### 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 **CP**^{1}, 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.