## Sam Mundy : Vanishing of Selmer groups for Siegel modular forms

- Number Theory ( 0 Views )Let ? be a cuspidal automorphic representation of Sp_2n over Q which is holomorphic discrete series at infinity, and ? a Dirichlet character. Then one can attach to ? an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that ? is ordinary at p, and that p does not divide the conductor of ?. I will describe work in progress which aims to prove that the Bloch--Kato Selmer group attached to the twist of ρ by ? vanishes, under some mild ramification assumptions on ?; this is what is predicted by the Bloch--Kato conjectures. The proof uses "ramified Eisenstein congruences" by constructing p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of nonclassical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of ρ by ?.

## Joshua Greene : Symplectic geometry and inscription problems

- Geometry and Topology ( 0 Views )The Square Peg Problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the plane contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb.

## Benjamin McKenna : Injective norm of real and complex random tensors

- Probability ( 0 Views )The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states. The proof is based on spin-glass methods, the Kac??Rice formula, and recent progress coming from random matrices. Joint work with Stéphane Dartois.

## Spencer Leslie : Relative Langlands and endoscopy

- Number Theory ( 0 Views )Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre-)stabilization of relative trace formulas.