### Saturday, April 24, 2021 (online)

### Jarod Alper (Washington)

**Title:** Iwahori decompositions and S-completeness

**Abstract: **We present a short and self-contained proof of Iwahori decompositions for reductive groups. The existence of Iwahori decompositions is the key algebraic input in the proof of the Hilbert–Mumford criterion in geometric invariant theory. We then relate the existence of Iwahori decompositions for a group G to a property of the classifying stack BG, which we refer to as S-completeness. We will discuss some of the spectacular properties of S-completeness and highlight its applications to moduli theory. This is joint work with Daniel Halpern-Leistner and Jochen Heinloth.

### Chandrashekhar Khare (UCLA)

**Title:** An analog of Serre’s conjecture for reducible mod p Galois representations.

**Abstract: **Serre’s conjecture asserts that 2-dimensional irreducible odd representations G_Q —> GL_2(k), with G_Q the absolute Galois group the rationals Q and k a finite field, arise from modular forms. As has been remarked by Wiles one can ask for an analog for reducible odd representations. In joint work with Fakhruddin and Patrikis we prove (almost all of) this analog, building on earlier work of S. Hamblen and R. Ramakrishna. I would like to explain the relevance of lifting mod p representations to geometric characteristic zero representations to the proofs of Serre’s conjecture (in earlier joint work with J-P. Wintenberger), and its analog for reducible representations.

### Bao Le Hung (Northwestern)

**Title:** TBA

**Abstract: **TBA

### Emily Norton (Clermont Auvergne)

**Title:** TBA

**Abstract: **TBA

### Lue Pan (Chicago)

**Title:** The Hecke action on overconvergent modular forms.

**Abstract: **Serre introduced the notion of *p*-adic modular forms in the study of congruences between modular forms. It is well-known that to get a better spectral theory of the Hecke operator at $p$ (U_p-operator), one should consider the subspace of overconvergent modular forms. In this talk, we will show that Hecke operators away from $p$ also have better convergence acting on overconvergent modular forms. As a direct consequence, we will present a new way to determine the Hodge-Tate-Sen weights of the Galois representation attached to an overconvergent eigenform, which previously was obtained by myself and Sean Howe independently by totally different methods.