## Discussion on Black Hole Entropy

Aron Wall

December 6, 2017

Aron Wall

December 6, 2017

Marius Lemm

California Institute of Technology; Member, School of Mathematics

December 6, 2017

In spin systems, the existence of a spectral gap has far-reaching consequences. So-called "frustration-free" spin systems form a subclass that is special enough to make the spectral gap problem amenable and, at the same time, broad enough to include physically relevant examples. We discuss "finite-size criteria", which allow to bound the spectral gap of the infinite system by the spectral gap of finite subsystems. We focus on the connection between spectral gaps and boundary conditions. Joint work with E. Mozgunov.

Xiaoliang Qi

December 5, 2017

Ian Mertz

University of Toronto

December 5, 2017

December 5, 2017

Alexander Goncharov

Yale University; Member, School of Mathematics and Natural Sciences

December 5, 2017

According to Langlands, pure motives are related to a certain class of automorphic representations.

Can one see mixed motives in the automorphic set-up? For examples, can one see periods of mixed motives in entirely automorphic terms? The goal of this and the next lecture is to supply some examples.

We define motivic correlators describing the structure of the motivic fundamental group $\pi_1^{\mathcal M}(X)$ of a curve. Their relevance to the questions raised above is explained by the following examples.

Alexei Kitaev

December 5, 2017

Jun Su

Princeton University

December 5, 2017

We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings’ B-G-G spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.

Daniel Jafferis

December 5, 2017

Daniel Holz

The University of Chicago

December 5, 2017