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 solves the classification problem, and is often a step in solving it. *A (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 * Classification theorems of surfaces ** Classification of two-dimensional closed manifolds ** Enriques–Kodaira classification of algebraic surfaces (complex dimension two, real dimension four) ** Nielsen–Thurston classification which characterizes homeomorphisms of a compact surface * Thurston's eight model geometries, and the geometrization conjecture * Berger classification * Classification of Riemannian symmetric spaces * Classification of 3-dimensional lens spaces * Classification of manifolds

Algebra

Classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or els ...

** Classification of Abelian groups ** Classification of Finitely generated abelian group ** Classification of Rank 3 permutation group ** Classification of 2-transitive permutation groups * Artin–Wedderburn theorem — a classification theorem for semisimple rings *

Classification of Clifford algebras In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In ...

Classification of low-dimensional real Lie algebras This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstract algebra. An English version and review of th ...

* Bianchi classification * ADE classification * Langlands classification

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 In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to ...

* Sylvester's law of inertia

Analysis

* Classification of discontinuities

Complex analysis

* Classification of Fatou components

Mathematical physics

* Classification of electromagnetic fields *

Petrov classification In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is ...

Segre classification The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy–momentum tensor (or the Ricci tensor) and primarily finds applicati ...