Programme
Horaire
Lundi, 8h30: accueil des participants
| Heure | Lundi | Mardi | Mercredi | Jeudi | Vendredi |
|---|---|---|---|---|---|
| 9:00-10:30 | Cours 1 | Cours 1 | Cours 1 | Cours 3 | Cours 3 |
| 10:30-11:00 | café | café | café | café | café |
| 11:00-12:30 | Cours 2 | Cours 2 | Cours 2 | Cours 4 | Cours 4 |
| 12:30-14:00 | Lunch | Lunch | Lunch | Lunch | Lunch |
| 14:00-15:30 | Exercises 1 | Exercises 1 | Cours 3 | Exercises 3 | |
| 15:30-16:00 | café | café | café | café | |
| 16:00-17:30 | Exercises 2 | Exercises 2 | Cours 4 | Exercises 4 |
Mini-cours 1
Conférencier: Kasra Rafi
Mirzakhani’s integration formula for the volume of moduli space
This is meant to be a self-contained mini-course with the goal of obtaining a recursive formula for the volume of moduli space of Riemann Surfaces. We start with with elementary hyperbolic geometry. We then examine the Fenchel-Nielsen coordinates on Teichmüller space and its relation to Wolpert’s volume form. We then review some of Mirzakhani’s work and the applications of the recursive formula.
Exercises 1Exercises 2
Mini-cours 2
Conférencière: Claire Burrin
Spectrum of the Laplacian and Selberg’s trace formula
The Laplacian plays a special role in studying the geometry of Riemannian manifolds. For instance, a diffeomorphism on a Riemannian manifold is an isometry if and only if it leaves the Laplacian invariant. In this mini-course I will present key elements of the spectral theory of compact hyperbolic surfaces, including Selberg’s trace formula, and — if time permits — Sunada’s method to construct isospectral surfaces in relation to the classical question « can one hear the shape of a drum? » The goal is to illustrate the interlinks between spectrum, orbits of the geodesic flow, coverings, and isometries.
Exercises 1Exercises 2
Mini-cours 3
Conférencier: Timothy Budd
Geometry of random planar maps and genus-0 hyperbolic surfaces
In this mini-course I will explain how some of the combinatorial techniques used in the study of random planar maps, i.e. embedded graphs in the sphere, have natural analogues for genus-0 hyperbolic surfaces with boundaries. In particular, this opens up the opportunity to study statistical properties of geodesic distances in hyperbolic surfaces with many boundaries or cusps sampled from the Weil-Petersson measure.
ExercisesNotes de cours
Mini-cours 4
Conférencier: Joe Thomas
Delocalisation phenomena of Laplacian eigenfunctions on large hyperbolic surfaces
In this mini-course, we will study eigenfunctions of the Laplacian operator and their observed delocalisation features on hyperbolic surfaces of large genus, motivated by the predictions of quantum mechanics for eigenfunctions of large energy. In particular, we will discuss tools that relate features of the eigenfunctions to the geometry of the surface. This will provide us with a framework to study geometric constraints that give rise to delocalisation phenomena such as quantum ergodicity and small sup norms. We will then discuss how typical these kinds of geometries are in the Weil-Petersson model of random hyperbolic surfaces and how they relate to similar observations on large regular graphs.
Exercises