### Charlotte Chan (Princeton University)

**Title:** Affine Deligne-Lusztig varieties at infinite level for GL_{n}

**Abstract: **Affine Deligne-Lusztig varieties have been of interest for some time because of their relation to Shimura varieties and the Langlands program. In this talk, we will construct a tower of affine Deligne Lusztig varieties for GL_{n} and its inner forms. We prove that its limit at infinite level is isomorphic to the semi-infinite Deligne-Lusztig variety of Lusztig and that its cohomology realizes certain cases of automorphic induction and Jacquet-Langlands. This is joint work with A. Ivanov.

### Tsao-Hsien Chen (University of Chicago)

**Title:** Kostant-Sekiguchi homeomorphisms

**Abstract: **The Kostant-Sekiguchi correspondence is a remarkable bijection between the real and symmetric nilpotent orbits. It is one of the fundamental theorems in Lie theory and plays an important role in representations of real groups. In my talk, I will explain a new approach to the Kostant-Sekiguchi correspondence which provides a lift of the correspondence to a stratified homeomorphism between the real and symmetric nilpotent cones. This new approach is inspired by the geometric Langlands theory. If time permits, I will discuss applications to Springer theory. Joint works with David Nadler.

### Sándor Kovács (University of Washington)

**Title:** Liftable local cohomology and deformations

**Abstract:** This is a report on joint work with János Kollár. We introduce a lifting property for local cohomology, which implies that the cohomology sheaves of the relative dualizing complex of a flat morphism are flat and commute with base change. We also establish that several well-known classes of singularities have this property and derive various consequences for moduli spaces and moduli functors. In particular, this implies, among other consequences, that a stable non-Cohen-Macaulay singularity, e.g., a cone over an abelian surface, is never smoothable. All of these results work in arbitrary characteristic.

### Baiying Liu (Purdue University)

**Title:** On Langlands functoriality and converse theorems

**Abstract: **Converse theorems have been playing important roles in the theory of Langlands functoriality. In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GL_{n}(*F*), where *F* is a non-archimedean local field. This is a joint work with Hervé Jacquet. Then I will briefly introduce our recent progress on establishing Langlands functorial descent (joint with Joseph Hundley) and the study of converse theorems (joint with Qing Zhang) for the split exceptional group of type G_{2}.

### Romyar Sharifi (UCLA)

**Title:** Iwasawa theory in higher codimension

**Abstract:** Classically speaking, Iwasawa theory concerns the growth of *p*-parts of class groups in towers of number fields of *p*-power degree. One considers the inverse limit of such groups as a finitely generated, torsion module over the Iwasawa algebra, which is a completed group ring for the Galois group of the tower. Often, this growth can be slow enough that the support of this unramified module lies in codimension two and higher, while invariants like characteristic ideals which one hopes might be described through *L*-values don’t measure anything beyond codimension one. The typical way around this is to allow ramification at enough primes over *p* to make the module larger. Yet, the original modules are certainly of arithmetic interest. I’ll discuss how, over CM fields, these unramified Iwasawa modules and others satisfy nice “reflection principles” and how *p*-adic *L*-functions tell us something about them. This is joint work with Bleher, Chinburg, Greenberg, Kakde, and M. Taylor, and in part with Pappas as well.