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In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. [1] The subspace is then called a retract of the original space.
He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg [6] on the complexity of knottedness detection. The connect-sum decomposition of 3-manifolds is also implemented in Regina , has exponential run-time and is based on a similar algorithm to the 3-sphere recognition algorithm.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological ...
The higher order directed homotopy theory can be developed through cylinder functor and path functor, all constructions and properties being expressed in the setting of categorical algebra []. This approach emphasizes the combinatorial role of cubical sets in directed algebraic topology.
In mathematics, specifically algebraic topology, the mapping cylinder [1] of a continuous function between topological spaces and is the quotient = (([,])) / where the denotes the disjoint union, and ~ is the equivalence relation generated by
Category theory is the language of modern algebra, and has been widely used in the study of algebraic geometry and topology. It has been noted that "the key observation of [10] is that the persistence diagram produced by [8] depends only on the algebraic structure carried by this diagram."
Computer-assisted research in various areas of mathematics, such as logic (automated theorem proving), discrete mathematics, combinatorics, number theory, and computational algebraic topology; Cryptography and computer security, which involve, in particular, research on primality testing, factorization, elliptic curves, and mathematics of ...
His research was then mainly in the area of manifolds, particularly geometric topology and related abstract algebra included in surgery theory, of which he was one of the founders. In 1964 he introduced the Brauer–Wall group of a field. His 1970 research monograph "Surgery on Compact Manifolds" is a major reference work in geometric topology.