Hecke operators act on many invariants associated to modular curves and their generalizations. For example, they act on modular forms and on cohomology groups of modular curves. In each of these cases, they generate a semi-simple, commutative algebra. In the first part of this talk, I will recall (in friendly, elementary, geometric terms) what Hecke operators are and how they act on the standard invariants. I will then show that they also act on loops in modular curves (aka, conjugacy classes in modular groups). In this case, the Hecke operators generate a non-commutative subalgebra of the vector space generated by the conjugacy classes, which leads to a very natural non-commutative generalization of the classical Hecke algebra. In the second part of the talk will discuss why one might want do construct such a Hecke action. As a prelude to this, I will explain why this Hecke action commutes with the natural action of the absolute Galois group after taking profinite completions. And, in the unlikely event that I have sufficient time, I will also explain how (after taking the appropriate completion) this Hecke action is also compatible with Hodge theory.

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.

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.

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.

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.

Theta functions are automorphic forms on the double cover of symplectic groups and are important for constructing automorphic liftings. For higher-degree covers of symplectic groups, there are generalized theta representations and it is natural to ask if these ``higher'' theta functions play a similar role in the theory of metaplectic forms. In this talk, I will discuss new lifting of automorphic representations on the 4-fold cover of symplectic groups using such theta functions. A key feature is that this lift produces counterexamples of the generalized Ramanujan conjecture, which motivates a connection to the emerging ``Langlands program for covering groups'' by way of Arthur parameters. The crucial fact allowing this lift to work is that theta functions for the 4-fold cover still have few non-vanishing Fourier coefficients, which fails for higher-degree covers.

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.

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.

POSTPONED

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.

We will describe new algorithms to compute an explicit finite simplicial model for compact, congruence locally symmetric spaces and Hecke actions thereon. Joint work with Aurel Page.

The remarkable Cheeger-Muller theorem, of differential geometric origin, provides an analytic means of studying torsion in the cohomology of Riemannian manifolds. We describe how this theorem can be applied to prove a numerical form of Langlands' base change functoriality for torsion in cohomology.

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.

The module theory of the Weyl algebra, known as the theory of $D$-modules, has profound applications in various fields. One of the most famous results is the Riemann-Hilbert correspondence, establishing equivalence between holonomic $D$-modules and perverse sheaves on smooth complex varieties. However, when dealing with singular varieties, such correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. In our ongoing work, we introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It should contain the Weyl algebra as a proper subring and shares many properties with the Weyl algebra on smooth varieties. In the talk, after a brief review of the Weyl algebra, I will explain how the new ring of differential operators arises as a consequence of an explicit form of the Poisson summation conjecture and discuss its properties.

In the first part of the talk I will describe a general context which, in some specific situations, permits us to give a cohomological interpretation to the Langlands-Shahidi theory of L-functions. In the second part of the talk, I will specialize to the context of the general linear group over a totally imaginary base field F, and discuss some recent results of mine on the special values of Rankin-Selberg L-functions for GL(n) x GL(m) over such an F. The talk is based on my preprint: https://arxiv.org/abs/2207.03393

(joint w/ B.Bakker) For an elliptic curve E over a field k, the p-torsion E[p] gives a 2-dimensional representation of the Galois group G_k over F_p. For k=Q and p>13, the Frey-Mazur conjecture famously states that one can recover the isogeny class of E from the representaiton E[p]. We state and prove a direct analogue of this question over function fields of complex algebraic curves. Specifically, for any complex algebraic curve C, let k(C) be its field of rational functions. Then there exists a constant A(C), such that for all primes p>A(C), isogeny classes of elliptic curves E over k(C) can be recovered from E[p]. Moreover, we show that A(C) can be made to depend only on the gonality of C, which can be thought of as the analogous notion of degree for number fields. The study of this question will lead us into the realm of moduli spaces and hyperbolic geometry. The use of the latter means that, unfortunately, these methods don't apply in finite characteristic.

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.

We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, we obtain a purely algebro-geometric proof of homological stability of configuration spaces.

Let M be a compact Riemannian manifold, and f an L^2 normalised Laplace eigenfunction on M. A popular question in semiclassical analysis is how well one can bound the other L^p norms of f, or its restriction to a submanifold. I will give an introduction to this problem, and describe how one can make progress on it using the additional assumptions that M is arithmetic and f is a Hecke-Maass form.

By studying universal motivic unipotent representations of fundamental group of varieties and comparing their different realizations, we combine Kim's recent method in Diophantine geometry with Deligne-Goncharov's theory of motivic fundamental groups to develop a machinery for approaching Diophantine problems concerning integral points.

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.

It is well known that for an exponential sum with a prime modulus the best bound for the sum comes from Weil's famous estimation. In this talk, we discuss when this bound can be improved on average over integral modulus in a number field. Investigations into exponential sums on average, or sums of exponential sums, have many applications including the Riemann hypothesis and the Ramanujan conjecture for automorphic forms. In particular, we will get an asymptotic for sums of quartic exponential sums over the Gaussian integers. Tools we will use to get this asymptotic include automorphic forms and the trace formula.

Let $F$ be a number field with ring of adeles $\mathbb{A}_F$. Let $r_1,r_2,r_3$ be a triple of positive integers and let $\pi:=\otimes_{i=1}^3\pi_i$ where the $\pi_i$ are all cuspidal automorphic representations of $\mathrm{GL}_{r_i}(\mathbb{A}_F)$. We denote by $L(s,\pi, \otimes^3)=L(s, \pi_1\times \pi_2 \times \pi_3)$ the corresponding triple product $L$-function. It is the Langlands $L$-function defined by the tensor product representation $\otimes^3:{}^L(\mathrm{GL}_{r_1} \times \mathrm{GL}_{r_2} \times \mathrm{GL}_{r_3}) \to \mathrm{GL}_{r_1r_2r_3}(\mathbb{C})$. In this talk I will present a family of Eulerian period integrals, which are holomorphic multiples of the triple product -function in a domain that nontrivially intersects the critical strip. We expect that they satisfy a local multiplicity one statement and a local functional equation. This is joint work with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie.

I will begin with an introduction to Galois deformation theory and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss Kisin's work on flat deformations and explain how to generalize to Galois representations valued in an arbitrary reductive group. Kisin's techniques led to the successful determination of the connected components of the flat deformation ring in the 2-dimensional case. If time permits, I will touch on difficulties of going beyond GL_2.

I will introduce a parabolic flow of Hermitian metrics which is a generalization of Kahler-Ricci flow. This flow preserves the pluriclosed condition, and its existence and convergence properties are closely related to the underlying topology of the given complex manifold. I will classify static solutions to the flow on various classes of complex surfaces, and show that no static solutions exist on Class VII surfaces, an important first step in using this flow to classify these surfaces. Joint with G. Tian.