Classification theorem: Difference between revisions

Revision as of 23:30, 6 March 2007

Latest revision as of 04:52, 15 September 2024

(48 intermediate revisions by 25 users not shown)

Line 1:

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

==Definition==

{{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.

==Related issues==

There are several related issues:

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

*A [[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 {{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 isomorphismequivalence 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.

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

==Examples==

There exist many '''classification theorems''' in [[mathematics]]:

===Geometry===

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

* '''Classification theorem of surfaces'''

** [[Platonic solid#Classification|Classification of two-dimensional closedPlatonic manifoldssolids]]

* '''Classification theoremtheorems of surfaces'''

** [[Enriques-Kodaira classification]] of [[algebraic surfaces]] (complex dimension two, real dimension four)

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

** [[Nielsen-Thurston classification]] which characterizes homeomorphisms of a compact surface

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

* Thurston's eight model geometries, and the [[geometrization conjecture]]

** [[Nielsen-Thurston{{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}}

* [[Finite dimensional vector spaces]] (by dimension)

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

* [[Structure theorem for finitely generated modules over a principal ideal ___domain]]

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

* [[Sylvester's law of inertia]]

** [[{{annotated link|Multiple transitivity|Classification of finiteRank 3 simplepermutation groups]]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==

{{disambig}} [[Category:Mathematical disambiguation]]

* [[Finite{{annotated link|Finite-dimensional vector spaces]]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}}

==Dynamical systems==

* {{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}}

{{disambig}} [[Category:Mathematical disambiguationtheorems]]

[[Category:Mathematical classification systems]]