View source for Classification theorem

{{short description|Describes the objects of a given type, up to some equivalence}}
In [[Mathematics|mathematics]], a '''classification theorem''' answers the [[Classification|classification]] problem: "What are the objects of a given type, up to some [[Equivalence relation|equivalence]]?". It gives a non-redundant [[Enumeration|enumeration]]: each object is equivalent to exactly one class.<ref>{{Cite web |date=2006-12-07 |title=An enormous theorem: the classification of finite simple groups {{!}} plus.maths.org |url=https://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups |access-date=2025-09-16 |website=plus.maths.org |language=en}}</ref>

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|complete set of invariants]], together with which invariants are realizable, 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 (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
* A [[Canonical form|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.

There exist many '''classification theorems''' in [[Mathematics|mathematics]], as described below.

==Geometry==
* [[Platonic solid#Classification|Classification of Platonic solids]]
* Classification theorems of surfaces
** {{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|Classification of manifolds}}

==Algebra==
* {{annotated link|Classification of finite simple groups}}
* {{annotated link|Artin–Wedderburn theorem}} &mdash; 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|Satake diagram|Classification of simple real Lie algebras}}
** {{annotated link|List of simple Lie groups|Classification of simple Lie groups}}
* {{annotated link|Astronomy:Bianchi classification}}
* {{annotated link|ADE classification}}
*{{annotated link|Langlands classification}}

==Linear algebra==
* {{annotated link|Rank–nullity theorem}} (by rank and nullity)
* {{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}}

==Dynamical systems==
* {{annotated link|Classification of Fatou components}}
* [[Ratner's theorems#Short description|Ratner classification theorem]]

==Mathematical physics==
* {{annotated link|Physics:Classification of electromagnetic fields}}
* {{annotated link|Petrov classification}}
* {{annotated link|Segre classification}}
* {{annotated link|Physics: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}}
[[Category:Mathematical theorems]]
[[Category:Mathematical classification systems]]

{{Sourceattribution|Classification theorem}}