I will propose a way to combine the theory of translation functors with the trace formula to study automorphic representations of connected semisimple anisotropic algebraic groups over the rational numbers whose Archimedean component is a limit of discrete series. I will explain the main ideas of the derivation of a trace formula which, modulo a conjecture on the decomposition of the tensor product of a limit of discrete series with a finite-dimensional representation into basic representations, allows to isolate the non-Archimedean parts of a finite family of C-algebraic automorphic representations containing the ones whose Archimedean component is a given limit of discrete series.

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.

In the 1980�??s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Ap ́ery-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. In this talk we discuss the distributions (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular, whereas the distribution for the 3F2 functions is the more exotic Batman distribution.

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.

Braverman and Kazhdan, L. Lafforgue, Ngo, and Sakellaridis have pursued a set of conjectures asserting that analogues of the Poisson summation formula are valid for all spherical varieties. If proven, these conjectures imply the analytic continuation and functional equations of quite general Langlands L-functions (and thus, by converse theory, much of Langlands functoriality). I will explain techniques for proving the conjectures in special cases that include the first known case where the underlying spherical variety is not a generalized flag variety.

We prove that, when monic quintic integral polynomials f\in \Z[x] with nonzero discriminant are ordered by height, the average number of solutions to y^2 = f(x) is bounded.

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.

In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.

I will first introduce some generalities about exponential sums, in particular that square-root cancellation is expected for many algebraic character sums over the rational points of an algebraic variety over a finite field. I will then set the stage for my work: we consider a family of exponential sums, which in our case is parameterized by the rational points of a variety (the parameter space). Our task is to obtain a good bound on the number of exceptional ("bad") parameters for which square-root cancellations fail. Following an idea of Michael Larsen, we consider even moments of the family of exponential sums. If the summands are of certain product form, a transformation can be applied to produce another family of exponential sums (of the same type). If the summands are products of multiplicative characters composed with certain polynomial functions, a weak bound can then be applied to the character sums in this family (with few bad parameters), yielding bounds for the moments. We know from the theory of l-adic sheaves that the parameter space for the original family have a stratification by smooth varieties, which is uniform in some sense as long as the degrees of the characters and polynomials are bounded. Moreover, on each stratum the character sum behave in certain uniform way, so that we can talk about good and bad strata. The bounds on moments yield bounds on dimensions of bad strata, which in turn yield bounds on the number of bad parameters (in any box) of the original family. Though not optimal, the bounds already imply nontrivial Burgess bounds for forms, in joint work with Lillian Pierce.

For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces over global function fields. In this talk, I will present the proofs via the intersection theory on GSpin Shimura varieties and also discuss various applications. These results are joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.

In his recent paper,\Beyond Endoscopy", Langlands proposed an approach to (ultimately) attack the general functoriality conjectures by means of the trace formula. For a (reductive algebraic) group G over a global field F and a representation of its L-group, the strategy, among other things, aims at detecting those automorphic representations of G for which the L-function, L(s;\pi ;\rho ), has a pole at s = 1. The method suggested using the the trace formula together with an averaging process to capture these poles. In this talk we will start by recalling the functoriality conjectures and brie y describe the method suggested by Langlands. Then, specializing on the group GL(2) we will discuss some recent work on Beyond Endoscopy. More precisely, we will discuss the elliptic part of the trace formula and the analytic problems caused by the volumes of tori, singularities of orbital integrals and the non-tempered terms. We will then describe how one can use an approximate functional equation in the trace formula to rewrite the elliptic part which resolves these issues. Finally, we will talk about applications of the resulting formula.

Fourier coefficients of automorphic forms are invariants which encode the analytic and arithmetic properties of automorphic forms. In this talk, we introduce the general notion of Fourier coefficients for automorphic representations of reductive groups and use them to construct explicit endoscopy correspondences, which construct via integral transforms with automorphic kernel functions members in global Arthur packets for classical groups. For instance, we will discuss with some details the recent work joint with Lei Zhang.

