Paul Bourgade: Random matrix universality
Dates: 22-24 September
Abstract: Following Wigner’s insight that universal statistical laws should govern complex correlated systems, random-matrix statistics are now known (or conjectured) to arise in a wide range of settings, including random linear algebra, random graphs, semiclassical analysis, and number theory. This mini-course will explain the dynamical proof of universality for Wigner matrices, following the method of Erdős, Yau, and collaborators, and will discuss related open problems. Topics will include concentration of measure, local laws, Dyson Brownian motion, and loop equations.
Schedule and location: 22, 23, 24 September at 14.00-16.00 in Room 137
Giuseppe Cannizzaro: Critical and Super-critical non-reversible continuum dynamics: large-scale behaviour and universality
Dates: 5-9 October and 23-27 November
Abstract: The goal of this course is to present an approach to determine the large-scale universal behaviour of a family of stationary statistical mechanics systems at and beyond the critical dimension. These are systems for which scaling or Renormalization group arguments suggest a Gaussian limit for large space-time scales, with logarithmic corrections to diffusivity at the critical dimension, but whose rigorous proof has long eluded mathematicians. Examples to which the approach applies, and that we will discuss in the lectures, include random walks/diffusions in random environment, interacting particle systems and, especially, Stochastic Partial Differential Equations. In particular, the SPDEs we will be concerned with lie beyond the range of applicability of Hairer’s theory of regularity structures and are not expected to admit a local solution theory.
The approach finds its roots in the celebrated work of C. Kipnis and S.R.S. Varadhan (1986) and further developed over the last 40 years, by T. Komorowski, C. Landim, S. Olla, J. Quastel, H.T. Yau and many others. What they realised is that the large-scale behaviour of certain systems can be derived by identifying a suitable martingale approximation of so-called additive functionals of Markov processes, i.e. sums/integrals along their trajectories of suitable functionals of their state.
After recalling the basics of Markov processes and Kipnis-Varadhan theory, we will turn to more recent developments with a special focus on critical and supercritical SPDEs such as the Stochastic Burgers Equation, relevant in the description of driven diffusive systems, the Anisotropic KPZ equation, that arises in the context of random surface growth in 2 dimensions, and the stochastic Navier-Stokes equation, describing the evolution of the velocity field of turbulent flows.
Schedule and location:
1st Week: 6 October 11.00-13.00 Room 005; 7 October 14.00-16.00 Room 133; 9 October 11.00-13.00 Room 134
2nd Week: 23 November 11.00-13.00 Room 134; 25 November 14.00-16.00 Room 133; 27 November 11.00-13.00 Room 134
Hao Shen: Stochastic approach to quantum gauge theories
Dates: 12-23 October
Abstract: TBA
Schedule and location: All in Room 133:
1st Week: 12 October 11.00-13.00; 14 October 14.00-16.00; 16 October 11.00-13.00
2nd Week: 19 October 11.00-13.00; 21 October 14.00-16.00; 23 October 11.00-13.00
Massimiliano Gubinelli: A constructive introduction to quantum field theory
Dates: 26 October - 6 November
Abstract: In these lectures I will review some mathematical problems associated to the definition and construction of quantum field theories. Mainly from the probabilistic viewpoint but also paying some attention to the quantum aspects. The goal is to give a wide perspective on the open problems and opportunities for research, in the era of large language models.
Schedule and location:
1st Week: 26 October 11.00-13.00 Room 136; 28 October 14.00-16.00 Room 133; 30 October 11.00-13.00 Room 136
2nd Week: 4 November 14.00-16.00 Room 133; 5 November 11.00-13.00 Room 136; 6 November 11.00-13.00 Room 136
Thomas Leblé: The infinite-volume limit of 1d log-gases: a statistical physics approach
Dates: 16-27 November
Abstract: The Sine-beta point process describes eigenvalues of large random Hermitian matrices belonging to the universality class of Gaussian beta-ensembles, at microscopic scale, in the bulk of the spectrum. Its mere existence was only proven in the late 2000s, thanks to independent breakthroughs by Valkó–Virág and Killip–Stoiciu.
It is known since the works of Dyson in the 1960s that the eigenvalues of Gaussian ensembles can be interpreted as particles of a certain one-dimensional toy model of statistical physics, the 1d Log-gas. This opens the door to a "statistical physics" perspective on their behavior, which has turned out to be useful. However, this perspective has so far taken the existence of the limiting process for granted.
The goal of these lectures is to sketch how a fully "physical" proof of the existence of Sine-beta would go. This should take us in several directions:
1. Obtaining local laws for the log-gas, down to the microscopic scale, by a careful bootstrap.
2. Deriving the Dobrushin–Lanford–Ruelle (DLR) equations for every limit point of the microscopic process.
3. Proving the "DLR variational principle" and connecting DLR equations to minimizers of a certain "free energy functional".
4. Showing that free energy minimizers are unique, by a convexity argument involving optimal transportation of measures in infinite dimension.
Schedule and location:
1st Week: 16 November 14.00-16.00 Room 133; 18 November 14.00-16.00 Room 133; 20 November 14.00-16.00 Room 134
2nd Week: 23 November 14.00-16.00 Room 134; 24 November 11.00-13.00 Room 136; 26 November 11.00-13.00 Room 136