Schedule, Notes/Handouts & Videos

We discuss the problem of global existence of smooth solutions for the space-homogeneous Boltzmann and Landau equations. We analyze how the monotonicity of the Fisher information is used to obtain key a priori estimates.

The aim of my mini-course is to show how recently developed techniques from the analysis of singular stochastic PDEs (in particular the theory of regularity structures) can be used to analyse scaling limits of interacting particle systems. In the end I want to explain a new proof of the emergence of the KPZ equation from the weakly asymmetric simple exclusion processes completed recently (arXiv:2505.00621). To this end I will review some of the key ideas from the theory of regularity structures in the context of the KPZ equation and then show how these can be modified to apply in the discrete setting.

In those introductory lectures, I plan to cover some aspects of numerical methods for kinetic equations and dimension reduction techniques. Designing computational methods for kinetic problems presents significant challenges, primarily due to the curse of dimensionality and the need to accurately capture physical laws across multiple scales. While there are mature numerical methods available, the field continues to evolve rapidly.

 

We will focus on selected topics in kinetic simulations in the talks. In the first part,  we focus on high order deterministic solvers. In particular, we will review the discontinuous Galerkin (DG) finite element method, which is a well-known technique for computing PDEs. It is especially suited for designing structure-preserving deterministic methods for transport dominated problems. In the second part, we will present some work on the design of sparse grid DG methods for computing high dimensional differential equations. We show incorporating ideas of sparse grid and multiresolution can help alleviate the curse of dimensions. In the final part, we present several approaches of dimension reduction techniques based on low-rank methods, reduced basis and machine learning.

In this minicourse, we will describe the main ideas of our recent joint work with Yu Deng (Chicago) and Xiao Ma (Michigan) aimed at giving the rigorous derivation of the Boltzmann equation from Newton's laws on colliding particle systems, for arbitrarily long times. This leads to the full execution of the so-called "Hilbert’s Program", proposed in Hilbert's sixth problem from 1900, and aimed at deriving the fundamental equations of fluid dynamics from Newton’s laws, with Boltzmann’s equation as an intermediate step. The result follows parallel progress by Yu Deng and myself in the wave setting, where colliding particles are replaced by interacting waves, which we will briefly discuss as well.

In recent years rough paths theory has taken a very applied turn, primarily focusing on data analysis and machine learning. To the best of my knowledge, one cannot see (yet) this strong trend in rough PDEs or regularity structures. In this talk I will thus explore some avenues for the application of rough PDEs in various contexts. (Time permitting) I will cover control, finance, identification, and numerics for PDEs, reinforcement learning, and image processing. I will try to provide a broad picture, without entering in technical details.

In this talk, I will present convergence results for scaled solutions of the Boltzmann equation in open-system settings, such as non-isothermal boundary problems. I will also discuss the case when the limiting fluid solutions lack uniqueness, highlighting recent progress on convergence toward less regular fluid behavior.

The aim of my mini-course is to show how recently developed techniques from the analysis of singular stochastic PDEs (in particular the theory of regularity structures) can be used to analyse scaling limits of interacting particle systems. In the end I want to explain a new proof of the emergence of the KPZ equation from the weakly asymmetric simple exclusion processes completed recently (arXiv:2505.00621). To this end I will review some of the key ideas from the theory of regularity structures in the context of the KPZ equation and then show how these can be modified to apply in the discrete setting.

In those introductory lectures, I plan to cover some aspects of numerical methods for kinetic equations and dimension reduction techniques. Designing computational methods for kinetic problems presents significant challenges, primarily due to the curse of dimensionality and the need to accurately capture physical laws across multiple scales. While there are mature numerical methods available, the field continues to evolve rapidly.

We will focus on selected topics in kinetic simulations in the talks. In the first part,  we focus on high order deterministic solvers. In particular, we will review the discontinuous Galerkin (DG) finite element method, which is a well-known technique for computing PDEs. It is especially suited for designing structure-preserving deterministic methods for transport dominated problems. In the second part, we will present some work on the design of sparse grid DG methods for computing high dimensional differential equations. We show incorporating ideas of sparse grid and multiresolution can help alleviate the curse of dimensions. In the final part, we present several approaches of dimension reduction techniques based on low-rank methods, reduced basis and machine learning.