Classification theorem

Short description: Describes the objects of a given type, up to some equivalence

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 (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 Platonic solids
• Classification theorems of surfaces
  • 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
• Classification of manifolds – Basic question in geometry and topology

Algebra

• Classification of finite simple groups – Massive theorem assigning all but 27 finite simple groups to a few infinite families
• 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 real Lie algebras – Term in mathematics
  • Classification of simple Lie groups – Wikipedia list article
• Astronomy:Bianchi classification – Lie algebra classification
• ADE classification – Mathematical classification
• Langlands classification

Linear algebra

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

Mathematical physics

• Physics:Classification of electromagnetic fields
• Petrov classification – Classification used in differential geometry and general relativity
• Segre classification – Algebraic classification of rank two symmetric tensors
• Physics:Wigner's classification – Classification of irreducible representations of the Poincaré group

See also

• Representation theorem – Proof that every structure with certain properties is isomorphic to another structure
• Comparison theorem
• List of manifolds
• List of theorems