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,^[clarify] solves the classification problem, and is often a step in solving it.
• 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 – Two-dimensional manifoldPages displaying short descriptions of redirect targets
  • 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
• Berger classification – Concept in differential geometry
• Classification of Riemannian symmetric spaces – A (pseudo-)Riemannian manifold whose geodesics are reversible.
• 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 – basic questionPages displaying wikidata descriptions as a fallback

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 – Five sporadic simple groupsPages displaying short descriptions of redirect targets
  • Classification of 2-transitive permutation groups
• Artin–Wedderburn theorem – Classification of semi-simple rings and algebrasPages displaying short descriptions of redirect targets — 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 – Direct sum of simple Lie algebras
  • Classification of simple real Lie algebras – graph encoding the structure of a reductive group over a non-algebraically-closed field, in which vertices are colored black or white according to whether they vanish on a maximal split torus, and the white vertices are acted upon by the Galois groupPages displaying wikidata descriptions as a fallback
  • 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 subgroupsPages displaying short descriptions of redirect targets
• Bianchi classification – Lie algebra classification
• ADE classification
• Langlands classification

Linear algebra

• Finite-dimensional vector space – Number of vectors in any basis of the vector spacePages displaying short descriptions of redirect targetss (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 – Statement in abstract algebra
• 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

• 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

See also

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

References