
Conference: Informal Talks on the Topology, Combinatorics, and Low and High Algebra of wKnots
Informal Talks on the Topology, Combinatorics, and Low and High Algebra of wKnots
Dror BarNatan's talk Date: 29.10.13 Time: 15.00  17.00 Room: Y03G95
Abstract: Taylor's theorem maps smooth functions to power series. In other words, it maps the smooth to the combinatorial and algebraic, which is susceptible to an inductive degreebydegree study. Surprisingly, there is a notion of "expansions" for topological things, which shares the spirit of the original Taylor expansion while having nothing to do with approximations of smooth functions.
"wKnots", or more generally "wknotted objects", are knotted 2dimensional objects in 4dimensional space (some restrictions apply). They have a rich theory of "expansions" which takes topology into combinatorics. That combinatorics, in itself, turns out to be the combinatorics of formulas that can be written universally in arbitrary finitedimensional Lie algebras ("low algebra"). Taylor's theorem for a certain class of wknotted objects turns out to be equivalent to some global statements about Lie algebras and Lie groups ("KashiwaraVergne", "high algebra"). I will do my best to talk about all these things.
"wKnotted objects" contain the usual "uknotted objects" (braids, knots, links, tangles, knotted graphs, etc.) and are quotients of the more general "vknotted objects". To within reason I will also speak about the relationship of "w" with "u" and "v", where the key words are "associators" and "Lie bialgebras", respectively.
Anna asked me to talk for up to 6 hours, and that's more than I can prepare in detail in advance. Hence the adjective "informal": I have a general idea of what I want to say and much of it I've said many times before. Beyond that things will flow, if they won't stand still, chaotically and randomly.
For more infos consult
http://www.math.toronto.edu/~drorbn/Talks/Zurich1310/