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.

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.

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.

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.

We will discuss how to generalize the Langlands quotient theorem to symmetric spaces. The key idea is to generalize so-called Casselman�?s criterion for temperedness to the context of symmetric spaces by using the work of Kato-Takano.

We propose general frameworks to inductively count number fields building on previously known counting results and good uniformity estimates in different flavors. By this method, we prove new results in counting number fields with Galois groups ranging from direct product to wreath product. We will also mention interesting applications en route. This involves my thesis and on going project with Melanie Matchett Wood and Robert J. Lemke Oliver.

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".

In a 1997 Fields Medalist Maxim Kontsevich suggested that the function F(q) = 1 + (1-q) + (1-q)(1-q^2) + (1-q)(1-q^2)(1-q^3)+�?, defined only for q a root of unity, is similar to certain functions arising from the computation of Feynman integrals in quantum field theory. In the last sixteen years this function has been connected to interval orders in decision making theory, ascent sequences and matchings in combinatorics, and Vassiliev invariants in knot theory. Don Zagier related the asymptotic properties of this function to the �?half-derivatives�? of modular forms and was led to define a notion of �?quantum modular form�?. In a trilogy of papers, my collaborators (Andrews, Bryson, Ono, Pitman, Zwegers) and I have connected this function to Ramanujan�?s mock theta functions and the combinatorics of unimodal sequences. I will tell the story of this function and these many relationships.

I will discuss the relative trace formula approach to the global Gan-Gross-Prasad conjectures for unitary groups. The focus will be on the spectral side. I will present the various terms that appear in the spectral development of the relative trace formula and discuss what is still missing. This is a joint work with Pierre-Henri Chaudouard.

In this talk, I will first review relative (unipotent) completions of discrete groups in general, and $\mathrm{SL}_2(\mathbb{Z})$ in particular. We then develop an explicit $\mathbb{Q}$-de Rham theory for the relative completion of $\mathrm{SL}_2(\mathbb{Z})$, which enables us to construct iterated integrals of modular forms of the second kind that provide its periods. Following Francis Brown, these periods are called `multiple modular values'. They contain periods of modular forms.

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.

Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas.

Rational maps from the Riemann sphere into itself can be described in terms of poles and principal parts; doing the same for maps into the flag manifold of a loop group gives insight into the topology of moduli of instantons and calorons. (joint work with Michael Murray)

We will give an explicit construction of the regular quotient of Morrissey-Ngô in the case of a symmetric pair. In the case of a quasisplit form (i.e. the regular centralizer group scheme is abelian), we will give a Galois description of the regular centralizer group scheme using parabolic covers. We will then describe how the nonseparated structure of the regular quotient recovers the spectral description of Hitchin fibers given by Schapostnik for U(n,n) Higgs bundles. This work is joint with B. Morrissey.

The Liouville function λ(n) is the completely multiplicative arithmetic function defined by λ(p) = −1 for each prime p. Pólya investigated its summatory function L(x) = Σ_n≤x λ(n), and showed for instance that the Riemann hypothesis would follow if L(x) never changed sign for large x. While it has been known since the work of Haselgrove in 1958 that L(x) changes sign infinitely often, oscillations in L(x) and related functions remain of interest due to their connections to the Riemann hypothesis and other questions in number theory. We describe some connections between the zeta function and a number of oscillation problems, including Pólya's question and some of its weighted relatives, and, in joint work with T. Trudgian, describe a method involving substantial computation that establishes new lower bounds on the size of these oscillations.

A page in Ramanujan's lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous ``circle problem'' and the other to the equally famous ``divisor problem.'' We discuss these classical unsolved problems. Each identity can be interpreted in three distinct ways. We discuss various methods that have been devised to prove the identities under these different interpretations. Weighted divisor sums naturally arise, and new methods for estimating trigonometric sums need to be developed. Trigonometric analogues and extensions of Ramanujan's identities to Riesz and logarithmic sums are discussed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu.