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.

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

We investigate the $p$-adic monodromy of certain kinds of abelian varieties in $\mathcal{A}_{g}$ and prove a formal density theorem for the locus of deformations with big monodromy. Also, we prove that the small monodromy locus of the deformation space of a supersingular elliptic curve is $p$-adic nowhere dense. The approach is based on a congruence condition of $p$-divisible groups and transform of data between the Rapoport-Zink spaces and deformation spaces.

A well-known theorem of Waldspurger relates central values of automorphic L-functions for GL(2) to automorphic period integrals over non-split tori. His result was reproved by Jacquet via the comparison of relative trace formulae. Guo-Jacquet�s conjecture aims to generalise Waldspurger�s result as well as Jacquet�s approach to higher dimensions. In this talk, we shall first recall the background of Guo-Jacquet trace formulae. Then we shall focus on an infinitesimal variant of these formulae and try to explain several results on the local comparison of most terms. Our infinitesimal study is expected to be relevant to the study of geometric sides of the original Guo-Jacquet trace formulae.

We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of positive definite integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. If time permits, we may discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local-global principle for certain Kleinian sphere packings.

Orbital integrals for the Lie algebra can be analyzed using the Hitchin fibration. In turn the Hitchin fibration can be analyzed via the morphism g^{reg} ----> g//G from the regular elements of the Lie algebra, to the GIT quotient by the adjoint action. In trying to generalize this story by replacing the action of G on g by the action of G on some sufficiently nice variety M, we must replace the GIT quotient with what we call the regular quotient. This talk will look at the reasons for this, and the difference between the GIT and regular quotients in the case of G acting on G by conjugation (when the derived group of G is not simply connected), G acting on the commuting scheme, and G acting on the Vinberg monoid.

Ribet’s method describes a way to construct a certain extension of fields from the existence of a suitable modular form. To do so, we consider the Galois representation of an appropriate cuspform, which gives rise to a cohomology class that cuts out our desired extension. The process of obtaining a cohomology class from such a representation is usually known as Ribet’s lemma. Several generalizations of this lemma have been stated and proved during the last decades, but the vast majority of them makes the assumption that the representation is residually distinguishable, meaning that the characters of its residual decomposition are non-congruent modulo the maximal ideal. However, recent applications of Ribet’s method, such as for the proof of the 2-part of the Brumer-Stark conjecture, have encountered the challenge that the representation we obtain does not satisfy this assumption. In my talk, I describe the limitations of the residually indistinguishable case and conjecture a new general version of Ribet’s lemma in this context, giving a proof in some particular cases.

For a smooth projective variety X over an algebraic number field a conjecture of Bloch and Beilinson predicts that the kernel of the Abel-Jacobi map of X is a torsion group. When X is a curve, this follows by the Mordell-Weil theorem. In higher dimensions however there is hardly any evidence for this conjecture. In this talk I will focus on the case when X is a product of smooth projective curves and construct infinitely many nontrivial examples that satisfy a weaker form of the Bloch-Beilinson conjecture. This relies on a recent joint work with Jonathan Love.

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.