Knot theory in four and two dimensions, and the Kashiwara-Vergne problem

Seminar/Forum

107
Peter Hall
We'll describe a machine, which translates a certain type of problem in knot theory (looking for a "universal finite type invariant") to a problem of solving equations in some infinite dimensional graded space. For "knotted trivalent tubes in $$\mathbb{R}^4$$", the equations output by this machine turn out to be interesting in their own right, studied by Lie theorists for decades: the Kashiwara-Vergne equations. We'll explain how, via this correspondence, topology sheds new light on Lie theory. Surprisingly, a different flavour of knot theory (this time in two dimensions), also leads to the Kashiwara-Vergne equations when fed into the same machine. What do the two topological problems have to do with each other? Nobody knows. (Based on joint work with Dror Bar-Natan, as well as work of Alekseev, Enriquez, Kawazumi, Kuno, Naef and Torossian.)