10:00 to 10:55 
Mark Rudelson (University of Michigan, USA) 
Geometric approach to invertibility of random matrices  III We Will survey recent developments in invertibility of random matrices including breakthrough results of Tikhomirov on random matrices with i.i.d. Bernoulli entries and of Campos, Jenssen, Michelen, and Saharasbudhe on random symmetric matrices. This progress relies on a combination of probabilistic and geometric ideas.



11:15 to 12:10 
Shirshendu Ganguly (UC Berkeley, USA) 
Line ensembles, resampling, and the tangent method The KardarParisiZhang (KPZ) equation is a canonical nonlinear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the onepoint distribution of its analog of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behavior has remained out of reach. In this talk, based on joint work with Milind Hegde, we will discuss a geometric approach, closely connected to the tangent method introduced by ColomoSportiello and rigorously implemented by Aggarwal for the sixvertex model. The approach relies on a Gibbs resampling property of the KPZ equation as well as classical correlation inequalities, and yields a sharp understanding for a large class of initial data.



12:15 to 13:10 
Kavita Ramanan (Brown University, USA) 
The $\psi_2$ behaviour of Schatten balls and its relation to the hyperplane conjecture The hyperplane conjecture in convex geometry is a statement about the volume of a convex body and its hyperplane sections. Taking a measuretheoretic perspective to this problem, Bourgain highlighted the importance of the notion of a $\psi_2$convex body, which captures integrability properties of linear images of the volume measure on the body. Despite this notion being introduced more than a quarter century ago, there are not many examples of such bodies. We describe several results on the $\psi_2$ (or more generally, $\psi_\alpha$) behavior of Schatten balls and their marginals. Along the way, we also establish some properties of the Haar measure on the orthogonal group that may be of independent interest. This is joint work with Grigoris Paouris.



14:30 to 15:25 
Benjamin Dadoun (New York University Abu Dhabi, UAE) 
Monotonicity of the logarithmic energy for random matrices It is wellknown that the semicircle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko−Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas for the logarithmic energy of the mentioned models which can be of independent interest. Joint work with D. Chafaï et P. Youssef.



15:30 to 16:25 
Erwin Bolthausen (University of Zurich, Switzerland) 
A onedimensional spin model with Kac type interaction, and a continuous symmetry. (Online talk) A celebrated problem in quantum mechanics concerns the effective mass of the Frohlich polaron in the strong coupling limit. Feynman gave a path integral formulation which relates it to a three dimensional Brownian motion with an attractive pair interaction of Kac type. The effective mass can be expressed as the inverse of the variance parameter of the selfinteracting Brownian. There is a long standing conjecture about the asymptotic behavior of the effective mass for which Spohn gave a heuristic argument. A key property is that the interaction is shift invariant. Despite of considerable recent progress by Mukherjee, Varadhan, Spohn, Lieb, and Seiringer, the key problem is still open.
We present a much simpler model having a similar probabilistic structure which can be analyzed rigorously, and where the Spohn picture can be analyzed rigorously. No claim is made that the ""true"" polaron behaves in the same way.
This is joint work with Amir Dembo.


