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 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.
Geometry
- Classification of Euclidean plane isometries
- Classification theorems of surfaces
- Classification of two-dimensional closed manifolds – Two-dimensional manifold
- 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 question
Algebra
- Classification of finite simple groups – Massive theorem assigning all but 27 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 groups
- Classification of 2-transitive permutation groups
- Artin–Wedderburn theorem – Classification of semi-simple rings and algebras — 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 group
- 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 space – Number of vectors in any basis of the 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 – result 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
Complex analysis
- Classification of Fatou components
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
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