Aktuelles & Events

Velocity averaging is a class of regularity results for the transport equation which says that macroscopic observables attached to the distribution function solution of a kinetic equation are more regular than the distribution function itself. It has important applications to the existence theory for the Boltzmann and Vlasov equations, and to hydrodynamic limits of the Boltzmann equation. This talk will discuss the possibility of applying velocity averaging in the context of quantum dynamics. Our analysis of this problem will lead us to discuss the so-called Madelung quantum hydrodynamic equations and the magnetic or gauge-invariant Wigner transform due to Stratonovich.

In this talk, I will present two results on the dynamics of token representations in Transformer models, using tools from probability and PDE analysis. In the first part, I will discuss the stochastic scaling limit of deep and wide Transformers, and show how noise can lead to synchronization between tokens. In the second part, I will study a class of nonlocal transport equations arising from self-attention and show that their solutions converge to the heat equation, with quantitative estimates that are uniform in time. Together, these results provide a mathematical description of how interactions between tokens can lead to different collective behaviors in deep Transformers.

This mini-course offers a welcoming introduction to the Boltzmann and Landau equations. We begin with the spatially homogeneous Boltzmann equation in the angular cutoff setting, introducing the basics and then presenting an existence result. We then study the grazing-collision limit from the non-cutoff Boltzmann equation to the Landau equation, explaining how frequent small-angle collisions give rise to diffusion in velocity. Finally, we discuss inhomogeneous equations, including renormalised solutions and compactness methods.

Modern neural networks are highly overparameterized, yet successful training can often be achieved within lower-dimensional random subspaces of parameter space. Motivated by this phenomenon, we formalize the intrinsic dimension of a learning problem through random affine intersections with its solution set. This perspective yields a geometric, task-dependent complexity measure that reflects both the complexity of the target function and redundancies in the network parameterization. We develop general tools for bounding the intrinsic dimension and then apply them to the teacher-student setting, obtaining upper and/or lower bounds for shallow and deep ReLU networks. The talk is based on joint work available at https://arxiv.org/abs/2610.39674.