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.
 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
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.