This week

joint with Martin Čech, Daniel Fiorilli, Kaisa Matomäki and Anders Södergren. I will discuss the distribution of low-lying zeros of L-functions in families of degree two, for which, thanks to good trace formulas, we are able to extend the unconditional support in the Katz--Sarnak prediction.

A random ensemble of cusp forms for the full modular group is introduced. For a weight-k cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be approximately \sqrt{\log(k)}, in line with the conjectured bounds. In addition, the exponential concentration of the supremum around its median is established. Contrary to the compact case, the global expected supremum, attained around the cusp, grows like k^{1/4}. This talk is based on a forthcoming work, joint with B. Huang, S. Lester and N. Yesha.

Murmurations, originally referring to the wave-like patterns in the movement of flocks of starlings, have acquired a new meaning in analytic number theory: a correlation between the Dirichlet coefficients of a family of L-functions and their root numbers. These phenomena were first observed in Elliptic curves by He, Lee, Oliver, and Pozdnyakov in 2022, using machine learning algorithms, and soon in other families of L-functions, including those of Dirichlet characters and automorphic forms, and have been actively studied ever since.

In this talk, I will present joint work with Jonathan Bober, Andrew R. Booker, David Lowry-Duda, Andrei Seymour-Howell, and Nina Zubrilina, demonstrating murmurations in holomorphic modular forms and Maass forms, with a focus on the archimedean aspect.

Understanding the non-vanishing of central values of L-functions is an important and notoriously hard problem in number theory. Moment methods from analytic number theory propose a general approach which leads to deep questions in exponential sums which unfortunately is out of reach for many natural families of L-functions. In this talk I will focus on ‘horizontal families’ given by twisting a fixed automorphic form by Dirichlet characters of square-free conductor (and subfamilies thereof). I will explain how in some cases one can use various regularity properties of automorphic periods to get new non-vanishing result using both archimedean and p-adic methods. This is partly based on joint work with Daniel Kriz.

In this talk I will discuss various results related to quantum unique ergodicity and its refinements with a focus on dimensions 2 and 3. This is joint work with D. Chatzakos, R. Frot and Y. Petridis, M. Risager.

Several classic arithmetic problems, such as the Gauss circle problem, congruent number problem, and the equidistribution of rational points on degree 2 curves, can be studied using Dirichlet series built from n-fold shifted convolutions of theta functions. In this talk, I will discuss point-counting problems on the sphere from this perspective.

The study of integer or rational solutions to polynomial equations with integer coefficients is one of the oldest areas of mathematics and remains a very active field of research. The most basic question we can ask about such an equation is whether its set of rational solutions is empty or not. This turns out to be a very hard question! I will discuss some modern methods for proving that the set of rational solutions is empty. Along the way, I will describe some joint work with Martin Bright concerning the wild part of the Brauer–Manin obstruction.