Geometry, Probability and their Synergies
Les Diablerets, March 13-17, 2023


Nalini Anantharaman
Ara Basmajian
Jeffrey Brock
Kevin Buzzard
Alessandra Faggionato
Anna Gusakova
Bram Petri
Andrea Seppi
and all GPS PRIDE candidates!

Hugo Parlier
Giovanni Peccati

All lectures will take place in the Hotel des Sources.

For the schedule please click here.
For the poster please click here.

Titles and Abstracts:

Anantharaman: Trace method for random hyperbolic surfaces, and spectral gap of the laplacian

I will be interested in the spectrum of the laplacian for a random compact hyperbolic surface, in the limit of large genus. Starting from the Selberg trace formula, I will present some techniques I develop with Laura Monk to study the first non trivial eigenvalue.

Basmajian: Hyperbolic surfaces and the geodesic flow

Lecture 1will be a basic introduction to hyperbolic geometry including the construction of surfaces with such a geometry, their geometric invariants, and the dynamics of the geodesic flow. Lecture 2 will focus on finding topological, geometric, and analytic conditions for which the geodesic flow exhibits random behavior.

Brock: Weil-Petersson geodesics with the same ending lamination

Buser: Some 3-dimensional hyperbolic polyhedra and manifolds built upon them

Buzzard: Computer proof verification in geometry and probability

Recently, so-called "computer proof assistants" have shown themselves to be capable of understanding certain parts of modern mathematics. However most of the more impressive achievements have been in more algebraic areas. Can computers understand arguments from geometry and probability, areas where arguments are sometimes more "heuristic"? I will give an overview of how things stand right now.

Gusakova: Random polytopes

Random polytope is one of the key random geometrical objects studied in stochastic geometry. In the classical settings a polytope is considered to be a subset of $\mathbb{R}^d$ and it can for example be generated as the convex hull of independent and identically distributed random points. One of most studied models is when the generating random points $X_1,\ldots, X_n$, $n\in \mathbb{N}$ are taken to be independent and uniformly distributed in a given compact convex set $K$. Denote by $K_n$ a random polytope, constructed as a convex hull of points $X_1,\ldots,X_n$. Different geometrical functional of $K_n$ such as the number of $k$-dimensional faces, volume, surface area are of interest, but it appeared to be extremely hard problem to say something precise about the distribution of these random variables. Instead we are interested in the asymptotic results, like asymptotics for the mathematical expectation and variance, and central limit theorems as the number of generating points $n$ tends to infinity. It should be noted that in $\mathbb{R}^d$ the study of the model when $K$ is a smooth convex body or a polytope is basically completed. The new trend in stochastic geometry is to move the classical Euclidean setting to non-Euclidean spaces, like hyperbolic and spherical. It turned out that we may define the analogue of random polytope $K_n$ in hyperbolic and spherical settings and this model admits some new types of asymptotic behavior (sometimes quite unexpected), which we do not expect to appear in Euclidean case.

Faggionato: Hydrodynamic limit of symmetric simple exclusion processes with random conductances on point processes (slides)

Interacting particle systems in random environments are models particularly suited to study transport in disordered systems. The symmetric simple exclusion process (SSEP) represents a fundamental model in this class and corresponds to a system of random walkers interacting only by site-exclusion. We will focus on SSEP's on random weighted graphs in R^d, where the weights (called conductances) equal the jump probability rates for a single walker. The class of graphs we will consider is very large, built on simple point processes and fulfilling the homogenization paradigm in the sense that they describe a medium microscopically disordered but macroscopically homogeneous (from a statistical viewpoint). By means of homogenization and duality we will derive the quenched hydrodynamic behavior of these SSEP's.

Petri: Extremal problems and probabilistic methods in hyperbolic geometry

In these talks, I will speak about various extremal problems in hyperbolic geometry. I will discuss analogies between these problems and questions from both graph theory and the theory of Euclidean sphere packings. Afterwards, I will talk about how ideas from these fields - often probabilistic in nature - can be applied in hyperbolic geometry.

Seppi: Counting surfaces in three-dimensional hyperbolic manifolds.

The goal of these two talks is to give an overview of the role of surfaces, in particular minimal surfaces, in the study of hyperbolic three-manifolds, with emphasis on asymptotic countings. In the first talk, I will give an introduction to quasi-Fuchsian manifolds and explain their importance in hyperbolic geometry. In the second talk, I will focus on minimal surfaces and present recent results on their asymptotic counting initiated by Calegari-Marques-Neves.

Advocaat: Overconvergent p-adic modular forms

Fisac: A Basmajian-type inequality for compact surfaces.

Maini: Covariograms in probability

Mesbah: Quasi-Fuchsian manifolds

Perissinotto: My research in Kummer theory

Smutek: Models of Random Laplace Eigenfunctions and Quantum Approximations

Trauthwein: Normal Approximation of Poisson Functionals

This conference was enabled by a FNR PRIDE Grant and the support of the Department of Mathematics of the University of Luxembourg.