Exx Inversion theorems
**40H05 Functional analytic methods in summability
*40F05 Absolute and strong summability
**40J05 Summability in abstract structures [See also 43A55, 46A35, 46B15]*41-XX Approximations and expansions  
*40Gxx Special methods of summability
*41-XX Approximations and expansions  
*40H05 Functional analytic methods in summability
*42-XX Fourier analysis
*40J05 Summability in abstract structures [See also 43A55, 46A35, 46B15]*41-XX Approximations and expansions  
** 42-04 Explicit machine computation and programs (not the theory of computation or programming)
====41-XX Approximations and expansions ====
*** [[Fourier series]]
====42-XX Fourier analysis====
** 42Axx Fourier analysis in one variable
* 42-04 Explicit machine computation and programs (not the theory of computation or programming)
** 42Bxx Fourier analysis in several variables {For automorphic theory, see mainly 11F30}
:: [[Fourier series]]
** 42Cxx Nontrigonometric Fourier analysis  
* 42Axx Fourier analysis in one variable
*43-XX Abstract harmonic analysis  
* 42Bxx Fourier analysis in several variables {For automorphic theory, see mainly 11F30}
*44-XX Integral transforms, operational calculus  
* 42Cxx Nontrigonometric Fourier analysis  
*45-XX Integral equations
====43-XX Abstract harmonic analysis ====
*46-XX Functional analysis  
====44-XX Integral transforms, operational calculus ====
*47-XX Operator theory
====45-XX Integral equations ====
*49-XX [[Calculus of variations]] and optimal control; optimization  
====46-XX Functional analysis ====
*51-XX [[Geometry]]  
====47-XX Operator theory====
*52-XX Convex and discrete geometry
====49-XX [[Calculus of variations]] and optimal control; optimization ====
*53-XX [[Differential geometry]]  
====51-XX [[Geometry]] ====
*54-XX [[General topology]]  
====52-XX Convex and discrete geometry====
*55-XX [[Algebraic topology]]
====53-XX [[Differential geometry]] ====
*57-XX Manifolds and cell complexes  
====54-XX [[General topology]] ====
*58-XX Global analysis, analysis on manifolds  
====55-XX [[Algebraic topology]] ====
*60-XX [[Probability]] theory and stochastic processes  
====57-XX Manifolds and cell complexes ====
**60Axx Foundations of probability theory  
====58-XX Global analysis, analysis on manifolds ====
**60Bxx Probability theory on algebraic and topological structures  
====60-XX [[Probability]] theory and stochastic processes ====
*60Axx Foundations of probability theory  
*60Bxx Probability theory on algebraic and topological structures  
**60C05 Combinatorial probability  
**60C05 Combinatorial probability  
**60D05 Geometric probability, stochastic geometry, random sets [See also 52A22, 53C65]  
**60D05 Geometric probability, stochastic geometry, random sets [See also 52A22, 53C65]  
**60Exx Distribution theory [See also 62Exx, 62Hxx]  
*60Exx Distribution theory [See also 62Exx, 62Hxx]  
**60Fxx Limit theorems [See also 28Dxx, 60B12]  
*60Fxx Limit theorems [See also 28Dxx, 60B12]  
**60Gxx Stochastic processes  
*60Gxx Stochastic processes  
**60Hxx Stochastic analysis [See also 58J65]  
*60Hxx Stochastic analysis [See also 58J65]  
**60Jxx [[Markov processes]]  
*60Jxx [[Markov processes]]  
**60Kxx Special processes
*60Kxx Special processes
*62-XX [[Statistics]]
====62-XX [[Statistics]] ====
*65-XX [[Numerical analysis]]
====65-XX [[Numerical analysis]]====
*68-XX Computer science                       (we leave it for the computers Workgorup ?)  
====68-XX Computer science====
*70-XX Mechanics of particles and systems     (we leave it for physics Workgroup??)
''<small>(do we leave it for the computers Workgorup ?) </small> ''
*74-XX Mechanics of deformable solids           (we leave it for physics Workgroup??)
====70-XX Mechanics of particles and systems====
*76-XX Fluid mechanics                                     (we leave it for physics Workgroup??)
''<small>(do we leave it for physics Workgroup??)</small>''
*78-XX Optics, electromagnetic theory {For quantum optics, see 81V80}          (we leave it for physics Workgroup??)
====74-XX Mechanics of deformable solids ====
*80-XX Classical thermodynamics, heat transfer                                             (we leave it for physics Workgroup??)
''<small>(do we leave it for physics Workgroup??)</small>''
*81-XX Quantum theory                                                      
====76-XX Fluid mechanics====
*82-XX Statistical mechanics, structure of matter                       (we leave it for physics Workgroup??)
''<small>(do we leave it for physics Workgroup??)</small>''
*83-XX Relativity and gravitational theory                                 (we leave it for physics Workgroup??)
====78-XX Optics, electromagnetic theory ====
*85-XX Astronomy and astrophysics                                         (we leave it for physics Workgroup??)
'' {For quantum optics, see 81V80}          (do we leave it for physics Workgroup??)''
*86-XX Geophysics                                       (we leave it for physics Workgroup??)
====80-XX Classical thermodynamics, heat transfer ====
*90-XX Operations research, mathematical programming
''(do we leave it for physics Workgroup??)''
*91-XX Game theory, economics, social and behavioral sciences   (we leave it for economy Workgroup??)
====81-XX Quantum theory ====
*92-XX Biology and other natural sciences                                 (we leave it for biology Workgroup??)
====82-XX Statistical mechanics, structure of matter ====
*93-XX Systems theory; control
''(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====

Revision as of 04:31, 3 March 2007


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:

The classification

00-XX General

(for calculus see 26-XX Real functions below)

  • elementary mathematics
trigonometric function

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 ; metric theory of algorithms
  • 11Lxx Exponential sums and character sums {For finite fields, see 11Txx}
  • 11Mxx Zeta and -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 -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
  • 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
Kummer surfaces

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

19-XX K-theory

20-XX (name?)

  • 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

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

schrodinger 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

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

(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??)

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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