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 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[clarification needed] (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

Classification of Euclidean plane isometries – Isometry of the Eluclidean plane
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

Algebra

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 in abstract algebra
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

Linear algebra

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

Analysis

Classification of discontinuities – Mathematical analysis of discontinuous points

Dynamical systems

Classification of Fatou components – Components of the Fatou set
Ratner classification theorem

Mathematical physics

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