The classical Moser-Trudinger inequality is a borderline case of Sobolev inequalities and plays an important role in geometric analysis and PDEs in general. Aubin in 1979 showed that the best constant in the Moser-Trudinger inequality can be improved by reducing to one half if the functions are restricted to the complement of a three dimensional subspace of the Sobolev space H1, while Onofri in 1982 discovered an elegant optimal form of Moser-Trudinger inequality on sphere. In this talk, I will present new sharp inequalities which are variants of Aubin and Onofri inequalities on the sphere with or without mass center constraints. Efforts have also been made to show similar inequalities in higher dimensions. We have improved Beckner’s inequality, the higher dimensional counterpart of Onofri’s inequality, for axially symmetric functions when the dimension n = 4, 6, 8. Numerical computations are exploited to provide rigorous proof. I will also present some new results on higher dimensional counterpart of Huber’s isoperimetric inequalities.

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.