TBD

The cohomology of arithmetic groups is built from certain automorphic forms, allowing for explicit computation of Hecke eigenvalues using topological techniques in some cases. For modular forms attached to the general linear group over a number field F of class number one, these cohomological forms can be described in terms an associated Voronoi polyhedron coming from the study of perfect n-ary forms over F. In this talk, we describe this relationship and report on some recent computational investigations of the modularity of elliptic curves over the cubic field of discriminant -23. This is joint work with Donnelly, Gunnells, and Klages-Mundt.

POSTPONED

In this talk I will describe my recent work with Mahesh Kakde on the Brumer-Stark Conjecture and certain refinements. I will give a broad overview that motivates the conjecture and gives connections to explicit class field theory. I will conclude with a description of recent work (joint w/ Kakde, Jesse Silliman, and Jiuya Wang) in which we complete the proof of the conjecture. Moreover, we deduce a certain special case of the Equivariant Tamagawa Number Conjecture, which has important corollaries. The key aspect of the most recent results, which allows us to handle the prime p=2, is the proof of a version of Ribet's Lemma in the case of characters that are congruent modulo p.

I will present a new trace formula approach to Waldspurger's formula for toric periods of automorphic forms on $PGL_2$. The method is motivated by interpreting Waldspurger's result as a period relation on $SO_2 \times SO_3$, which leads to a strange comparison of relative trace formulas. I will explain the local results needed to carry out this comparison, and discuss some small progress towards extending these results to high rank orthogonal groups.

Over the last decades, the study of "spin glasses" in physics has stimulated a large amount of theoretical activity in physics, and led to several breakthroughs. While the original puzzle of spin glass materials is still not fully solved, their theoretical analysis has created powerful techniques as well as a rich conceptual framework, to study emergent properties of strongly disordered and interacting systems. In this talk, I will use these tools and discuss how apparently unrelated complex problems such as: how to pack many objects in a given volume, how to color a graph with a given number of colors, why a liquid is turning into a glass when the temperature is lowered, and why some computational (classical and quantum) problems are hard while other are easy — actually (and surprisingly) do share many characteristics when looking at them through the (spin) glass.

In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the non-vanishing of central values of certain Rankin-Selberg L-functions to the non-vanishing of certain explicit integrals of automorphic forms, called 'automorphic periods', on classical groups. They have been subsequently refined by Ichino-Ikeda and Neal Harris into precise conjectural identities relating these two invariants thus generalizing a famous result of Waldspurger for toric periods on GL(2). In the case of unitary groups, those have been established by Wei Zhang under some local restrictions. I will review the current state of the art on this and in particular how certain results in local harmonic analysis allow to remove almost all the local restrictions made by Zhang.

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.

We will discuss the conjecture of Kolyvagin on the indivisibility of Heegner points and its role in constructing rational points on elliptic curves over rational numbers, particularly in the proof of a recent result of this type: "the Selmer rank being one implies that the Mordell--Weil rank being one".

Recently, Jiang made a conjecture investigating the connection between two fundamental invariants of an automorphic representation \pi appearing in the discrete spectrum of quasi-split classical groups G(A). The first invariant is the wave front of \pi, WF(\pi), which is the set of maximal unipotent orbits of G, such that \pi admits a non-trivial Fourier coefficients with respect to them. The second invariant is the Arthur parameter \psi of \pi to which one can associate a unipotent orbit \underline{p}(\psi) of the dual group of G. The conjecture says that in any Arthur packet associated to \psi, the Barbasch-Vogan duality of the orbit \underline{p}(\psi) is a sharp upper bound for the wave front of the representations of the packet. This is an important conjecture that vastly generalizes Shahidi's conjecture which claims that in every tempered packet there exists a generic representation. In this talk, I will review this conjecture and present some recent progress towards it. This is a joint work in progress with Dihua Jiang.

