Mathematics. Dictionary

A Hausdorff topological space that looks locally like the "ordinary" Euclidean space \mathbb{R}^n.

The group of self-homeomorphisms of a topological space, modulo homotopy.

The projection of a joint probability distribution onto one of its component random variables.

A method of proof which, in terms of a predicate P, could be stated as: if P(0) is true and if for any natural number n \ge 0, P(n) implies P(n + 1), then P(n) is true for any natural number n.

A subfield of logic and mathematics consisting of both the mathematical study of logic and the application of this study to other areas of mathematics, exemplified by questions on the expressive power of formal logics and the deductive power of formal proof systems.

An abstract mathematical representation of a process, device or concept using a number of variables to represent inputs, outputs and internal states, and sets of equations and inequalities to describe their interaction.

A process by which a matrix (a rectangular table of numbers or abstract quantities that can be added and multiplied) is broken down into simpler numerical building blocks.

Any of various theorems that saliently concern mean values.

A branch of mathematical analysis, concerned with the theory of integration, that generalizes the intuitive notions of length, area and volume.

Any set on which a metric is defined, giving a distance between any two elements.

For a given square matrix M over some field K, the smallest-degree monic polynomial over K which, when applied to M, yields the zero matrix.

Any surface that has the least possible surface area of those into which it could transform; typically the shapes of soap bubbles that span wire frames

A relationship between Calabi-Yau manifolds where two such manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

Any formal system that attempts to deal with modalities, such as possibility and necessity, but also obligation and permission.

Any system of arithmetic for integers which, for some given positive integer n, is equivalent to the set of integers being mapped onto the finite set {0, ... n} according to congruence modulo n, and in which addition and multiplication are defined consistently with the results of ordinary arithmetic being so mapped.

A set which is closed under an associative binary operation, and which contains an element which is an identity for the operation.

A function that either never decreases or never increases as its independent variable increases.

(formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.

The number of values for which a given condition holds.