Eyal Goren (McGill University)
Title: Theta operators for unitary modular forms
Abstract: This is joint work with Ehud De Shalit (Hebrew University). We shall consider p-adic modular forms on a unitary Shimura variety associated to a quadratic imaginary field, where p is inert in the field, and the mod p reduction of this variety. In this case, theta operators were constructed by Eischen and Mantovan, and by De Shalit-Goren, independently and using different approaches. I will describe our approach that makes heavy use of Igusa varieties. The main two theorems are (i) a formula for the effect of a theta operator on q-expansions and (ii) its analytic continuation from the ordinary locus to the whole Shimura variety in characteristic p. Along the way interesting questions about filtrations of automorphic vector bundles arise and, to the extent time allows, I will discuss these questions in light of our work on foliations on unitary Shimura varieties.
Christopher Hacon (University of Utah)
Title: Birational boundedness of algebraic varieties
Abstract: The minimal model program (MMP) is an ambitious program that aims to classify algebraic varieties. According to the MMP, there are 3 building blocks: Fano varieties, Calabi-Yau varieties and varieties of general type. In this talk I will recall the general features of the MMP and discuss recent advances in our understanding of Fano varieties and varieties of general type.
Gonçalo Tabuada (MIT)
Title: A noncommutative approach to the Grothendieck, Voevodsky, and Tate conjectures
Abstract: The Grothendieck standard conjectures, the Voevodsky nilpotence conjecture, and the Tate conjecture, play a key central role in algebraic geometry. Notwithstanding the effort of several generations of mathematicians, the proof of these celebrated conjectures remains elusive. The aim of this talk is to give an overview of a recent noncommutative approach which has led to the proof of the aforementioned important conjectures in several new cases.
Yunqing Tang (Princeton University)
Title: Exceptional splitting of reductions of abelian surfaces with real multiplication
Abstract: Zywina showed that after passing to a suitable field extension, every abelian surface A with real multiplication over some number field has geometrically simple reduction modulo 𝔭 for a density one set of primes 𝔭. One may ask whether its complement, the density zero set of primes 𝔭 such that the reduction of A modulo 𝔭 is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod 𝔭 isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.