Consider a sequence of independent and identically distributed SL(2, R) matrices. There are several classical results by Le Page, Tutubalin, Benoist, Quint, and others that establish various forms of the central limit theorem for the products of such matrices. In our work, we generalize these results to the non-stationary case. Specifically, we prove that the properly shifted and normalized logarithm of the norm of a product of independent (but not necessarily identically distributed) SL(2, R) matrices converges to the standard normal distribution under natural assumptions. A key component of our proof is the regularity of the distribution of the unstable vector associated with these products.

Kahler Gradient Ricci Solitons (KGRS) are fundamental in the theory of Ricci flows. This talk's focus is on complex dimension two and will first review the recent beautiful classification of the shrinking possibly non-compact case. Then we'll discuss an approach to detect symmetry applicable to all cases. This differs from and should complement the popular perspective based on asymptotic behaviors. The idea is inspired by Morse-theoretic aspects of symplectic geometry and involves the understanding of singular sets of a moment map. Precisely, we'll show that a KGRS in complex dimension two is an integrable Hamiltonian system; if the system is generic, then it admits a holomorphic 2-torus action.

Robust Statistics is a classical topic that dates back to the seminal work of Huber in the 1980s. In essence, the main goal of the field is to account for the effect of outliers when performing estimation tasks, such as mean estimation. A recent line of research, inspired by the seminal work of Catoni, has revisited some classical problems in robust statistics from a non-asymptotic perspective. The goal of this short seminar series is to introduce the key ideas related to robust estimation and discuss various notions of robustness, including heavy-tailed distributions and adversarial contamination. The primary example will be the mean estimation problem, and if time permits, I will also cover covariance estimation

The classical Sylvester-Gallai theorem says that if a finite set of points in the Euclidean plane has the property that the line joining any two points contains a third point from the set, then all the points must be collinear. More generally, a Sylvester-Gallai type configuration is a finite set of geometric objects with certain "local" dependencies. A remarkable phenomenon is that the local constraints give rise to global dimension bounds for linear SG-type configurations, and such results have found far reaching applications to complexity theory and coding theory.

In this talk we will discuss non-linear generalizations of SG-type configurations which consist of polynomials. We will discuss how the commutative-algebraic principle of Stillman uniformity can shed light on low dimensionality of SG-configurations. I’ll talk about recent progress showing that these non-linear SG-type configurations are indeed low-dimensional as conjectured by Gupta. This is based on joint work with R. Oliveira.

In this talk, I will present some development of the corona problem
of serval complex variables and discuss its relation to the solution of the
Cauchy-Riemann equations. It contains the Carleson and Wol 's solution of
Carleson's corona theorem for one complex variable, a short survey on the
solutions of Cauchy-Riemann equations. Which includes the Hormander's
weighted L2-estimates, sup-norm estimates for the `del-bar-operator', the Berndtsson's conjecture
and its connection to the corona problem in several complex variables.