Classification theorem: Difference between revisions

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{{short description|Describes the objects of a given type, up to some equivalence}}

{{Unreferenced|date=December 2009}}

In [[mathematics]], a '''classification theorem''' answers the classification problem "What are the objects of a given type, up to some [[Equivalence relation|equivalence]]?". It gives a non-redundant [[enumeration]]: each object is equivalent to exactly one class.

A few issues related to classification are the following.

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==Geometry==

* [[{{annotated link|Euclidean plane isometry#Classification of Euclidean plane isometries|Classification of Euclidean plane isometries]]}}

* Classification theorems of surfaces

** [[{{annotated link|Classification of two-dimensional closed manifolds]]}}

** [[{{annotated link|Enriques–Kodaira classification]]}} of [[algebraic surfaces]] (complex dimension two, real dimension four)

** [[{{annotated link|Nielsen–Thurston classification]]}} which characterizes homeomorphisms of a compact surface

* Thurston's eight model geometries, and the [[{{annotated link|geometrization conjecture]]}}

* [[{{annotated link|Holonomy#The Berger classification|Berger classification]]}}

* [[{{annotated link|Symmetric space#Classification result|Classification of Riemannian symmetric spaces]]}}

* [[{{annotated link|Lens space#Classification of 3-dimensional lens spaces|Classification of 3-dimensional lens spaces]]}}

* [[{{annotated link|Classification of manifolds]]}}

==Algebra==

* [[{{annotated link|Classification of finite simple groups]]}}

** [[{{annotated link|Abelian group#Classification|Classification of Abelian groups]]}}

** [[{{annotated link|Finitely generated abelian group#Classification|Classification of Finitely generated abelian group]]}}

** [[{{annotated link|Multiple transitivity|Classification of Rank 3 permutation group]]}}

** [[{{annotated link|Rank 3 permutation group#Classification|Classification of 2-transitive permutation groups]]}}

* [[{{annotated link|Artin–Wedderburn theorem]]}} — a classification theorem for semisimple rings

* [[{{annotated link|Classification of Clifford algebras]]}}

* [[{{annotated link|Classification of low-dimensional real Lie algebras]]}}

* Classification of Simple Lie algebras and groups

** [[{{annotated link|Semisimple Lie algebra#Classification|Classification of simple complex Lie algebras]]}}

** [[{{annotated link|Satake diagram|Classification of simple real Lie algebras]]}}

** [[{{annotated link|Simple Lie group#Full classification|Classification of centerless simple Lie groups]]}}

** [[{{annotated link|List of simple Lie groups|Classification of simple Lie groups]]}}

* [[{{annotated link|Bianchi classification]]}}

* [[{{annotated link|ADE classification]]}}

*[[{{annotated link|Langlands classification]]}}

==Linear algebra==

* [[{{annotated link|Finite-dimensional vector space]]}}s (by dimension)

* [[{{annotated link|Rank–nullity theorem]]}} (by rank and nullity)

* [[{{annotated link|Structure theorem for finitely generated modules over a principal ideal ___domain]]}}

* [[{{annotated link|Jordan normal form]]}}

* [[{{annotated link|Frobenius normal form]]}} (rational canonical form)

* [[{{annotated link|Sylvester's law of inertia]]}}

==Analysis==

* [[{{annotated link|Classification of discontinuities]]}}

==Complex analysis==

* [[{{annotated link|Classification of Fatou components]]}}

==Mathematical physics==

* [[{{annotated link|Classification of electromagnetic fields]]}}

* [[{{annotated link|Petrov classification]]}}

* [[{{annotated link|Segre classification]]}}

* [[{{annotated link|Wigner's classification]]}}

==See also==

* [[{{annotated link|Representation theorem]]}}

* [[{{annotated link|Comparison theorem]]}}

*[[ {{annotated link|List of manifolds]]}}

{{DEFAULTSORT:Classification Theorem}}