We present recent developments on the quantitative stability of the Sobolev inequalities, as well as the stability of critical points of their Euler–Lagrange equations.  In particular, we introduce our recent joint work with H. Chen (Hanyang University) and J. Wei (The Chinese University of Hong Kong) on the stability of the Yamabe problem, the fractional Lane–Emden equation for all possible orders, and the Brezis-Nirenberg problem.

The talk is divided into two parts. In the first part, we review the concept of phase transition in probability theory and mathematical physics, focusing on the standard +/- Ising model. In the second part, we discover why one may expect metastability in the low-temperature regime, and look at some concrete examples that exhibit this phenomenon.

H. Föllmer introduced in 1981 a version of Itô's formula without any probabilistic assumptions. It has been generalized in several aspects, including pathwise Tanaka's formula, high-order, and functional change-of-variable formula. Its drawbacks and a brief application to mathematical finance will also be presented.

Molecular simulations serve as fundamental tools for understanding and predicting the system of interest at atomic level. It is significant for applications like drug and material discovery, but often cannot scale to real-world problems due to the computational bottleneck. In this seminar, I will briefly introduce this area and recent machine learning algorithms that have shown great promise in accelerating the molecular simulations. I will also introduce some of my recent research in this direction. First work is about structure prediction of metal-organic frameworks using geometric flow matching (or neural ODE on SO(3) manifolds) and (2) simulating chemical reactions / transition paths through RL-like training of diffusion models (or log-divergence minimization between path measures).