报告题目：Knot Theorist adventures in the world of homological algebra
报告人：Prof. Jozef H. Przytycki (George Washington University)
2017年6月26日下午3:00-4:00， 后主楼 1220
2017年6月28日下午3:00-4:00， 后主楼 1129
2017年6月30日下午3:00-4:00， 后主楼 1220
报告摘要：We start from the historical introduction to knot theory, starting from Leibniz and Vandermonde, saying few words on Gauss and his student Listing and the British/Scottish physicists: Maxwell, Kelvin and Tait. I will complete the introduction with Reidemeister moves (1926) and Fox 3-coloring, introduced by Ralph Hartzler Fox (1913 -1973) around 1956 when he was explaining knot theory to undergraduate students at Haverford College.
Further, I will show how Fox coloring can be naturally generalized to Yang-Baxter weighted colorings and Yang-Baxter operators. This in turn can be used to define most of the quantum invariants of links, including the Jones and Hompflypt polynomials.
Then I will define Khovanov homology following O.Viro very elementary approach, using the Kauffman bracket polynomial and show the long exact sequence of homology related to the skein relation of the Kauffman bracket polynomial. I will observe that Khovanov homology can be obtained as a homology of a small category with coefficient in the functor -- the Khovanov functor to k-modules.
Next part will be devoted to distributive structures coming from knot theory (racks and quandles), and their homology theory (parallel to group homology). I will show how to generalize this homology to Yang-Baxter operators and speculate how to connect Y-B homology with Khovanov type homology.
The path I plan to take is as follows: Khovanov homology is the categorification of the Jones polynomial. Jones polynomial can be obtained using the specific Yang-Baxter operator. Yang-Baxter operator generalizes distributivity.
I do not assume a deep knowledge neither of knot theory nor homological algebra. We start from the basis, from 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. We should see, at my lectures, many open problems, which may become research problems for participants.