Asked by some graduate students what my current research is about, I will try to introduce the very basic ideas and definition of Segal--Bargmann transform, which intertwines the Heisenberg and Fock pictures.

A \textbf{repetition} in an edge-colored graph is a path in which the sequence of colors in the first half of the path is identical to that in the second half. In 2002 Alon, Grytczuk, Haluszczak, and Riordan showed that every $k$-ary tree has a repetition-free edge-coloring with at most $4k$ colors. We present a simple procedure for obtaining repetition-free edge-colorings of $k$-ary trees with at most $3k+1$ colors. We will also discuss some related vertex-coloring questions.

In this talk, we discuss the Diederich--Forn\ae ss index in several complex variables. A domain $\Omega\subset\mathbb{C}^n$ is said to be pseudoconvex if $-\log(-\delta(z))$ is plurisubharmonic in $\Omega$, where $\delta$ is a signed distance function of $\Omega$. The Diederich--Forn\ae ss index has been introduced since 1977 as an index to refine the notion of pseudoconvexity. After a brief review of pseudoconvexity, we discuss this index from the point of view of geometric analysis. We will find an equivalent index associated to the boundary of domains and with it, we are able to obtain accurate values of the Diederich--Forn\ae ss index for many types of domains.

Arising frequently in sciences and engineering, ill-posed problems remain a formidable challenge and a frontier in scientific computing because their solutions appear to be infinitely sensitive to data perturbations. Many common algebraic problems are ill-posed, such as matrix rank, singular linear and nonlinear systems, polynomial factorizations and Jordan Canonical Forms. On the other hand, the instability of such problems may be a ``misconception'', as argued by W. Kahan 40 years ago, since the solutions are well behaved when certain structures of the problems are preserved. Furthermore, the hypersensitivity is not random but one-directional: Tiny perturbations can only decrease the singularity of the problem and never increase it. From a geometric perspective, ill-posed problems with a specific structure form a smooth manifold that is embedded in similar manifolds of lower codimensions. Based on this property and the Tubular Neighborhood Theorem, ill-posed algebraic problems can be regularized to remove the instability through two optimization problems: Maximizing the codimension of nearby manifolds and minimizing the distance from the manifold to the data point. In this talk we shall elaborate the computing strategy based on the regularization approach along with algorithms/software for finding accurate solutions of many ill-posed algebraic problems from empirical data.

High dimensional time series data arise in a wide range of disciplines, including finance, signal processing, neuroscience, meteorology, seismology and many other areas. For low dimensional time series there is a well-developed estimation and inference theory. Inference theory in the high dimensional setting has been rarely studied. In this talk, I will give an overview of the work that is proposed to develop and advance statistical inference theory for high dimensional time series data analysis including parameter estimation, construction of simultaneous confidence intervals, prediction, model selection, Granger causality test, hypothesis testing and spectral domain estimation.

We explain a recent work with Huang where we combine various results in complex analysis including Fefferman's invariant theory, the Chern-Moser theory, the Cheng-Yau solution of Kahler-Einstein metrics, to provide an affirmative solution of a conjecture posed by Cheng-Yau. The conjecture stated that the Bergman metric of a bounded strongly pseudoconvex domain is Einstein if and only if the domain is holomorphically equivalent to the ball.

Regularization techniques are widely employed in the solution of inverse problems in data analysis and scientific computing due to their effectiveness in addressing difficulties due to ill-posedness. In their most common manifestation, these methods take the form of penalty functions added to the objective in optimization-based approaches for solving inverse problems. The purpose of the penalty function is to induce a desired structure in the solution, and these functions are specified based on prior domain-specific expertise. We consider the problem of learning suitable regularization functions from data in settings in which precise domain knowledge is not directly available; the objective is to identify a regularizer to promote the type of structure contained in the data. The regularizers obtained using our framework are specified as convex functions that can be computed efficiently via semidefinite programming. Our approach for learning such semidefinite regularizers combines recent techniques for rank minimization problems along with the Operator Sinkhorn iteration. (Joint work with Yong Sheng Soh).

I will discuss stochastic interacting particle systems in the KPZ universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable examples of such systems, i.e., for which certain observables can be written in exact form suitable for asymptotic analysis. Examples include a continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, density limit shapes can be described in an explicit way. We also obtain asymptotics of fluctuations, in particular, around slow bonds and infinite traffic jams caused by slowdowns.