Thursday, March 1, 2018
03/01/2018 - 4:00pm to 5:00pm
Two fundamental dividing lines between "tame" and "wild" structures in model theory, combinatorics, and algebra are the order property and the independence property. In structures (or, more generally, set systems) without the order property, we can associate a realnumber called VC density with each definable set which can be interpreted as a measure of complexity. Thicket density is a new quantity we can assign to each definable set which relates to the order property in exactlythe same way VC density relates to the independence property, complete with a new notion of shatter function which satisfies a Sauer-Shelah typebound. In this talk, we will introduce and state the basic properties of thicket density, describe some recent work connecting it to extant VC-theory, and sketch applications to combinatorics and logic.
03/01/2018 - 4:30pm to 5:30pm
We will talk about a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is naturally obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As anapplication, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve. This is based on my joint work with Gufang Zhao.