Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. Ben-Zvi, Sakellaridis and Venkatesh conjecture that the local unramified part of this theory can be categorified to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. In this talk I will explain the proof of this equivalence and some of its properties. I will also discuss connections to conjectures of Braverman-Kazhdan on constructions of general automorphic L-functions. This is joint work with Tsao-Hsien Chen (in preparation).

Let G be the group of rational points of a linear algebraic group over a local field. A representation of G is distinguished by a subgroup H if it admits a non-zero H-invariant linear form. A Galois symmetric pair (G,H) is such that H=Y(F) and G=Y(E) where E/F is a quadratic extension of local fields and Y is a reductive group defined over F. In this talk we show that for a Galois symmetric pair, often the necessary condition for H-distinction of a parabolically induced representation, emerging from the geometric lemma of Berenstein-Zelevinsky, are also sufficient. In particular, we obtain a characterization of H-distinguished representations induced from cuspidal in terms of distinction of the inducing data. We explicate these results further when Y is a classical group and point out some global applications for Galois distinguished automorphic representations of SO(2n+1). This is joint work with Nadir Matringe.

So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture on the number of rational points of bounded anticanonical height on Fano varieties. Work of B. Birch back in 1962 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In this talk we are interested in subvarieties of biprojective space. There is not much known so far, unless the underlying polynomials are of bidegree (1,1) or (1,2). In this talk we present recent work which combines the circle method with the generalised hyperbola method developed by V. Blomer and J. Bruedern. This allows us to verify Manin's conjecture for certain smooth hypersurfaces in biprojective space of general bidegree.

This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will extend ideas of Julg-Valette to show how this construction can be used to study K-amenability and K-homology of groups acting on CAT(0)-cube complexes. This talk is based on joint work with Erik Guentner and Nigel Higson.

In the previous talk, we discussed the algebraic de Rham theory for unipotent fundamental groups of elliptic curves. In this talk, we generalize it to a Q-de Rham theory for the relative completion of the modular group, the (orbifold) fundamental group of the modular curve. Using Chen's method of power series connections, we construct a connection on the modular curve that generalizes the elliptic KZB connection on an elliptic curve. By Tannaka duality, it can be viewed as a universal relative unipotent connection with a regular singularity at the cusp. This connection enables us to construct iterated integrals of modular forms, possibly 'of the second kind', that provide periods called 'multiple modular values' by Brown. These periods include multiple zeta values and periods of modular forms.

The classification of discrete series is one main subject in Langlands program with numerous applications. We first explain the result on the classification of discrete series of odd GSpin groups, generalizing the Mœglin-Tadi ́c classification for classical groups. Note that our approach will give alternate proof for classical groups. This is a joint work with Ivan Mati ́c. We also explain its application on the generic local Langlands correspondence via Langlands-Shahidi method. If time permits, we will explain possible generalization of those to other groups, which is work in progress

Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.