In this talk, I will discuss recent results on the free energy of logarithmically interacting charges in the plane in an external field. Specifically, at a particular inverse temperature $\beta=2$, this system exhibits the distribution of eigenvalues of certain random matrices, forming a determinantal point process. I will explain how the large N expansion of the free energy depends on the geometric and topological properties of the region where particles condensate, considering the disk, annulus, and sphere cases. I will further discuss the conditional Ginibre ensemble as a non-radial example confirming the Zabrodin-Wiegmann conjecture regarding the spectral determinant emerging at the O(1) term in the free energy expansion. This talk is based on joint works with Sung-Soo Byun, Meng Yang, and  Nam-Gyu Kang.

Momentum-based acceleration of first-order optimization methods, first introduced by Nesterov, has been foundational to the theory and practice of large-scale optimization and machine learning. However, finding a fundamental understanding of such acceleration remains a long-standing open problem. In the past few years, several new acceleration mechanisms, distinct from Nesterov’s, have been discovered, and the similarities and dissimilarities among these new acceleration phenomena hint at a promising avenue of attack for the open problem. In this talk, we discuss the envisioned goal of developing a mathematical theory unifying the collection of acceleration mechanisms and the challenges that are to be overcome.

Dyson Brownian motion, the eigenvalues of matrix-valued Brownian motion, has become the most standard and well-established approach to universalities for local (i.e. microscopic) eigenvalue statistics of Hermitian random matrices. When combined with a noble characteristic flow method, it can also help study the eigenvalue statistics on a mesoscopic scale. In this talk, we demonstrate this mechanism via yet another simplification of the proof of local laws for Wigner matrices and discuss some generalities.