# Hausdorff-Kolloquium 2013

Date: April 24, 2013 - July 10, 2013

Venue: Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

## Wednesday, April 24

 15:15 Jean Bertoin (Zürich): Giant and almost giant clusters for percolation on large trees. 16:45 Jürg Kramer (Humboldt, Berlin): Mathematical concepts from school to current research: an example.

## Wednesday, June 12

 15:15 Gérard Laumon (Paris Sud, Orsay): On the counting of Hitchin bundles 16:45 Laure Saint-Raymond (ENS, Paris): The irreversibility in gas dynamics, a matter of probability

## Wednesday, July 10

 15:15 Martin Hairer (Warwick): Dynamics near criticality 16:45 Nalini Anantharaman (Paris Sud, Orsay): Quantum Ergodicity on Large Regular Graphs

## Abstracts:

##### Nalini Anantharaman: Quantum Ergodicity on Large Regular Graphs

"Quantum ergodicity" usually deals with the study of eigenfunctions of the Laplacian on Riemannian manifolds, in the high-frequency asymptotics. The rough idea is that, under certain geometric assumptions (like negative curvature), the eigenfunctions should become spatially uniformly distributed, in the high-frequency limit. I will review the many conjectures in the subject, some of which have been turned into theorems recently. Physicists like Uzy Smilansky or Jon Keating have suggested looking for similar questions and results on large (finite) discrete graphs. Take a large graph $G=(V, E)$ and an eigenfunction $\psi$ of the discrete Laplacian -- normalized in $L^2(V)$. What can we say about the probability measure$|\psi(x)|^2$ (x\in V)? Is it close to uniform, or can it, on the contrary, be concentrated in small sets? I will talk about ongoing work with Etienne Le Masson, in the case of large regular graphs.

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##### Jean Bertoin: Giant and almost giant clusters for percolation on large trees

Motivated by a celebrated work of Erdös and Rényi, we consider Bernoulli bond percolation on a tree with size $n \gg 1$, where the parameter of percolation depends on the size of the tree.
Our purpose is to investigate the asymptotic behavior of the sizes of the largest clusters for appropriate regimes.
We shall first provide a simple characterization of tree families and percolation regimes which yield giant clusters, answering a question raised by David Croydon.
In the second part, we will review briefly recent results concerning two natural families of random trees with logarithmic heights, namely recursive trees and scale-free trees.
We shall see that the next largest clusters are almost giant, in the
sense that their sizes are of order $\frac{n}{\ln{n}}$, and obtain precise limit theorems in terms of certain Poisson random measures. A common feature in the analysis of percolation for these models is that, even though one addresses a static problem, it is useful to consider dynamical versions in which edges are removed, respectively vertices are inserted, one after the other in certain order as time passes.

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##### Martin Hairer: Dynamics near criticality

Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed. I will give an overview of a new theory of "regularity structures" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of "renormalisation group techniques" in this context. At the technical level, the main novel idea involves a complete rethinking of the notion of "Taylor expansion" at a point for a function or even a distribution. The resulting structure is useful for encoding "recipes" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.

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##### Jürg Kramer: Mathematical concepts from school to current research: an example

In our talk we plan to illustrate by means of the well known example of the irrationality of the square root of 2 (which is usually discussed in grades 8 or 9 in schools to motivate the real numbers) how this topic provides the foundations of far reaching concepts in number theory beginning with the (classical) study of rational points on algebraic curves and leading to current developments in Arakelov geometry to study cycles on higher dimensional algebraic varieties.

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##### Gérard Laumon : On the counting of Hitchin bundles

On a curve over a finite field, the total mass of vector bundles (of a given rank and of a given degree) is finite. A beautiful formula established by Siegel, expresses this mass in terms of values of the zeta function of the curve.

On the opposite, the total mass of the Hitchin bundles (also called Higgs bundles or Hitchin pairs) on the curve is infinite and we have no analog of the Siegel formula. Nevertheless there are only finitely many semistable Hitchin bundles and it makes sense to try to find a formula for their total mass.

With Pierre-Henri Chaudouard, we apply to this question some techniques developed by Langlands and Arthur for the trace formula. Our main motivations are the conjecture of Hausel and Rodriguez-Villegas, and the understanding of the nilpotent part of the Arthur-Selberg race formula.

In the talk I will present the method that we use and some results that we have obtained.

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##### Laure Saint-Raymond: The irreversibility in gas dynamics, a matter of probability

The goal of this lecture is to present a derivation of the Boltzmann equation starting from the hamiltonian dynamics of particles in the Boltzmann-Grad limit, i.e. when the number of particles $N\to\infty$ and their size $\eps \to 0$ with $N\eps^2 = 1$. We will especially discuss the origin of irreversibility and the phenomenon of relaxation towards equilibrium, which are apparently paradoxical properties of the limiting dynamics.

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