One of the main topics in Random Matrix Theory(RMT) is universality. In this talk, we focus on edge universality in Wigner matrices. With overall description of significant findings including spiked Wigner matrix and BBP transition, we introduce our recent topic, fluctuations of the largest eigenvalues of transformed spiked Wigner matrices. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes. This is a joint work with Prof. Ji Oon Lee (KAIST).

In undergraduate PDE course, one may have learned that the (classical) diffusion equation can be expressed as $u_t=D \Delta u$, where $D$ is a constant diffusivity. This is true for homogeneous environment. However, for (spatially) heterogeneous environment, $D$ is no longer a constant, and diffusion phenomena in those environments such as fractionation, or Soret effect, cannot be explained with the classical diffusion equation. In this talk, I will first discuss how to model and derive some of the diffusion equations in heterogeneous environment by using basic random walk theory. We will see that the heterogeneity of components, such as speed, walk length, sojourn time, etc, can explain the diffusion phenomena. Then, I will give some specific examples how such models can be applied in science, based on my recent works.

Graph coloring is one of the central topics in graph theory, and there have been extensive studies about graph coloring and its variants. In this talk, we focus on the structural and algorithmic aspects of graph coloring together with their interplay. Specifically, we explain how local information on graphs can be transformed into global properties and how these can be used to investigate coloring problems from structural and algorithmic perspectives. We also introduce the notion of dicoloring, a variant of coloring defined for directed graphs, and present our recent work on dicoloring for a special type of directed graph called tournaments.