We will formulate a conjectural identity relating the Fourier--Jacobi periods on unitary groups and the central value of certain Rankin--Selberg $L$-functions. This refines the famous Gan--Gross--Prasad conjecture. We will give some examples supporting this conjecture.

We will discuss the notion of Artin root numbers attached to an n-dimensional continuous Frobenius-semisimple complex representation of the Weil-Deligne group and show their equalities with those defined by Langlands-Shahidi method through local Langlands correspondence for GL(n) and the exterior and symmetric square representation of the L-group GL(n,C) of GL(n). The proof is a robust deformation argument using local-global techniques, complemented with suitable asymptotic expansions for partial Bessel functions inspired by certain generalized Shalika germ expansions of Jacquet and Ye. This is a joint work with J. Cogdell and T.-L. Tsai.

Whittaker functions are certain special functions that play a central role in automorphic representation theory. When dealing with automorphic forms on covering groups, new methods are needed to compute these functions. In this talk, I will outline how the study of this problem has uncovered connections with geometric representation theory and crystal graphs. I also explain my work in making this connection practical, allowing for new computations of spherical Whittaker functions for covering groups.

Recently Alexandrov, Banerjee, Manschot and Pioline [ABMP] constructed generalizations of Zwegers theta functions for lattices of signature (n-2,2). They also suggested a generalization to the case of arbitrary signature (n-q,q) and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\CC$ of negative vectors, are obtained by `completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. In joint work with Jens Funke, we show that their completed modular series arises as integrals of the q-form valued theta functions, defined in old joint work of the author and John Millson, over a certain singular $q$-cube determined by the data $\CC$. This gives an alternative construction of such series and a conceptual basis for their modularity. If time permits, I will discuss the simplicial case and a curious `convexity' problem for Grassmannians that arises in this context.

In this talk we describe a partially successful attempt to describe a characteristic p > 0 analog of the locally symmetric spaces for GL(n,R), by interpreting this as a "moduli space" for abelian varieties with real structure.

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.

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.

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.

A result of Gelfand and Graev shows that the supercuspidal representations of SL_2 are neatly parameterized by characters of elliptic tori, and that the stable character data for all such representations may be collected into a single function by means of a Fourier Transform. Using recent advances in the computation of characters of supercuspidal representations, we prove analogous results for the group SL_n.

I will discuss the relation between singularities in formal arc spaces of certain reductive monoids and local factors of automorphic L-functions.

By a non-standard comparison between (relative) trace formulas we mean one where the scalar "transfer factors" are substituted by non-scalar "transfer operators". The problem of global triviality of transfer factors now becomes a problem of proving a Poisson summation formula for such non-scalar operators. I will discuss the adelic analysis behind such a non-standard comparison, that leads to a new proof of Waldspurger's theorem on toric periods for GL(2) and the analytic continuation of the quadratic base change L-function in the spirit of "Beyond Endoscopy".

This is joint work with Julee Kim and Nicolas Templier. Irreducible smooth representations of a p-adic reductive group are said to be supercuspidal if they do not appear in any induced representation from a proper parabolic subgroup. While it is still an open problem to obtain a precise character formula for them (apart from some special cases), I will explain that we can prove a reasonable upper bound and a limit formula as the formal degree tends to infinity, for a large class of supercuspidal representations. An expected application is an equidistribution result as well as a low-lying zero statistics for L-functions in a new kind of families of automorphic representations.

We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications (mainly due to other researchers) include new proofs of some important results in Brill-Noether theory, a generalization of the Eisenbud-Harris theory of limit linear series, and new bounds for the number of rational points on algebraic curves over number fields.

Shnirelman discovered in the 1970s that the eigenfunctions of the Laplacian on a compact Riemannian manifold whose flow is ergodic with respect to Liouville measure exhibit an analogue of classical ergodicity at the quantum level. This phenomenon became known as "Quantum Ergodicity" and Schnirelman's proof was completed by Zelditch and Colin de Verdiere in the 1980s. Following a brief introduction to the subject, I will show that Quantum Ergodicity can also hold in systems which are essentially integrable, provided they have some arithmetic structure. Finally, in the absence of such an arithmetic structure, a very different phenomenon occurs: scarring. This talk is based on joint work with Par Kurlberg, KTH Stockholm.

