In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 related issues to classification are the following.

• The equivalence problem is "given two objects, determine if they are equivalent".
• A complete set of invariants
  Complete set of invariants
  In mathematics, a complete set of invariants for a classification problem is a collection of mapsf_i : X \to Y_i \,, such that x ∼ x' if and only if f_i = f_i for all i...
  
  , together with which invariants are realizable, solves the classification problem, and is often a step in solving it.
• A computable complete set of invariants
  Complete set of invariants
  In mathematics, a complete set of invariants for a classification problem is a collection of mapsf_i : X \to Y_i \,, such that x ∼ x' if and only if f_i = f_i for all i...
  
   (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
• A canonical form
  Canonical form
  Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
  
   solves the classification problem, and is more data: it not only classifies every class, but gives a distinguished (canonical) element of each class.

There exist many classification theorems in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, as described below.

Geometry

• Classification of Euclidean plane isometries
• Classification theorem of surfaces
  • Classification of two-dimensional closed manifolds
  • Enriques-Kodaira classification
    Enriques-Kodaira classification
    In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space...
    
     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
  Geometrization conjecture
  Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed canonically into submanifolds that have geometric structures. The geometrization conjecture is an analogue for 3-manifolds of the uniformization theorem for surfaces...
  
  

Algebra

• Classification of finite simple groups
  Classification of finite simple groups
  In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
  
  
• Artin–Wedderburn theorem
  Artin–Wedderburn theorem
  In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
  
   — a classification theorem for semisimple rings

Linear algebra

• Finite-dimensional vector spaces (by dimension)
• rank-nullity theorem
  Rank-nullity theorem
  In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix over some field, thenThis applies to linear maps as well...
  
   (by rank and nullity)
• Structure theorem for finitely generated modules over a principal ideal domain
  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 finitely generated modules can be uniquely decomposed in...
  
  
• Jordan normal form
  Jordan normal form
  In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...
  
  
• Sylvester's law of inertia
  Sylvester's law of inertia
  Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...