Pierre Colmez (CNRS, Université Pierre et Marie Curie)
Title: p-adic étale cohomology of the Drinfeld tower and p-adic local Langlands correspondence
Abstract: It is now classical that the l-adic étale cohomology of the Drinfeld tower, for l not p, encodes both the local Langlands and Jacquet-Langlands correspondences. I will explain that, in dimension 1, the p-adic étale cohomology of this tower encodes part of the p-adic local Langlands correspondence (this is joint work with Gabriel Dospinescu and Wieslawa Niziol).
Davesh Maulik (MIT)
Title: Gopakumar-Vafa invariants via vanishing cycles
Abstract: Given a Calabi-Yau threefold X, one can count curves on X using various approaches, for example using stable maps or ideal sheaves; for any curve class on X, this produces an infinite sequence of invariants, indexed by extra discrete data (e.g. by the domain genus of a stable map). Conjecturally, however, this sequence is determined by only a finite number of integer invariants, known as Gopakumar-Vafa invariants.
In this talk, I will propose a direct definition of these invariants via sheaves of vanishing cycles, building on earlier approaches of Kiem-Li and Hosono-Saito-Takahashi. Conjecturally, these should agree with the invariants as defined by stable maps. I will also explain how to prove the conjectural correspondence in various cases. This is joint work with Yukinobu Toda.
Wieslawa Niziol (CNRS, ENS-Lyon/IAS)
Title: Cohomology of p-adic Stein spaces
Abstract: I will discuss a comparison theorem that allows us to recover p-adic (pro-)étale cohomology of p-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. Present applications include a computation of the p-adic étale cohomology of the Drinfeld half-space in any dimension and of its coverings in dimension 1. This is a joint work with Pierre Colmez and Gabriel Dospinescu.
Nicolas Templier (Cornell University)
Title: Mirror symmetry for minuscule flag varieties
Abstract: We prove cases of Rietsch mirror conjecture that the quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. In particular we link the purity of Berenstein-Kazhdan crystals to the Ramanujan property of certain Hecke eigensheaves. The talk will keep the prerequisite knowledge to a minimum by introducing the above concepts of “mirror” and “crystal” with the examples of CP1, projective spaces and Grassmannians. Work with Thomas Lam.
Yihang Zhu (Columbia University)
Title: Arithmetic fundamental lemma in the minuscule case
Abstract: The arithmetic Gan-Gross-Prasad conjecture generalizes the Gross-Zagier formula to Shimura varieties associated to unitary or orthogonal groups. The arithmetic fundamental lemma (AFL), formulated by Wei Zhang in the unitary case, is a key local ingredient in the relative trace formula approach towards arithmetic GGP. The AFL compares arithmetic intersection numbers on Rapoport-Zink spaces with derivatives of orbital integrals. We prove an explicit formula for the arithmetic intersection numbers in both unitary and orthogonal cases, under a minuscule assumption. In particular, our work gives a new proof of the theorem of Rapoport-Terstiege-Zhang on the AFL in the unitary case. This is joint work with Chao Li.