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Mobius transformation action of the Bianchi group $\operatorname{PSL}(2,\mathcal{O}_K)$. The arrangement takes the form of a dense collection of intricately nested circles. Aspects of the number theory of $\mathcal{O}_K$ can be characterised by properties of this picture: for example, the arrangement is connected if and only if $\mathcal{O}_K$ is Euclidean. I'll explore this structure and its connection to Apollonian circle packings. Specifically, the Schmidt arrangement for the Gaussian integers is a disjoint union of all primitive integral Apollonian circle packings. Generalizing this relationship to all imaginary quadratic $K$, the geometry naturally defines some new circle packings and thin groups of arithmetic interest.

Hyperbolic 3-manifolds have been studied intensely by topologists since the mid-1970's. When the fundamental group arises from a certain number theoretic construction (in this case, the manifold is called "arithmetic"), the manifold acquires extra features that lead to important connections with number theory. Accordingly, arithmetic hyperbolic 3-manifolds have been studied by number theorists (perhaps not as intensely as the topologists) with different motivations. Very recently, number theorists have started to study the torsion in the homology of arithmetic hyperbolic 3-manifolds. The aim of the first half of this introductory talk, where we will touch upon notions like "arithmeticity", "Hecke operators", will be to illustrate the importance of torsion from the perspective of number theory. In the second half, I will present new joint work with N.Bergeron and A.Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion in the homology. I will especially focus on the interesting use of the celebrated "Cheeger-Mueller Theorem" from global analysis.

Random multiplicative functions naturally serve as models for number theoretic objects such as the Mobius function. After fixing a particular model, there are many interesting questions one can ask. For example, what is the distribution of their partial sums? Harper has recently made remarkable progress for partial sums of certain random multiplicative functions with values that lie on the complex unit circle. He settled the correct order of magnitude for their low moments and surprisingly established that one expects better than square-root cancellation in their partial sums. I will discuss an extension of Harper's analysis to a wider class of multiplicative functions such as those modeling the coefficients of automorphic $L$-functions.

POSTPONED

I will discuss recent results towards the weight part of Serre's conjecture for GL_n as formulated by Herzig. The conjecture predicts the set of weights where an odd n-dimensional mod p Galois representation will appear in cohomology (modular weights) in terms of the restriction of the representation to the decomposition group at p. We show that the set of modular weights is always contained in the predicted set in generic situations. This is joint work with Daniel Le and Bao V. Le Hung.

Classically the analytic properties of L-functions, in particular the functional equation, have been related to summation formulae of Poisson type. On the other hand, analytic properties of automorphic L-functions could be used to deduce functorial lifting of automorphic forms to general linear groups via the converse theorem. In his recent work, L. Lafforgue showed that a conjectural nonlinear Poisson summation formula on reductive groups is equivalent to the existence of functorial liftings to general linear groups. Here the Fourier transform is on a nonlinear space and involves nonstandard test functions. In my talk I will explain a toy model of such a summation formula for an integral transform between nonstandard spaces of test functions. The integral transform in question is the Harish-Chandra transform operating on the space of orbital integrals, and the summation formula follows from a trace formula on Lie algebras.

Given an elliptic curve $E/\mathbb{Q}$, let $E[2^k]$ denote the set of points on $E$ that have order dividing $2^k$. The coordinates of these points are algebraic numbers and using them, one can build a Galois representation $\rho : G_{\mathbb{Q}} \to \GL_{2}(\mathbb{Z}_{2})$. We give a classification of all possible images of this Galois representation. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

For a finite 2-generated group G, one can consider the moduli of elliptic curves equipped with G-structures, which is roughly a G-Galois cover of the elliptic curve, unramified away from the origin. The resulting moduli spaces are quotients of the upper half plane by possibly noncongruence subgroups of SL(2,Z). When G is abelian, it is easy to see that such level structures are equivalent to classical congruence level structures, but in general it is difficult to classify the groups G which yield congruence level structures. In this talk I will focus on a recent joint result with Pierre Deligne, where we show that for any metabelian G, G-structures are congruence in an arithmetic sense. We do this by studying the monodromy action of the fundamental group of the moduli stack of elliptic curves (over Q) on the pro-metabelian fundamental group of a punctured elliptic curve.

We prove an analogue of the Prime Number Theorem for short intervals on a smooth proper curve of arbitrary genus over a finite field. Our main result gives a uniform asymptotic count of those rational functions, inside short intervals defined by a very ample effective divisor E, whose principal divisors are prime away from E. In this talk, I will discuss the setting and definitions we use in order to make sense of such a count, and will give a rough sketch of the proof. This is a joint work with Tyler Foster.

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.

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.