CLASSIFICATION THEOREM

In mathematics, a 'classification theorem' answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
A few related issues to classification are the following.

★ The isomorphism 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 computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the isomorphism problem.

★ A canonical form solves the classification problem, and is more data: it not only classifies every class, but gives a distinguished (canonical) element of each class.
There exist many 'classification theorems' in mathematics, as described below.

Contents

Geometry

Algebra

Linear algebra

Complex analysis

Geometry

★ 'Classification theorem of surfaces'

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★ Classification of two-dimensional closed manifolds

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★ Enriques-Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)

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★ Nielsen-Thurston classification which characterizes homeomorphisms of a compact surface

★ Thurston's eight model geometries, and the geometrization conjecture

Algebra

★ Classification of finite simple groups

★ Artin–Wedderburn theorem — a classification theorem for semisimple rings

Linear algebra

★ Finite-dimensional vector spaces (by dimension)

★ rank-nullity theorem (by rank and nullity)

★ Structure theorem for finitely generated modules over a principal ideal domain

★ Jordan normal form

★ Sylvester's law of inertia

Complex analysis

★ Classification of Fatou components