Joint meeting, June 17-18 2025, Jussieu, room 15-25.502
Speakers:
Claire Burrin (University of Zürich)
Sary Drappeau (Université d'Aix-Marseille)
Manfred Einsiedler (ETH Zürich)
Florent Jouve (Université de Bordeaux)
Manuel Lüthi (ETH Zürich)
Asbjørn Nordentoft (Paris-Saclay)
June 17
- 9:30-10:30: Manfred Einsiedler Equidistribution of closed adelic orbits
- 11:00-12:00: Sary Drappeau Baladi-Vallée type theorems for Fuchsian groups and additive twists of GL(2) L-functions
- 14:30-15:30: Manuel Lüthi Some measure rigidity for certain non-stiff random walks
We discuss an effective adelic equidistribution result of closed orbits on homogeneous spaces. This generalises a result due to Mozes and Shah from 1995, which relied heavily on the fundamental work of Ratner in unipotent dynamics. Our theorem is joint work with Lindenstrauss, Mohammadi, and Wieser and is stronger in two important aspects. First of all, the theorem is effective. In fact the error term provided is polynomial in terms of a complexity parameter for intermediate orbits. Secondly, the theorem avoids a splitting condition by using dynamics at a varying prime (satisfying a logarithmic bound). Unlike prior work with Margulis, Mohammadi, and Venkatesh we avoid any kind of maximality condition for the subgroup but assume (currently) that the ambient space is compact. The proof relies on Prasad’s volume formula, Clozel’s property (tau), and an effective version of a theorem of Greenberg.
As an application we strengthen a theorem of Ellenberg and Venkatesh of representations of quadratic forms by quadratic forms.
The additive twists we'll talk about are series of the shape L(x, s) = \sum_{n\geq 1} a_n \e^{2\pi i n x} / n^s where (a_n) are the Fourier coefficients of an automorphic form for a Fuchsian group Gamma of the first kind for SL(2, R), and x\in \R varies among Gamma-orbits of cusp. As an example one may think of a_n = r_2(n), the number of ways to write n as sums of two squares, or to Fourier coefficients of a Maass form.
In a joint works with Jungwon Lee, Asbjørn Nordentoft and Sandro Bettin, we prove a central limit-type theorem for L(x, 1/2) as x varies among cusps of bounded "denominators". This generalizes various earlier results (holomorphic forms, or SL(2, Z) forms) in particular due to Petridis-Risager, Nordentoft and Lee-Sun.
The method relies on a generalization of a theorem of Baladi-Vallée for rational orbits of the Gauss map, where "Gauss" is replaced by "Bowen-Series". The talk will review this theorem and the challenges posed by its application to L(x, s) (in particular verifying its hypotheses, and computing the parameters of the limit law).
I will discuss recent joint work with Osama Khalil and Barak Weiss, where we provide a more conceptual proof of equidistribution of pushes of fractal measures in the space of S-adic lattices via (partial) measure rigidity for the associated random walk. The random walk in question has a diagonal component and, hence, there is no uniform expansion which can be exploited. In this talk, I will discuss the classification in the very special case arising from the middle third Cantor set, quickly recall the exponential drift argument as developed in the seminal work by Yves Benoist and Jean-François Quint, and finally provide an outline of our strategy which eventually allows us to employ this argument for the classification.
- 9:30-10:30: Florent Jouve Algebraic properties of graph polynomials
- 11:00-12:00: Claire Burrin Rational points on the sphere
- 14:30-15:30: Asbjørn Nordentoft A mixing version of QUE
To a given graph G, one can attach several integer polynomials that encode various structural properties of G (the chromatic polynomial of a graph is a well known example). To approach the difficult "inverse question" of determining which integer polynomials can be realized as relevant graph polynomials, one may first ask what are the typical properties of such graph polynomials e.g do these polynomials behave like random integer polynomials? I will report on joint work with Jean-Sébastien Sereni and Andrew Goodall in which we study a conjecture of Bohn--Cameron--Müller (2010) claiming that the Tutte polynomial (a bivariate integer polynomial specializing to the chromatic polynomial) of a 2-connected graph should have maximal monodromy. We notably exploit an equidistribution result of S. D. Cohen to obtain a probabilistic statement confirming the conjecture and we also prove that the conjecture holds for some particular families of graphs.
We consider rational points on the 2-sphere and investigate their equidistribution in shrinking spherical caps.
In this talk I will describe a refinement of the quantum uniqueness ergodicity (QUE) conjecture in the level aspect given by pushing forward to the product of two modular curves via the Hecke correspondence. This gives rise to a two variable equidistribution problem resembling in many ways the mixing conjecture of Michel-Venkatesh. I will explain how to resolve the conjecture in the cocompact case under GRH. This is joint work in preparation with Radu Toma