Additional research seminars given by Hong-Bin Chen (IHES) and Shizan Fang (de Bourgogne & NYUSH). 

Schedule: 

Main courses and research seminars from 13—17 May and on 20 May. 

Free discussions from 21—25 May. 

Course 1: 

Lecturers: Giuseppe Cannizzaro and Fabio Toninelli

Title: Large-scale behavior of (super-)critical stochastic PDEs

Abstract: This course will focus on some singular stochastic PDEs (SPDEs) motivated by out-of-equilibrium statistical physics, in particular driven diffusive systems and stochastic interface growth. We will deal with equations that are "critical" or "super-critical", for which scaling or Renormalization group arguments suggest a  Gaussian limit for large space-time scales (with logarithmic corrections to diffusivity in the critical dimension). In particular, this class includes the "Anisotropic KPZ equation" in dimension d=2 and the stochastic Burgers equation in dimension $d\ge 2$. We will explain how a careful analysis of the generator of the processes allows to prove Gaussian scaling limits in the super-critical case, and in the critical case in the weak-coupling regime.

Course 2: 

Lecturers: Francesco Caravenna, Rongfeng Sun and Nikos Zygouras

Abstract: In these lectures, we review recent progress in the study of the stochastic heat equation (SHE) and its discrete analogue, the directed polymer model (DPM), in the critical spatial dimension 2. A phase transition exists on an intermediate disorder scale, with Gaussian (Edwards-Wilkinson) fluctuations in the sub-critical regime. In the critical window, a unique scaling limit has been identified and is called the critical 2d stochastic heat flow (SHF), which gives a meaning to the solution of the SHE in the critical dimension 2 that lies beyond existing solution theories for singular SPDEs. We will outline the proof ideas for these results and explain the key ingredients.

Research talks: 

Speaker: Hong-Bin Chen
Title: Dynamic polymers: invariant measures and ordering by noise
Abstract: Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We consider polymer dynamics in 1+1 dimension as a stochastic gradient flow and we explore the connection of this question with ergodic properties of the dynamics. After establishing the phenomenon of ordering by noise, we prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by the unique infinite volume polymer measure. This is based on a joint work with Yuri Bakhtin. 
Speaker: Shizan Fang
Title: Navier-Stokes equation and geometric interpretation
Abstract: We will give a geometric meaning for the helicity and the strain tensor associated to the Navier-Stokes equation.

Registration  (The deadline is May 7. )