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^[clarify] (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 of Platonic solids
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 – Three dimensional analogue of uniformization conjecture
Berger classification
Classification of Riemannian symmetric spaces
Classification of 3-dimensional lens spaces – 3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)
Classification of manifolds

Algebra

Classification of finite simple groups – Massive theorem assigning all but 26 finite simple groups to a few infinite families
- 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
- Classification of 2-transitive permutation groups
Artin–Wedderburn theorem — 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
- Classification of simple real Lie algebras
- 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 subgroups
Bianchi classification – Lie algebra classification
ADE classification
Langlands classification

Linear algebra

Finite-dimensional vector spaces (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
Jordan normal form – Form of a matrix indicating its eigenvalues and their algebraic multiplicities
Frobenius normal form (rational canonical form)
Sylvester's law of inertia

Analysis

Classification of discontinuities – Mathematical analysis of discontinuous points

Dynamical systems

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

References