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.

Hyperbolicity is a fundamental concept that connects differential geometry and algebraic geometry. It is in general very hard to determine whether a given manifold or variety is hyperbolic or not. A key tool for verifying hyperbolicity is symmetric differentials; more precisely, the positivity of the cotangent bundle. In this talk, I will introduce various notions of hyperbolicity and explore their geometric properties. I will also discuss how the cotangent bundle, or more generally the syzygy bundle, plays a crucial role in this context.

In this talk, we explore some ordinary and partial differential equations (ODEs and PDEs) in a class of completely integrable systems. We begin by introducing Hamiltonian systems in classical mechanics and their integrability. We then discuss completely integrable ODEs and introduce the Lax pair formulation, a powerful framework for analyzing complete integrability. As a concrete example, we examine the classical Calogero-Moser system, a well-known completely integrable many-body system with remarkable mathematical properties. We then investigate the Calogero-Moser derivative nonlinear Schrödinger equation (CM-DNLS), which is a completely integrable PDE that arises as the continuum limit of the classical Calogero-Moser system. Finally, we present recent developments in the study of CM-DNLS, such as well-posedness and long-time dynamics.