The Stefan problem is a free boundary problem describing the interface between water and ice. It has PDE and probabilistic aspects. We discuss an approach to this problem, based on optimal transport theory. This approach is related to the Skorokhod problem, a classical problem in probability regarding the Brownian motion.

For a uniformly supersonic flow past a convex cornered wedge with the pressure being given for the surrounding quiescent gas at the downstream, as shown in experimental results, it is expected to form a shock followed by a contact discontinuity, which is also called the jet flow. By the shock polar analysis, it is well-known that there are two possible shocks, one a strong shock and the other one a weak shock. The strong shock is always transonic, while the weak shock could be transonic or supersonic. We prove the global existence, asymptotic behaviors, uniqueness, and stability of the subsonic jet with a strong transonic shock under the perturbation of the upstream flow and the pressure of the surrounding quiescent gas, for the two-dimensional steady full Euler equations.

We consider a family of nonlocal diffusion equations with a prescribed equilibrium state, which includes the fractional heat equation as well as a nonlocal equation of Fokker-Planck type. This family of equations will be shown to arise as the gradient flow of the relative entropy with respect to a version of the nonlocal Wasserstein metric introduced by Erbar. Such equations may also be viewed as the evolutionary Gamma-limit of a certain sequence of heat flows on discrete Markov chains. I will discuss criteria for existence, uniqueness, and stability of solutions, and sufficient criteria on the equilibrium state which ensure fast convergence to equilibrium.

In this talk, we consider nonlinear elliptic equations of the $p$-Laplacian type with lower order terms which involve nonnegative potentials satisfying a reverse H\"older type condition. We establish interior and boundary $L^q$ estimates for the gradient of weak solutions and the lower order terms, independently, under sharp regularity conditions on the coefficients and the boundaries. In addition, we prove interior estimates for Hessian of strong solutions and the lower order terms for nondivergence type elliptic equations. The talk is based on joint works with Jihoon Ok and Yoonjung Lee.

The spectrum of a general non-Hermitian (non-normal) matrix is unstable; a tiny perturbation of the matrix may result in a huge difference in its eigenvalues. This instability is often quantified as eigenvalue condition numbers in numerical linear algebra or as eigenvector overlap in random matrix theory. In this talk, we show that adding a smoothly random noise matrix regularizes this instability, by proving a nearly optimal upper bound of eigenvalue condition numbers. If time permits, we will also discuss the effect of the noise matrix on a macroscopic scale in terms of the Brown measure of free circular Brownian motion. This talk is based on joint works with László Erdős.

We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

The talk will focus on a part of the frontier of our current understanding of nonlinear stability of traveling waves of partial differential equations, especially on how spectral stability implies nonlinear stability and which kind of dynamics may be expected. We shall highlight main expected difficulties related to the stability of discontinuous waves of hyperbolic systems, and show a few significant steps obtained by the speaker with respectively Vincent Duchêne (Rennes), Gregory Faye (Toulouse) and Louis Garénaux (Karslruhe).

In this lecture, we aim to delve deep into the emerging landscape of 'Foundation Models'. Distinct from traditional deep learning models, Foundation Models have ushered in a new paradigm, characterized by their vast scale, versatility, and transformative potential. We will uncover the key differences between these models and their predecessors, delving into the intricate mechanisms through which they are trained and the profound impact they are manifesting across various sectors. Furthermore, the talk will shed light on the invaluable role of mathematics in understanding, optimizing, and innovating upon these models. We will explore the symbiotic relationship between Foundation Models and mathematical principles, elucidating how the latter not only underpins their functioning but also paves the way for future advancements.

In this talk, we study the non-cutoff Boltzmann collision kernel for the inverse power law potentials $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $d=3$. We will study the formal derivation of the non-cutoff collision kernel. Then we will prove the limit of the non-cutoff kernel to the hard-sphere kernel and check the angular singularity would vanish. We will also see precise asymptotic formulas of the singular layer near $\theta\simeq 0$ in the limit $s\to \infty$. Consequently, we will also see that solutions to the homogeneous Boltzmann equation converge to the respective solutions weakly in $L^1$ globally in time as $s\to \infty$.

In physics, Bohr’s correspondence principle asserts that the theory of quantum mechanics can be reduced to that of classical mechanics in the limit of large quantum numbers. This rather vague statement can be formulated explicitly in various ways. In this talk, focusing on an analytic point of view, we discuss the correspondence between basic inequalities and that between measures. Then, as an application, we present the convergence from quantum to kinetic white dwarfs.