Time: 3:30 PM - 4:30 PM

Speaker: Ananth Shankar (IAS and Northwestern University)

Title: $p$-adic hyperbolicity for Shimura varieties and period images

Abstract: Borel proved that every holomorphic map from a product of punctured unit discs to a complex Shimura variety extends to a map from a product of discs to its Bailey-Borel compactification. In joint work with Oswal, Zhu, and Patel, we proved a p-adic version of this theorem over discretely valued fields for Shimura varieties of abelian type. I will speak about work with Bakker, Oswal, and Yao, where we prove the analogous $p$-adic extension theorem for compact non-abelian Shimura varieties and geometric period images for large primes $p$.

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Time: 5:00 PM - 6:00 PM

Speaker: Tony Feng (UC Berkeley)

Title: Mirror symmetry and the Breuil—Mezard Conjecture: an update

Abstract: The Breuil—Mezard Conjecture predicts a precise indexing of cycles in moduli spaces of local Galois representations by modular representations of finite groups of Lie type. A couple years ago, Bao Le Hung and I introduced a new approach to the Breuil—Mezard Conjecture based on a connection to an instance of mirror symmetry, which in that instance predicts a precise indexing of Lagrangians in a symplectic variety by representations of a quantum group. Recently, we used this to prove the Breuil—Mezard Conjecture in the generic range for arbitrary unramified groups, including exceptional groups. My intent is to review this and also work-in-progress with Le Hung and Zhongyipan Lin, which aims to extend the result to ramified groups. The key new aspect of the ramified case is a nascent theory of "Spectral Langlands functoriality", an analogue of Langlands functoriality for the spectral (i.e., "Galois") side of the Langlands correspondence.