I will discuss how Gibbs measures concentrate and exhibit two distinct central limit theorems around multi-vortex manifold in QFT, especially in comparison with point vortices in incompressible fluids/Coulomb gases. Joint work with Martin Hairer.

We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. This talk is based on the joint work with Kunlun Qi.

Inverse scattering problems aim to identify the geometric and material properties of scatterers from measured data. Despite their wide range of applications, these problems are inherently nonlinear and ill-posed. In this talk, we introduce the basics of inverse scattering problems, with a particular focus on acoustic obstacle scattering governed by the Helmholtz equation. After a brief overview of inverse problems, we discuss several types of inverse scattering problems and the main challenges arising in inverse obstacle scattering. We then study some commonly used reconstruction methods and approaches for these problems. In particular, we present layer potential theory, which serves as a fundamental tool in the analytical study of inverse problems.

The Korteweg-de Vries-Burgers (KdVB) equation is a fundamental model capturing the interplay of nonlinearity, viscosity (dissipation), and dispersion, with broad physical relevance. It is well known that the KdVB equation admits traveling wave solutions, called viscous-dispersive shocks. These shock profiles are monotone in the viscosity-dominant regime, while they exhibit infinitely many oscillations when dispersion dominates. In this talk, we study the stability of such viscous-dispersive shocks, focusing on an L2 contraction property under arbitrarily large perturbations, up to a time-dependent shift. We begin with viscous shocks of the viscous Burgers equation (i.e., the KdVB equation without dispersion), then treat monotone viscous-dispersive shocks and finally address oscillatory shocks. We also present detailed structural properties of the oscillatory profiles. This is joint work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).