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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 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 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 computable complete set of invariants (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.

There exist many classification theorems in mathematics, as described below.

• Geometry

• Algebra

• Linear algebra

• Analysis

• Dynamical systems

• Mathematical physics

• See also

• References

• Classification of Euclidean plane isometries

• Classification of Platonic solids

• Classification theorems of surfaces

• Classification of two-dimensional closed manifolds – Two-dimensional manifoldPages displaying short descriptions of redirect targets

• Enriques–Kodaira classification – Mathematical classification of surfaces of algebraic surfaces (complex dimension two, real dimension four)

• Nielsen–Thurston classification – Characterizes homeomorphisms of a compact orientable surface which characterizes homeomorphisms of a compact surface

• Thurston's eight model geometries, and the geometrization conjecture – Three dimensional analogue of uniformization conjecture

• Berger classification – Concept in differential geometry

• Classification of Riemannian symmetric spaces – (pseudo-)Riemannian manifold whose geodesics are reversible

• Classification of 3-dimensional lens spaces – Class of topological space

• Classification of manifolds – Basic question in geometry and topology

• Classification of finite simple groups – Theorem classifying finite simple groups

• Classification of Abelian groups – Commutative group (mathematics)

• Classification of Finitely generated abelian group – Commutative group where every element is the sum of elements from one finite subset

• Classification of Rank 3 permutation group – Five sporadic simple groupsPages displaying short descriptions of redirect targets

• Classification of 2-transitive permutation groups

• Artin–Wedderburn theorem – Classification of semi-simple rings and algebrasPages displaying short descriptions of redirect targets — a classification theorem for semisimple rings

• Classification of Clifford algebras

• Classification of low-dimensional real Lie algebras

• Classification of Simple Lie algebras and groups

• Classification of simple complex Lie algebras – Direct sum of simple Lie algebras

• Classification of simple real Lie algebras – Term in mathematics

• Classification of centerless simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroups

• Classification of simple Lie groups – Connected non-abelian Lie group lacking nontrivial connected normal subgroupsPages displaying short descriptions of redirect targets

• Bianchi classification – Lie algebra classification

• ADE classification – Mathematical classification

• Langlands classification – Mathematical theory

• Finite-dimensional vector space – Number of vectors in any basis of the vector spacePages displaying short descriptions of redirect targetss (by dimension)

• Rank–nullity theorem – In linear algebra, relation between 3 dimensions (by rank and nullity)

• Structure theorem for finitely generated modules over a principal ideal domain – Statement in abstract algebra

• Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities

• Frobenius normal form – Canonical form of matrices over a field (rational canonical form)

• Sylvester's law of inertia – Theorem of matrix algebra of invariance properties under basis transformations

• Classification of discontinuities – Mathematical analysis of discontinuous points

• Classification of Fatou components

• Ratner classification theorem

• Classification of electromagnetic fields

• Petrov classification – Classification used in differential geometry and general relativity

• Segre classification – Algebraic classification of rank two symmetric tensors

• Wigner's classification – Classification of irreducible representations of the Poincaré group

• Representation theorem – Proof that every structure with certain properties is isomorphic to another structure

• Comparison theorem

• List of manifolds

• List of theorems