Knot theory in four and two dimensions, and the KashiwaraVergne problem
Seminar/Forum
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 KashiwaraVergne 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 KashiwaraVergne 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 BarNatan, as well as work of Alekseev, Enriquez, Kawazumi, Kuno, Naef and Torossian.)
Presenter

Dr Zsuzsanna Dancso, University of Sydney