Abstract

There will be four mini-courses of four hours each, and research talks of one hour each during the week.

Mini-courses:

Geometric Inverse Problems, by Colin Guillarmou (Université Paris-Saclay). 

We will explain two rigidity problems for Riemannian manifolds, that consist in the determination of the metric from the lengths of its geodesics. The geodesics we can measure are either geodesics with finite lengths and endpoints on the boundary on a manifold with boundary or closed (periodic) geodesics on a closed manifold. These are called boundary rigidity and marked length spectrum rigidity.
Combining microlocal techniques with energy methods based on the the differential geometry of the unit sphere bundle, these problems can be studied from an analytic point of view.  We shall present some applications of these techniques to certain solutions of these rigidity questions.

Pollicott-Ruelle resonances in hyperbolics dynamics, by Malo Jézéquel (MIT). 

Statistical properties of certain hyperbolic dynamical systems may be studied using functional analytic methods (in particular, spectral theoretic tools). This approach gives rise to the notion of Pollicott-Ruelle resonances. They are "generalized eigenvalues" of an operator associated to the dynamics, and may be used for instance to describe the asymptotics of correlation functions associated to smooth observables.

In this minicourse, I will define Pollicott-Ruelle resonances for smooth Anosov diffeomorphisms and then explain how the asymptotics of correlations is deduced from this notion.

Probabilistic methods in hyperbolic geometry, by Bram Petri (Sorbonne Université).

Hyperbolic manifolds are Riemannian manifolds whose metric is complete and has constant sectional curvature equal to -1. Another way to phrase the latter property is to say that they are locally isometric to hyperbolic space (of the right dimension). The (spectral) geometry of such manifolds plays a role in various domains of mathematics. In this mini course I will talk about how probabilistic methods can help to solve questions on the geometry of hyperbolic manifolds.

Introduction to microlocal analysis, by Jared Wunsch (Northwestern University).

This course will be a "user's guide" to pseudodifferential operators, laying out the motivation for their use and discussing their properties more or less axiomatically, without proof.  Two main directions of application are to the construction of elliptic parametrices (both globally and microlocally) and to studying regularity of solutions to PDE.  We will especially focus on the latter direction.  The end of the class will feature a discussion of propagation of singularities for operators of real principal type, with applications to solvability of PDE, as well as a nod to the recent advances of Vasy on using propagation estimates to construct global Fredholm problems for hyperbolic equations.

Research talks:

Viviane Baladi (Sorbonne Université),  Anisotropic spaces applied to Sinai billiards

Sinai billiards (or dispersive billiards) are (discrete or continuous time) dynamical systems having a simple and natural definition, but offering challenging technical difficulties. These difficulties are due to the presence of singularities, where derivatives blow up. The basic results of the theory were developed nearly 60 years ago by Sinai (ergodicity and mixing of the "physical" invariant measure). Very recently, we were able to adapt functional analytic tools (Ruelle transfer operators acting on anisotropic Banach spaces) to study in particular the properties of other invariant measures. I will briefly describe these tools, and I will present some recent results, ending with the construction of the measure of maximal entropy of generic finite horizon Sinai billiards flows, and some open questions. (Joint work with J. Carrand and M. Demers.)

Yannick Bonthonneau (Université Paris 13), Narrow capture problem

A classical problem in the study of the Brownian motion is to estimate the time it takes to reach an obstacle, starting from a given point. Since the generator of the Brownian motion is the Laplacian, this is equivalent to studying a particular kind of Laplace equation, with nice boundary conditions. A more general problem comes when replacing the brownian motion by a Levy process, which has jumps, instead of continuous trajectories.

While this question has been extensively studied in the case of open sets of R^n, not much was done on manifold. Recently with Chaubet Lefeuvre and Tsou, we investigated the case of stable Levy processes on manifolds. While in the flat case, the generator is a fractional power of the laplacian (and is thus quite explicit), in the general manifold case, it is related to the geodesic flow, and only ``looks like'' the fractional laplacian. Using the machinery of microlocal analysis introduced for the study of Ruelle-Pollicott resonances and geometric inverse problems, we were able to obtain some asymptotics formulæ in the case of negative curvature.

Maxime Fortier-Bourque (Université de Montréal), From sphere packings to extremal problems on hyperbolic surfaces

Joint work with V. Delecroix and P. Zograf.

We will tell how Maxim Kontsevich and Paul Norbury have counted metric  ribbon graphs and how Maryam Mirzakhani has counted simple closed geodesic multicurves on hyperbolic surfaces. Both counts use  Witten-Kontsevich correlators (they will be defined in the lecture with no appeals to quantum gravity).

We will present a formula for the asymptotic count of square-tiled  surfaces of any fixed genus g tiled with at most N squares as N tends  to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large  genus asymptotic analysis of this formula, performed by Amol Aggarwal, and the uniform large genus asymptotics of intersection numbers of  Witten-Kontsevich correlators, proved by Aggarwal, combined with the results of Kontsevich, Norbury and Mirzakhani, allowed us to describe the structure of a random multi-geodesic on a hyperbolic surface of large genus and of a random square-tiled surface of large genus. For instance we show that the number of components of a randon multi-geodesic asymptotically behaves as the number of cycles of a random permutation of 3g-3 elements taken with respect to a very explicit probability distribution.

If time permits, we will then count oriented meanders on surfaces of any genus and an asymptotic probability to get a meander by a random identification of endpoints of a random braid on a two-component  surface of any genus. 

Maxime Ingremeau (Université Côte d'Azur), Semiclassical evolution of Lagrangian states into random waves

In 1977, Berry conjectured that eigenfunctions of the Laplacian on manifolds of negative curvature behave, in the high-energy (or semiclassical) limit, as a random superposition of plane waves. This conjecture, central in quantum chaos, is still completely open. In this talk, we will consider a much simpler situation. On a manifold of negative curvature, we will consider a Lagrangian state (the generalization of a plane wave on a manifold). We aim to show that, when evolved during a long time by the Schrödinger equation, these functions do behave, in the semiclassical limit, as a random superposition of plane waves. We will  show such a result in two situations: when the phase of the Lagrangian state is generic, and when the Laplace-Beltrami operator is perturbed by a small generic potential.

This is based on join work with Alejandro Rivera, and with Martin Vogel

Gabriel Rivière (Université de Nantes), Orthospectrum of convex bodies and Poisson formulas

I will start by decribing two "variants" of the Poisson formula that are due to Guinand and Meyer. I will then show how these two formulas can be interpreted in terms of the orthospectrum (a family of characteristic lengths) of a convex body with respect to some lattice. With that interpretation in mind, Guinand-Meyer's formulas correspond to the case where the convex body is reduced to a point in dimension 3. Using tools from differential geometry and from Fourier analysis, I will explain how this formula can be extended to general (strictly) convex bodies and how it encodes certain geometric quantities (mixed volumes). This is a joint work with Nguyen Viet Dang and Matthieu Léautaud.

Gunther Uhlmann (University of Washington), Microlocal Analysis and Geometric Inverse Problems

This talk is a personal journey through several geometric inverse problems where techniques of microlocal analysis have been useful. These include Calderón's inverse problem, the boundary rigidity and lens rigidity problem, and some inverse problems arising in general relativity.