11 a.m. - 11:50 a.m. Location: FN 2.102
Józef H. Przytycki
George Washington University
Adventures of a knot theorist in the word of homological algebra
We start this talk by giving a few historical remarks concerning the area of knot theory. In particular, we mention pioneering work in this area done by Leibniz, Vandermonde, Gauss and his student Listing, as well as British/Scottish physicists Maxwell, Kelvin and Tait. We finish the less formal part of the talk by discussing Reidemeister moves (1926) and Fox 3-coloring. We show that Fox 3-coloring can naturally be generalized to Yang-Baxter weighted colorings and Yang-Baxter operators. This, in turn, can be used to define the vast majority of the quantum invariants of links (including the Jones and Hompflypt polynomials).
Even thought I took a class in homological algebra at the Warsaw University as an undergraduate student, in my study of knots, I was rather using standard algebraic topology. My adventures with homological algebra started in early May of 2005, after I realized that Hochschild homology seems to generalize the Khovanov homology for links.
Since we start from the basics, i.e. Fox 3-coloring, Reidemeister moves, and Jones polynomial on one hand, and chain complexes, (pre)simplicial, and (pre)cubic sets and chain homotopy from the homological algebra side (no advanced knowledge of neither knot theory nor homological algebra is assumed), my talk will be addressed to mathematics students. In particular, I plan to formulate many open problems, which could become interesting as research projects for participants.
Coffee to be served in FN 2.102 at 10:30 AM.
Sponsored by the Department of Mathematical Sciences