The symbol length problem is a longstanding question concerning the Brauer group of a field. In the case of fields of positive characteristic, every Brauer class is split by a finite extension of height 1. This observation suggests a connection between the symbol length problem and the Galois theory of purely inseparable extensions, where the restricted Lie algebra naturally arise. In this talk, we will explore how various symbol length problems in Brauer groups relate to restricted Lie algebras and introduce a moduli-theoretic description of restricted subspaces in a restricted Lie algebra.

Classification of the steady states for 2D Euler equation is a classical topic in fluid mechanics. In this talk, we consider the rigidity of the analytic steady states in  bounded simply-connected domains. By studying an over-determined elliptic problem in Serrin type, we show the stream functions for the steady state are either radial functions or solutions to semi-linear elliptic equations.  This is the joint work with Tarek Elgindi, Ayman Said and Chunjing Xie.

I will introduce the notion of a new equivalence relation on the class of countable discrete groups, called von Neumann orbit equivalence (vNOE). I will also discuss the stability of vNOE under the operations of taking free products and graph products of groups. This is based on a joint work with Aoran Wu.

Let $G_{n,p}$ denote the random $n$-vertex graph obtained by including each edge independently and with probability $p$. Given a graph $F$, let $\mathrm{ex}(G_{n,p},F)$ denote the size of a largest $F$-free subgraph of $G_{n,p}$. When $F$ is non-bipartite, the asymptotic behavior of $\mathrm{ex}(G_{n,p},F)$ is determined by breakthrough work done independently by Conlon-Gowers and by Schacht, but the behavior for bipartite $F$ remains largely unknown.

We will discuss some recent developments that have been made for bipartite $F$, with a particular emphasis on the case of theta graphs.  Based on joint work with Gwen McKinley.

One of the central problems in Arithmetic Statistics is counting number field extensions of a fixed degree with a given Galois group, parameterized by discriminants. This talk focuses on C2 ≀ H extensions over an arbitrary base field. While Jürgen Klüners has established the main term in this setting, we present an alternative approach that provides improved power-saving error terms for the counting function.

Optimization models that incorporate uncertainty and hierarchical structures have attracted much attention in data science. Recent advances in polynomial optimization offer promising methods to certify global optimality for these complex models. In this talk, I will use two-stage stochastic optimization as a major model to demonstrate how polynomial optimization can be efficiently applied to data science optimization under uncertainty.

Drugs of abuse, such as opiates, have been widely associated with enhancing susceptibility to HIV infection, intensifying HIV replication, accelerating disease progression, diminishing host-immune responses, and expediting neuropathogenesis. In this talk, I will present a variety of mathematical models to study the effects of the drugs of abuse on several aspects of HIV infection and replication dynamics. The models are parameterized using data collected from simian immunodeficiency virus infection in morphine-addicted macaques. I will demonstrate how mathematical modeling can help answer critical questions related to the HIV infection altered due to the presence of drugs of abuse. Our models, related theories, and simulation results provide new insights into the HIV dynamics under drugs of abuse. These results help develop strategies to prevent and control HIV infections in drug abusers.

Many data-driven problems in the modern world involve solving nonconvex optimization problems. The large-scale nature of many of these problems also necessitates the use of first-order optimization methods, i.e., methods that rely only on the gradient information, for computational purposes. But the first-order optimization methods, which include the prototypical gradient descent algorithm, face a major hurdle for nonconvex optimization: since the gradient of a function vanishes at a saddle point, first-order methods can potentially get stuck at the saddle points of the objective function. And while recent works have established that the gradient descent algorithm almost surely escapes the saddle points under some mild conditions, there remains a concern that first-order methods can spend an inordinate amount of time in the saddle neighborhoods. It is in this regard that we revisit the behavior of the gradient descent trajectories within the saddle neighborhoods and ask whether it is possible to provide necessary and sufficient conditions under which these trajectories escape the saddle neighborhoods in linear time. The ensuing analysis relies on precise approximations of the discrete gradient descent trajectories in terms of the spectrum of the Hessian at the saddle point, and it leads to a simple boundary condition that can be readily checked to ensure the gradient descent method escapes the saddle neighborhoods of a class of nonconvex functions in linear time. The boundary condition check also leads to the development of a simple variant of the vanilla gradient descent method, termed Curvature Conditioned Regularized Gradient Descent (CCRGD). The talk concludes with a convergence analysis of the CCRGD algorithm, which includes its rate of convergence to a local minimum of a class of nonconvex optimization problems.

As we know, Kaehler-Ricci flow can be reduced to solve a class of  parabolic   complex Monge-Amp\`ere equations for Kaehler potentials and  the solutions usually depend on the Kaehler class of initial metric.   Thus there  gives a  degeneration of Kaehler metrics arising from the Kaehler-Ricci flow.  For a class of $G$-spherical manifolds,   we can  use  the local estimate  of  Monge-Amp\`ere equations as well as  the H-invariant for $C^*$-degeneration  to determine the limit of  Kaehler-Ricci flow after resales.  In particular,  on such manifolds,  the flow will develop the singularities of  type II.  

We present a highly effective algorithmic approach, PMM, for generating differentially private synthetic data in a bounded metric space with near-optimal utility guarantees under the 1-Wasserstein distance. In particular, for a dataset in the hypercube [0,1]^d, our algorithm generates synthetic dataset such that the expected 1-Wasserstein distance between the empirical measure of true and synthetic dataset is O(n^{-1/d}) for d>1. Our accuracy guarantee is optimal up to a constant factor for d>1, and up to a logarithmic factor for d=1. Also, PMM is time-efficient with a fast running time of O(\epsilon d n). Derived from the PMM algorithm, more variations of synthetic data publishing problems can be studied under different settings.

Riemannian geometry, the study of smooth manifolds and how to define distances and angles on them, can be viewed either intrinsically or extrinsically. In this talk, we discuss how Nash unified these views starting with his 1954 paper “C1 Isometric Imbeddings,” where the isometric embedding and the solution to the corresponding system of partial differential equations is constructed as the limit of iteratively defined subsolutions. This technique is cited as one of the first instances of what is now known as convex integration, and is used to construct solutions to many problems in geometry and PDE.