Department of

Mathematics


Seminar Calendar
for events the day of Wednesday, September 18, 2019.

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Wednesday, September 18, 2019

3:00 pm in 241 Altgeld Hall,Wednesday, September 18, 2019

Polish groups whose measure preserving actions are whirly

Pavlos Motakis (UIUC Math)

Abstract: Let $\mathrm{MALG}(X)$ denote the measure algebra of a standard probability space $(X,\mu)$. A measure preserving action of a Polish group $G$ on $\mathrm{MALG}(X)$ is called whirly if for any $A, B$ in $\mathrm{MALG}(X)$ with positive measure and for any open neighborhood $U$ of the identity of $G$ there exists $g\in U$ so that $(gA)\cap B$ has positive measure. We follow a paper of Glasner–Tsirelson–Weiss to show that if $G$ is certain type of Polish group, namely a Lévy group, then any non-trivial Borel action on $\mathrm{MALG}(X)$ is whirly. We also show that the Polish group $L_0(\mathbb{T})$ of all measurable functions $[0,1] \to \mathbb{T}$ is Lévy using a suitable concentration inequality.
This is a follow up to the lecture of D. Ihli on 09/11/2019.

4:00 pm in 447 Altgeld Hall,Wednesday, September 18, 2019

Intro to the Gorsky-Negut wall-crossing conjecture

Josh Wen (Illinois Math)

Abstract: The Hilbert scheme of points on the plane is a space that by now has been connected to many areas outside of algebraic geometry: e.g. algebraic combinatorics, representation theory, knot theory, etc. The equivariant K-theory of these spaces have a few distinguished bases important to making some of these connections. A new entrant to this list of bases is the Maulik-Okounkov K-theoretic stable bases. They depend in a piece-wise constant manner by a real number called the slope, and the numbers where the bases differ are called the walls. Gorsky and Negut have a conjecture relating the transition between bases when the slope crosses a wall to the combinatorics of q-Fock spaces for quantum affine algebras. I'll try to introduce as many of the characters of this story as I can as well as discuss a larger picture wherein these stable bases are geometric shadows of things coming from deformation quantization.