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.

In this talk, we will discuss the current state and future prospects of multimodal AI. In particular, we will focus on the key challenges in ensuring reliability and efficiency in multimodal AI, explaining why addressing these factors is crucial for the successful real-world deployment of next-generation intelligent systems.

This talk concerns the classification problem of long-term dynamics for critical evolutionary PDEs. I will first discuss critical PDEs and soliton resolution for these equations. Building upon soliton resolution, I will further introduce the classification problem. Finally, I will also touch on a potential instability mechanism of finite-time singularities for some critical PDEs, suggesting the global existence of generic solutions.