Eisenstein integerIn mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form where a and b are integers and is a primitive (hence non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. The Eisenstein integers are a countably infinite set.
Morse theoryIn mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Vector bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manifold, or an algebraic variety): to every point of the space we associate (or "attach") a vector space in such a way that these vector spaces fit together to form another space of the same kind as (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over .
Square-free integerIn mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are Every positive integer can be factored in a unique way as where the different from one are square-free integers that are pairwise coprime.
Surgery theoryIn mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery" on a differentiable manifold M of dimension , could be described as removing an imbedded sphere of dimension p from M. Originally developed for differentiable (or, smooth) manifolds, surgery techniques also apply to piecewise linear (PL-) and topological manifolds.
Integer factorizationIn number theory, integer factorization is the decomposition, when possible, of a positive integer into a product of smaller integers. If the factors are further restricted to be prime numbers, the process is called prime factorization, and includes the test whether the given integer is prime (in this case, one has a "product" of a single factor). When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist.
Topological quantum field theoryIn gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory.
Partially ordered setIn mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, transitive and antisymmetric.
Axiom of choiceIn mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every .
EquinumerosityIn mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y. Equinumerous sets are said to have the same cardinality (number of elements). The study of cardinality is often called equinumerosity (equalness-of-number). The terms equipollence (equalness-of-strength) and equipotence (equalness-of-power) are sometimes used instead.
