2019 Bott Lectures

On April 9 and 10, 2019 the CMSA hosted two lectures by Mina Aganagic (UC Berkeley). This was the second annual Math Science Lecture Series held in honor of Raoul Bott.

“Two math lessons from string theory”

Lecture 1:  “Lesson on Integrability”

Video

Abstract: The quantum Knizhnik-Zamolodchikov (qKZ) equation is a difference generalization of the famous Knizhnik-Zamolodchikov (KZ) equation. The problem to explicitly capture the monodromy of the qKZ equation has been open for over 25 years. I will describe the solution to this problem, discovered jointly with Andrei Okounkov. The solution comes from the geometry of Nakajima quiver varieties and has a string theory origin.

Part of the interest in the qKZ monodromy problem is that its solution leads to integrable lattice models, in parallel to how monodromy matrices of the KZ equation lead to knot invariants. Thus, our solution of the problem leads to a new, geometric approach, to integrable lattice models. There are two other approaches to integrable lattice models, due to Nekrasov and Shatashvili and to Costello, Witten and Yamazaki. I’ll describe joint work with Nikita Nekrasov which explains how string theory unifies the three approaches to integrable lattice models.

Lecture 2: “Lesson on Knot Categorification”

Video

Abstract: An old problem is to find a unified approach to the knot categorification problem. The new string theory perspective on the qKZ equation I described in the first talk can be used to derive two geometric approaches to the problem.The first approach is based on a category of B-type branes on resolutions of slices in affine Grassmannians. The second is based on a category of A-branes in a Landau-Ginzburg theory. The relation between them is two dimensional (equivariant) mirror symmetry. String theory also predicts that a third approach to categorification, based on counting solutions to five dimensional Haydys-Witten equations, is equivalent to the first two.This talk is mostly based on joint work with Andrei Okounkov.


Information about last year’s Math Science Bott lecture can be found here. 

Asymptotically flat extensions with charge

New publication by Aghil Alaee et al.

Title: Asymptotically flat extensions with charge

Abstract: The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

 

ArXiv Link: 1903.09014

Kummer Rigidity for K3 Surface Automorphism vis Ricci-Flat Metrics

New Publication by Valentino Tosatti:

Title: Kummer Rigidity for K3 Surface Automorphism vis Ricci-Flat Metrics
Abstract: We give an alternative proof of a result of Cantat & Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.
ArXiv: 1808.08673