mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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

, 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 In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. Main themes Overview * Low-dimensional manifolds are classified by geometric struct ...

Algebra

* Classification of finite simple groups ** 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 In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitel ...

* Jordan normal form * Sylvester's law of inertia

Analysis

Classification of discontinuities Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of ...

Complex analysis

* Classification of Fatou components

Mathematical physics

Classification of electromagnetic fields In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. It is used in the study of solutions of Maxwell's equations and has ap ...

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 mos ...

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 ...