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.