Geometry and analysis of locally symmetric spaces

Jean Philippe Anker: Smooth barycentric decompositions of Weyl chambers - a helpful tool to handle inverse spherical Fourier transforms on noncompact symmetric spaces of higher rank
The study of invariant dispersive PDE on noncompact symmetric spaces, such as the wave equation or the Schrödinger equation, requires to analyze oscillating integrals arising from the inverse spherical Fourier transform. While this can be achieved by classical though nontrivial tools in rank one, a major problem in higher rank lies in the fact that the Plancherel density is not a differentiable symbol in general, and thus integration by parts produces no additional global decay at infinity. In this talk, we will explain a way to overcome this problem by introducing a smooth barycentric decomposition of Weyl chambers, which leads eventually to the same dispersive and Strichartz estimates as in rank one. This work started 15 years ago as a joint project with S. Meda, V. Pierfelice, M. Vallarino and was finally achieved in collaboration with H.-W. Zhang.

Edgar Assing: On the unipotent mixing conjecture
It is a classical fact that low-lying horocycles equidistribute in the modular curve. Going a step further, one can consider shifted pairs of low-lying horocycles and ask whether they equidistribute simultaneously. This question was recently addressed by Blomer and Michel using a beautiful mix of tools from number theory and dynamical systems. In this talk, I will explain the work of Blomer and Michel and discuss some extensions thereof.

Claire Burrin: Pairs of saddle connections
Starting with Gauss, there is a collection of classical problems at the intersection of number theory and analysis that concern the distribution of lattice points in space. In this talk I will explain how counting saddle connections on Veech surfaces stands as a natural ‘higher genus’ analogue to the classical primitive Gauss circle problem. I will then discuss work with Samantha Fairchild and Jon Chaika on pairs of saddle connections.

Benjamin Delarue: Axiom A flows for projective Anosov subgroups
Projective Anosov subgroups generalize fundamental groups of convex cocompact hyperbolic manifolds to higher rank. Key properties of these groups are reflected by the dynamical properties of a suitable "abstract geodesic flow" such as Sambarino's refraction flow or a Gromov-Mineyev flow. In the rank one case of a fundamental group of a convex cocompact hyperbolic manifold M the "abstract geodesic flow" is simply the restriction of the actual geodesic flow on the unit tangent bundle SM to its non-wandering (or "trapped") set K given by the closure of the set of periodic points in SM. I will present joint work with Daniel Monclair and Andrew Sanders in which we show that also in higher rank every projective Anosov subgroup comes with a smooth flow whose restriction to its non-wandering set is an "abstract geodesic flow" in the required sense. This flow represents a higher rank generalization of the geodesic flow which is analytic, contact, and has the Axiom A property. We deduce a general exponential mixing result and the existence of a discrete spectrum of Ruelle-Pollicott resonances with associated (co-)resonant states. If time allows, I will outline applications to pseudo-Riemannian manifolds and Benoist subgroups and might give a glimpse at our ongoing work on the general Anosov case.

Manfred Einsiedler: Disjointness in Number Theory
In recent/ongoing joint work with Aka, Luethi, Michel, Wieser we use an effective version of a relatively simple disjointness result in Ergodic Theory to prove joint equidistribution results in number theory. The method applies to a couple of problems: monomials on low-lying rational points on closed horocycles, monomials in CM-points (conditionally on GRH), and others. We will explain the method and present a few of the applications.

William Hide: Small eigenvalues of hyperbolic surfaces
We study the spectrum of the Laplacian on finite-area hyperbolic surfaces of large volume, focusing on small eigenvalues i.e. those below 1/4. We shall look at the number of small eigenvalues of random hyperbolic surfaces. I will discuss some different constructions of random surfaces and explain recent developments in this area. Based on joint works with Michael Magee and with Joe Thomas.

Min Lee: Murmurations of automorphic forms in archimedean families
In April 2022, He, Lee, Oliver and Pozdnyakov made an interesting discovery using machine learning – a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions. They coined this correlation 'murmurations of elliptic curves'. Naturally, one might wonder whether we can identify a common thread of 'murmurations' in other families of L-functions. In this talk, I will introduce joint works with Jonathan Bober, Andrew R. Booker, David Lowry-Duda, Andrei Seymour-Howell and Nina Zubrilina, demonstrating murmurations in holomorphic modular forms and Maass forms in archimedean families.

Christopher Lutsko: Spectral bounds and temperedness of locally symmetric spaces.
Consider a semi-simple Lie group, $G$ and a discrete torsion-free subgroup $\Gamma < G$. When $G$ has rank one, there is an important, well-understood connection between the growth rate of elements of $\Gamma$, bounds on the spectrum, and temperedness (or more generally properties of the unitary $L^2$ representation (due, in part, to work of Elstrodt and Patterson). In this talk I will present an extension of this connection to higher rank. This is joint work with Tobias Weich and Lasse Wolf.

Jens Marklof: Directional Statistics of Lattice Points and Escape of Mass for Embedded Horospheres
I will discuss escape of mass estimates for $\mathrm{SL}(d,\mathbb{R})$-horospheres embedded in the space of affine lattices, which depend on the Diophantine properties of the shortest affine lattice vector. These estimates can be used, in conjunction with Ratner's theorem, to prove the convergence of moments in natural lattice point problems, including the statistics of directions in lattices, inhomogeneous Farey factions, and the distribution of smallest denominators. Based on joint work with Wooyeon Kim (ETH).

Simon Marshall: Asymptotics of eigenfunctions on locally symmetric spaces
Understanding the asymptotic behaviour of Laplace eigenfunctions with large eigenvalue on a compact locally symmetric space is a well-studied problem in harmonic analysis, with interesting connections to number theory. When the symmetric space is of noncompact type, one expects these eigenfunctions to exhibit chaotic behaviour based on the correspondence principle from mathematical physics. I will present results (some joint with Farrell Brumley) that help us understand to what extent this chaotic behaviour actually occurs.

Jasmin Matz: Towards a symplectic version of Duke's theorem
In generalization of Duke's seminal work on the equidistribution of Heegner points and closed geodesics on the modular surface, one can ask about the distribution of closed torus orbits on more general locally symmetric spaces. The case of $\mathrm{SL}(n)$ over an arbitrary number field was studied by Einsiedler, Lindenstrauss, Michel, and Venkatesh. In joint ongoing work with Farrell Brumley we now study the case of $\mathrm{GSp}(4)$.

Uri Shapira: Badly approximable grids and $k$-divergence
Let $v$ be a point on the $m$-torus and assume it generates a dense subgroup there. A central theme in Diophantine approximation is to analyze and quantify the speed at which this group becomes dense. Vaguely, the torus is split into two sets: The points towards which multiples of the generator approach fast and the points towards which multiples of the generator approach slowly. The latter set, $\mathrm{Bad}(v)$, is called the set of Badly approximable targets for $v$. In this talk I will discuss the measure of $\mathrm{Bad}(v)$ with respect to some very natural probability measures on the torus (algebraic measures) and relate this question to a novel concept in dynamics called "$k$-divergence". The discussion naturally leads to dynamical questions on the space of lattices.