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The low-level language of topology, which is not really considered a separate "branch" of topology. Point-set topology, also called set-theoretic topology or general ...

A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (i.e., the multiples of p) is prime whenever p ...

The mathematical study of combinatorial objects in which a certain degree of order must occur as the scale of the object becomes large. Ramsey theory is named after Frank ...

The space groups in two dimensions are called wallpaper groups. In three dimensions, the space groups are the symmetry groups possible in a crystal lattice with the ...

For every even dimension 2n, the symplectic group Sp(2n) is the group of 2n×2n matrices which preserve a nondegenerate antisymmetric bilinear form omega, i.e., a symplectic ...

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation, and has the form ...

A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R^n, where every element ...

The tensor product of two vector spaces V and W, denoted V tensor W and also called the tensor direct product, is a way of creating a new vector space analogous to ...

Let (K,L) be a pair consisting of finite, connected CW-complexes where L is a subcomplex of K. Define the associated chain complex C(K,L) group-wise for each p by setting ...

A zonotope is a set of points in d-dimensional space constructed from vectors v_i by taking the sum of a_iv_i, where each a_i is a scalar between 0 and 1. Different choices ...