Title: First explicit reciprocity law for unitary Friedberg—Jacquet periods Abstract: In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many […]

Title: Unipotent Representations of Real Reductive Groups and Mixed Hodge Modules Abstract: Let Pi(G) denote the set of irreducible unitary representations of a semisimple Lie group G. A fundamental problem in representation theory is to describe the structure of this set. In previous joint work with Losev and Matvieievskyi, we have defined a class of […]

Title: Nonabelian Fourier kernels on SL₂ and GL₂ Abstract: I will present my collaborative work with Ngô on explicit formulas for the nonabelian Fourier kernels on SL₂ and GL₂, as conjectured by Braverman and Kazhdan. Additionally, I will discuss its connection to the orbital Hankel transform studied by Ngô.

Title: The local theta correspondence and functoriality Abstract: In a letter to Howe, Langlands conjectured that the local theta correspondence is an instance of Langlands functoriality, e.g., it should preserve L-packets. Unfortunately, this was false. As a remedy, Adams conjectured that instead of L-packets, the local theta correspondence should preserve Arthur packets. Mœglin showed that […]