We consider the roots x mod m of the quadratic congruence x^2 = D mod m for a fixed, squarefree integer D. Besides these roots being a classical object of study, statistical information on their distribution can be crucial input into methods of analytic number theory. In joint work with Jens Marklof, we study the fine-scale distribution of these roots by seeing them as return times of the horocycle flow for a specific section in SL(2, Z) \ SL(2, R), analogous to Athreya-Cheung's interpretation of the Boca-Cobeli-Zaharescu map for Farey fractions.

Eisenstein series are key objects in the theory of automorphic forms. They play an important role in the study of automorphic $L$-functions, and they figure out in the spectral decomposition of the $L^2$-space of automorphic forms. In recent years, new constructions of global integrals generating identities relating Eisenstein series were discovered. In 2018 Ginzburg and Soudry introduced two general identities relating Eisenstein series on split classical groups (generalizing Maeglin 1997, Ginzburg-Piatetski-Shapiro-Rallis 1997, and Cai-Friedberg-Ginzburg-Kaplan 2016), as well as double covers of symplectic groups (generalizing Ikeda 1994, and Ginzburg-Rallis-Soudry 2011).
We consider the Kronecker product embedding of two general linear groups, $\mathrm{GL}{m}(\mathbb{A})$ and $\mathrm{GL}{n}(\mathbb{A})$, in $\mathrm{GL}{mn}(\mathbb{A})$. Now, similarly to Ginzburg and Soudry's construction, we use a degenerate Eisenstein series of $\mathrm{GL}{mn}(\mathbb{A})$ as a kernel function on $\mathrm{GL}{m}(\mathbb{A}) \otimes \mathrm{GL}{n}(\mathbb{A})$. Integrating it against a cusp form on $\mathrm{GL}{n}(\mathbb{A})$, we obtain a 'semi-degenerate' Eisenstein series on $\mathrm{GL}{m}(\mathbb{A})$. Locally, we find an interesting relation to the local Godement-Jacquet integral.
This construction demonstrates the rise of interesting $L$-functions from integrals of doubling type, as suggested by the philosophy of Ginzburg and Soudry.

This will be a survey talk about recent progress on pointwise convergence problems for multiple ergodic averages along polynomial orbits and their relations with the Furstenberg-Bergelson-Leibman conjecture.

Given a model M of an axiomatic theory A, and a model N of an axiomatic theory B, we say that they are bi-interpretable if, roughly speaking, they have the same definable sets: that is, there are definable maps that move definable sets in one to definable sets in the other. One interesting question we might ask, given an axiomatic theory A, is which of its models are bi-interpretable with the integers (seen as a model of the first-order theory of rings)? As self-interpretations of the integers are particularly simple, this gives a lot of information about properties of the model. In this talk, we will consider arithmetic groups like SL(n, Z) and discuss recent progress in understanding when such groups are bi-interpretable with arithmetic and what consequences this has when it occurs.

The Bateman-Horn conjecture, the polynomial Chowla conjecture and the Hasse principle for norm form equations are all notorious open questions. I'll discuss how they can be approached on average, for 100% of integer polynomials of fixed degree up to a given height. This is based on joint work with Efthymios Sofos and Joni Teravainen.

We give a purely topological formula for the square class of the central value of a symplectic representation on a curve. This is related to the theory of epsilon factors in number theory and Meyer's signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

We prove, in this joint work with Maksym Radziwill, a 1978 conjecture of S. Patterson (conditional on the Generalised Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846. One important byproduct of our proof is that we show Heath-Brown's cubic large sieve is sharp under GRH. This disproves the popular belief that the cubic large sieve can be improved. An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term.

I will describe a method for constraining Laplacian spectra of hyperbolic surfaces and d-manifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of SO(1,d) and linear programming, the method yields rigorous upper bounds on the spectral gap. In several examples, the bound is nearly sharp. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. The ideas were inspired by recent developments in the conformal bootstrap. The linear program is similar to the Cohn+Elkies linear program for bounding sphere packing density. Based on https://arxiv.org/abs/2111.12716 with P. Kravchuk and S. Pal and work in progress with the same collaborators and J. Bonifacio.

