Encyclopedia
Classification theorem
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 computable complete set of invariants (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.
01Geometry
- Classification of Euclidean plane isometries , Isometry of the Euclidean plane
- 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 , (pseudo-)Riemannian manifold whose geodesics are reversible
- Classification of 3-dimensional lens spaces , Class of topological space
- Classification of manifolds , Basic question in geometry and topology
02Algebra
- Classification of finite simple groups , Theorem classifying finite simple groups
- 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 in abstract algebra
- 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 , Term in mathematics
- 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 , Mathematical classification
- Langlands classification , Mathematical theory
03Linear 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
04Analysis
- Classification of discontinuities , Mathematical analysis of discontinuous points
05Dynamical systems
- Classification of Fatou components , Components of the Fatou set
- Ratner classification theorem
06Mathematical 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
Sources and credits
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