Stavans, Application of

algebraic topology to homologous recombination of DNA, iScience, 4, (2018), 64-67

Thomas presents her lecture notes for a graduate course for students with a background in basic group theory including group actions, a first course in

algebraic topology, and some familiarity with Riemannian geometry, particularly the geometry of the hyperbolic plane.

This is a tool developed from

algebraic topology, which summarizes the whole multiscale representation compactly in the form of a persistence diagram.

[2] Hatcher, A.:

Algebraic Topology, Cambridge University Press, Cambridge, 2002.

Jean-Pierre Serre's mathematical contributions, leading to a Fields Medal in 1954, were largely in the field of

algebraic topology, but his later work ranged widely--within algebraic geometry, group theory, and especially number theory.

The Hodge conjecture is a major unsolved problem in the field of algebraic geometry that relates the

algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety.

A few years later, Poincare stated the Poincare duality and provided a preliminary proof (2) in his 1895 seminal paper Analysis situs, which marked the formal establishment of

algebraic topology (Ferreiros, 2010b).

User's Guide to

Algebraic Topology. Dordrecht, Netherlands: Kluwer.

He assumes students are familiar with homological algebra,

algebraic topology based on different forms, and de Rham cohomology.

Proved by the work of French mathematician Jean-Pierre Serre (who has made fundamental contributions to

algebraic topology, algebraic geometry, and algebraic number theory) and American mathematician John Torrence Tate, Jr.

Within the discipline of mathematics, Gray argues that Poincare's chief contribution was the formulation of

algebraic topology. Also significant were his studies of algebraic geometry, the theory of transformation groups, and number theory.