Programme

Lundi, 10 juin

  • 9:00-10:15 == Julien Berestycki 1
  • 10:15-10:45 == Pause café
  • 10:45-12:00 == Nina Holden 1
  • 12:00-1:30 == Lunch
  • 1:30-2:45 == Julien Berestycki 2

Mardi, 11 juin

  • 9:00-10:15 == Nina Holden 2
  • 10:15-10:45 == Pause café
  • 10:45-12:00 == Julien Berestycki 3
  • 12:00-1:30 == Lunch
  • 1:30-2:45 == Nina Holden 3

Mercredi, 12 juin

  • Libre

Jeudi, 13 juin

  • 9:00-10:15 == Louigi Addario-Berry 1
  • 10:15-10:45 == Pause café
  • 10:45-12:00 == Louigi Addario-Berry 2
  • 12:00-1:30 == Lunch
  • 1:30-2:45 == Jean-Dominique Deuschel 1

Vendredi, 14 juin

  • 9:00-10:15 == Jean-Dominique Deuschel 2
  • 10:15-10:45 == Pause café
  • 10:45-12:00 == Louigi Addario-Berry 3
  • 12:00-1:30 == Lunch
  • 1:30-2:45 == Jean-Dominique Deuschel 3

Mini-cours

Free boundary problems arising from branching systems with competition

Conférencier: Julien Berestycki (Oxford)

Résumé: In a branching Brownian motion, particles independently move in space as Brownian motions and branch at rate one (imagine a growing cloud of diffusive reproducing particles). This particle system is in duality with the celebrated F-KPP equation, a reaction-diffusion equation which is the prototypical example of a non-linear parabolic PDE. Going back at least to the works of Watanabe, Neveu, McKean and Bramson, the study of this relation has a rich history. In the first part of this lecture I will present some classical results in this vein and some recent developments on the fine asymptotics of the front position for a F-KPP type equation. In a second part, I will focus on some variants of the branching Brownian motion which include sone form of selection. One such example, the so-called N-BBM, keeps the number of particle constant equal to some N by removing the leftmost particle from the system each time a particle wants to branch. I will in particular focus on the question of the hydrodynamic limit of this system when N is large. Finally, I will present some recent results concerning another variant of this model which is sometimes referred to as Brownian bees.

Introduction to Schramm-Loewner Evolutions

Conférencière: Nina Holden (ETH)

Résumé: The Schramm-Loewner evolution (SLE) is a family of random fractal curves which arise as the scaling limit of statistical mechanics models. They are uniquely characterized by two properties known as conformal invariance and the domain Markov property. This mini-course will give an introduction to SLE and some of its basic properties.

Invariance principle for random walks in degenerate random environment

Conférencier: Jean-Dominique Deuschel (TU Berlin)

Résumé: We will be looking at two types of random walks in random environments: the random conductance model and balanced random walks, both in static and dynamical settings. We derive quenched invariance principle and local limit theorems in non elliptic situations showing percolative effects. The environment viewed from the walk will play a central role. While random conductance model deals with reversible walks, for which the invariant distribution is given, in balanced walks the invariant measure is unknown. On the other hand balanced walks are martingales while random conductance model are not. Our method uses both probabilistic (coupling, ergodic theorem, CLT for martingales) and analytical tools (Sobolev inequalities, spectral gaps, maximum principle, Harnack inequalities).

Random trees: algorithms and asymptotics

Conférencier: Louigi Addario-Berry (McGill)