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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.
In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ.
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. [1] Category theory is used in almost all areas of mathematics.
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 ...
He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg [7] 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 in which tools from abstract algebra are used to study topological spaces The main article for this category is Algebraic topology . Contents
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 ...
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.