The zeta function of a curve C over a finite field can be expressed in terms of the characteristic polynomial of its Frobenius endomorphism. We will see how one can compute the trace of high powers of this endomorphism in various moduli spaces. Finally, we will discuss how one can use this information to compute the one-level density -- which concerns low-lying zeros of the zeta function -- in the case of cubic covers of the projective line.

For a non-degenerate integral quadratic form F(x1,...,xd) in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace Ω⊂Rd of the affine quadric F(x1,...,xd)=1. Suppose that we are given a small ball B of radius 0 < r < 1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies N ≫ (r−1m)4+ε for any ε > 0. Finally assume that we are given an integral vector (λ1, . . . , λd) mod m. Then we show that there exists an integral solution x = (x1, . . . , xd) x of F(x)=N such that xi ≡λi mod m and √N ∈B, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form F (x1 , . . . , x4 ) in 4 variables we prove the same result if N ≥ (r−1m)6+ε and N is not divisible by 2k where 2k ≫ Nε for any ε. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form F(X) in 4 variables with the optimal exponent 4.

We express the bias of global root numbers of Hilbert new forms of cubic level via special values of Dedekind L-functions attached to CM extensions determined by the level. In particular, our formula includes the case when weight 2 appears. We establish the formula by 1) a limit form of Jacquet-Zagier trace formula on PGL_2 associated to certain not necessarily integrable test functions at Archimedean places (when weight 2 occurs), and 2) showing the meromorphic continuation of certain Dirichlet series with coefficients given by special value of Dedekind L-functions via spectral side of the Jacquet-Zagier trace formula. This is a joint work with Q. Pi and H. Wu. arXiv: 2110.08310.

I will give a very informal talk on some work I am doing now with Mahesh Kakde. We hope to make progress on the Brumer-Stark conjecture using the theory of group-ring families of modular forms. I will motivate and state the conjecture, and describe the flavor of our approach.

In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.

For a (squarefree) integer d the negative Pell equation is given by: X^2 - d Y^2 = -1. It is easy to see that this equation has no solution over the integers, if d is negative or d is congruent to 3 modulo 4. In this talk we would like to study the asymptotic behavior of integers d such that this equation is solvable. This question is related to the behavior of the class group of the quadratic field generated by a square root of d. The distribution of those class groups is described by the Cohen-Lenstra heuristics.

We consider the family of $D_4$-quartic fields ordered by the Artin conductors of the corresponding 2-dimensional irreducible Galois representations. In this talk, I will describe ways to compute the number of such $D_4$ fields with bounded conductor. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination of geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of $D_4$ fields since counting quartic fields containing a quadratic subfield of large discriminant is difficult. However, when ordering by conductor, these techniques can be utilized due to additional algebraic structure that the Galois closures of such quartic fields have, arising from the outer automorphism of $D_4$. This result is joint work with Ali Altug, Arul Shankar, and Kevin Wilson.

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we were able to prove the multiplicity one theorem on Vogan L-packet for the Ginzburg-Rallis model. In some cases, we also proved the epsilon dichotomy conjecture which gives a relation between the multiplicity and the value of the exterior cube epsilon factor.

The local Langlands correspondence is a dictionary between representations of two kinds of groups: reductive p-adic groups (such as the general linear group) and the absolute Galois groups of p-adic fields. One entry in the dictionary is a conjectural formula of Hiraga, Ichino, and Ikeda for the size of a representation of a p-adic group, its "formal degree", in terms of the corresponding representation of a Galois group. In this talk, after reviewing the broad shape of p-adic representation theory, I'll explain why the conjecture is true for almost all supercuspidals, the fundamental building blocks of the subject.

