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.

We prove that the twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the annulus, is stable to perturbations. In fact, it is possible to prove the stability of the whole of the lifted dynamics to non-autonomous perturbations, though single particle paths are generically unstable. These all-time stability facts are used to establish a number of results related to the long-time behavior of fluid flows. (Joint work with T. Drivas and T. Elgindi)