Benjamin Antieau (UIC)
Title: On the Brauer group of the moduli stack of elliptic curves
Abstract: Mumford proved that the Picard group of the moduli stack of elliptic curves is a finite group of order 12, generated by the Hodge bundle of the universal family of elliptic curves. After giving background on Brauer groups and on the moduli of elliptic curves, I will talk about recent work with Lennart Meier, which computes the Brauer group of the moduli stack over various arithmetic base schemes and shows in particular that the Brauer group of the integral moduli stack vanishes. This talk will focus on the concrete computational and arithmetic aspects of the proof.
Joel Kamnitzer (University of Toronto)
Title: Monodromy of Bethe vectors and crystals
Abstract: Let g be a semisimple Lie algebra. A long-standing problem is to decompose of tensor products of representations of g. One approach to this problem is the Gaudin system, which provides a family of commuting operators acting on tensor product multiplicity spaces — the eigenvectors for this Gaudin system are called Bethe vectors. The Gaudin system depends on a parameter which lives in the moduli space of genus 0 curves and we can study how the Bethe vectors change as we vary this parameter. This provides the action of the cactus group, a certain finitely-generated group analogous to the braid group. We prove that this action of this cactus group can be obtained combinatorially, through the theory of crystals.
Kai-Wen Lan (University of Minnesota)
Title: Nearby cycles of automorphic étale sheaves
Abstract: I will explain that, in many cases where integral models are available in the literature, the automorphic étale cohomology of a (possibly noncompact) Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles in positive characteristics. If time permits, I will also talk about some applications or related results. (This is joint work with Stroh.)
Christian Schnell (Stony Brook University)
Title: Pluricanonical bundles and maps to abelian varieties
Abstract: Suppose we have a morphism from a smooth projective variety X to an abelian variety A (over the complex numbers). If we push forward the canonical bundle of X, we get a coherent sheaf on A with many special properties, all coming from Hodge theory. In the talk, I am going to explain what happens for pluricanonical bundles (= powers of the canonical bundle), based on joint work with Luigi Lombardi and Mihnea Popa.