We begin this talk by introducing the general notion (due to Helgason) of generalized Radon transforms for homogeneous spaces in duality, together with some motivating examples of such transforms (the classical Radon transform and the Funk transform) and a brief discussion of the types of problems about such transforms. The rest of the talk is devoted to the primitive Siegel transform of (sufficiently nice) functions $f :\mathbb{R}^n \to \mathbb{R},$ which is a particular kind of generalized Radon transform. The primitive Siegel transform $\widehat{f}$ of such a function $f$ is a pseudo-Eisenstein series on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SL}_n(\mathbb{Z}),$ the space of full-rank lattices in $\mathbb{R}^n$ of covolume one. After briefly discussing the history of this transform in the geometry of numbers, we show how classical formulae for the mean, due to C. L. Siegel (1945), and inner product when $n \geq 3$, due to C. A. Rogers (1955), of such transforms immediately yield whenever $n \geq 3$ the injectivity of this transform on even functions and an inversion formula. This injectivity and inversion formula were initially obtained by more direct means and in a context more general than the preceding one by Ghosh-Kelmer-Yu (2020). We then briefly discuss why the case $n=2$ is so different from the case $n \geq 3.$

The Galois group of an additive polynomial over over a finite field is contained in a finite general linear group. We will discuss three different probability distributions on these polynomials, and estimate the probability that a random additive polynomial has a "large" Galois group. Our computations use a trick that gives us characteristic polynomials of elements of the Galois group, so we may use our knowledge of the maximal subgroups of GL(n,q). This is joint work with Lior Bary-Soroker and Alexei Entin.

Euler primes are primes of the form $p = x^2+Dy^2$ with $D>0$. In analogy with Linnik's theorem, we can ask if it is possible to show that $p(D)$, the least prime of this form, satisfies $p(D) \ll D^A$ for some constant $A>0$. Indeed Weiss showed this in 1983, but it wasn't until 2016 that an explicit value for $A$ was determined by Thorner and Zaman, who showed one can take $A=694$. Their work follows the same outline as the traditional approach to proving Linnik's theorem, relying on log-free zero-density estimates for Hecke L-functions and a quantitative Deuring-Heilbronn phenomenon. In an ongoing work (as part of my PhD thesis) we propose an alternative approach to the problem via sieve methods that avoids the use of the above technical results on zeros of the Hecke L-functions. We hope that such simplifications may result in a better value for the exponent $A$.

The use of orthogonality relations in number theory goes back to Dirichlet's proof of his theorem on primes in arithmetic progressions. I will discuss analogs of the classical orthogonality relation for characters of finite groups to reductive groups, including some history and applications. I will also discuss new work with Goldfeld and Stade in which we give strong bounds with polynomial error terms for the group GL(n).

The Markoff equation is the affine cubic x^2 + y^2 + z^2 - 3xyz = 0. This equation first appeared in the work of Markoff in 1879 on Diophantine approximation and quadratic forms, but has since appeared in a number of contexts, ranging from hyperbolic geometry to the derived categories of varieties. It was first conjectured by Baragar in 1991 that the equation satisfies strong approximation -- that is, its integral points surject onto its mod p points for every prime p. In 2016, Bourgain, Gamburd, and Sarnak made significant progress towards the conjecture, proving it for all but a very sparse "exceptional" set of primes. Their work also proved a certain logarithmic lower bound for the sizes of orbits of mod p points under a certain group action, and showed that finiteness of the exceptional set would follow if one could prove a polynomial lower bound. In this talk we will show how to supply the polynomial lower bound by relating the problem to the geometry of "Hurwitz" moduli spaces of SL(2,p)-covers of elliptic curves. These moduli spaces are in fact noncongruence modular curves, and the result can also be understood in terms of computing the dimensions of spaces of modular forms for certain noncongruence subgroups of SL(2,Z).

Since the early 20th century, spectral methods have been used to obtain effective counting theorems for various objects of interest in number theory, geometry and group theory. In this talk I'll start by introducing two classical problems: the Gauss circle problem, and the Apollonian counting problem. By surveying results on these problems (and some generalizations), I'll demonstrate how to use spectral methods to obtain effective asymptotics for some very classical problems.