Week 03.03.2024 – 09.03.2024

We study the distribution of singular values for the product of random matrices related to the analysis of deep neural networks. The matrices are similar to the product of sample covariance matrices of statistics, but an important difference is that in statistics the population covariance matrices are assumed to be non-random or random but independent of the random data matrix, while now they are certain functions of the random data matrices (matrices of synaptic weights in the terminology of deep neural networks). The problem was treated recently by J. Pennington et al. assuming that the weight matrices are Gaussian and using the methods of free probability theory. Since, however, free probability theory deals with population covariance matrices that do not depend on data matrices, its applicability to this case must be justified. We use a version of the random matrix theory technique to prove the results of J. Pennington et al. in the general case where the entries of weight matrices are independent identically distributed random variables with zero mean and finite fourth
moment. This, in particular, extends the property of the so-called macroscopic universality to the random matrices in question.

I will outline the recent proof of the (global, unramified) geometric Langlands conjecture, obtained in collaboration with Arinkin, Chen, Gaitsgory, Faergeman, Lin, Raskin and Rozenblyum.

The talk is aimed at non-specialists: in particular, I will highlight some key geometric ingredients that might be useful in other situations.

Vlasov-Poisson type systems are well established kinetic models for dilute plasma. The precise structure of the model differs according to which species of charged particle (electrons or ions) it describes, with the most well known version of the system describing electrons. The model for ions, however, has been studied only more recently, owing to an additional exponential nonlinearity in the equation for the electrostatic potential that creates several mathematical difficulties not encountered in the electron case.

Quantitative stability estimates in Wasserstein distances have played a crucial role in the understanding of equations of Vlasov type. I will discuss recent developments in Wasserstein estimates in the context of the ionic Vlasov-Poisson system, including applications to the well-posedness theory, the quasineutral limit (in which the Debye length tends to zero) and the derivation of the equation from a particle system.

Based on joint works with Mikaela Iacobelli (ETH Zürich).