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.

*The equivalence problem is "given two objects, determine if they are equivalent".

*A [[complete set of invariants]], together with which invariants are {{clarify span|realizable,|reason=This notion should be introduced.|date=October 2020}} solves the classification problem, and is often a step in solving it. (A combination of invariant values is realizable if there in fact exists an object whose invariants take on the specified set of values)

*A {{clarify span|computable complete set of invariants|reason=Shouldn't this be "finite set of computable invariants"? Computability (whatever this is supposed to mean on a set of functions) is of no help if infinitely many functions must be evaluated or if an uncomputable function must be evaluated.|date=October 2020}} (together with which invariants are realizable) solves both the classification problem and the equivalence problem.

* A [[canonical form]] solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class.

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==Geometry==

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

* [[Platonic solid#Classification|Classification of Platonic solids]]

* 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]]}}

==ComplexDynamical analysissystems==

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

* [[Ratner's theorems#Short description|Ratner classification theorem]]

==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]]}}

* [[List of theorems]]

==References==

{{reflist}}

{{DEFAULTSORT:Classification Theorem}}