Physicists Green, Russo, and Vanhove have discovered solution to differential equations involving automorphic forms appear at the coefficients to the 4-graviton scattering amplitude in type IIB string theory. Specifically, for \Delta the Laplace-Beltrami operator and E_s(g) a Langlands Eisenstein series, solutions f(g) of (\Delta-\lambda) f(g) = E_a(g) E_b(g) for a and b half-integers on certain moduli spaces G(Z)\G(R)/K(R) of real Lie groups appear as coefficients to the analytic expansion of the scattering amplitude. We will briefly discuss different approaches to finding solutions to such equations and focus on a shifted convolution sum of divisor functions which appears as the Fourier modes associated to the homogeneous part of the solution. Initially, it was thought that, when summing over all Fourier modes, the homogeneous solution would vanish but recently we have found an exciting error term. This is joint work with Stephen D. Miller, Danylo Radchenko and Ksenia Fedosova.

In this talk we use generalizations of classical geometric constructions of Kummer and Humbert to construct new higher Chow cycles on Abelian surfaces and K3 surfaces over p-adic local fields, generalising some work of Collino. The existence of these cycles is predicted by the poles of the local L-factor at p of the L-function of the Abelian surface. The techniques involve using some recent work of Bogomolov, Hassett and Tschinkel on the deformations of rational curves on K3 surfaces. As an application we use these cycles to prove an analogue of the Hodge-D-conjecture for Abelian surfaces.

Our starting point will be the relationship between hydrodynamics from the macroscopic continuum point of view and its atomistic interpretation in statistical mechanical terms. Then we extend stationary state (equilibrium and nonequilibrium) Molecular Dynamics to time dependent situations, including response and relaxation. We call the procedure Dynamical Non-Equilibrium Molecular Dynamics (D-NEMD), to distinguish it from standard NEMD, rigorously useful only to simulate stationary nonequilibrium states. It is, in essence, a generalization of linear response theory. The idea, formulated by Onsager in the thirties in metaphysical language, has received a solid foundation in the fifties by the work of Kubo (in the linear and nonlinear regimes). Adapted to MD simulations by G.Jacucci, I.R.Mac Donald and myself in the seventies, it has been baptized as the (nonlinear) Kubo-Onsager relation, connecting dynamical nonequilibrium averages or dynamical relaxations to initial distribution which can be sampled in MD by stationary processes. Finally we go back to hydrodynamics, to illustrate the method, by studying the hydrodynamic relaxation of an interface between two immiscible liquids.

Henniart derived the following theorem from his numerical local Langlands correspondence: If $F$ is a non-archimedean local field and if $\pi$ is an irreducible representation of $GL(n,F)$, then, after a finite series of cyclic base changes, the image of $\pi$ contains a fixed vector under an Iwahori subgroup. This result was indispensable in all demonstrations of the local correspondence. Scholze gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower (and this result also appears, in a somewhat different form, in proofs based on the global correspondence for function fields). An analogous theorem should be valid for every reductive group, but the known proofs only work for GL(n). I will sketch a different proof, based on properties of L-functions and assuming the existence of cyclic base change, that also applies to classical groups; I will also explain how the analogous result for a general reductive group is related to the local parametrization of Genestier-Lafforgue.

I will provide some general class of automorphic forms or representations on a general orthogonal group, having a non-tempered non-archimedean local component. We call them non-tempered automorphic forms or representations. It is a fundamental problem to find non-tempered cusp forms, which are nothing but counterexamples to the Ramanujan conjecture. The general class above includes the cusp forms given by the Oda-Rallis-Schiffmann lifting to O(2,m) and non-holomorphic lifting to O(1,8n+1) recently given by the joint work with Yingkun Li and Ameya Pitale. Such general class is given by means of the notion of the special Bessel model and the local Maass relation.

Beatty sequences are generalized arithmetic progressions which have been studied intensively in recent years. Thanks to the work of Vinogradov, it is known that every Beatty sequence S contains "appropriately many" prime numbers. For a given pair of Beatty sequences S and T, it is natural to wonder whether there are "appropriately many" primes in S for which the next larger prime lies in T. In this talk, I will show that this is indeed the case if one assumes a certain strong form of the Hardy-Littlewood conjectures. This is recent joint work with Victor Guo.