Dynamics and Hodge theory: Uniformization and Bialgebraic Geometry

Research in dynamical systems aims to understand the long-term behavior of a structure that changes according to some prescribed law. The structure and the law can come from diverse fields such as physics, economics, biology, etc. and as a consequence dynamical systems pervade most areas of science and its applications. The PI will study a broad class of low-dimensional dynamical systems using tools from other areas of mathematics, notably algebraic geometry and Hodge theory. The tools and techniques that will be developed will cross-fertilize these mathematical disciplines, addressing longstanding problems and opening new directions of investigation. The research program will also be suitable for engaging and training early career mathematicians, including graduate students and postdocs. In one direction, the PI will study the relationship between Hodge theory and a class of dynamical systems called Anosov representations. These connections were only recently unraveled by the PI and promise to enrich both fields. In particular, longstanding obstacles related to the mysterious differential geometry of period maps in Hodge theory can be overcome by considering other target spaces in flag manifolds, suggested by dynamics, and which allow for uniformization results of a new kind. In a second direction, the PI will study dynamics in moduli spaces of translation surfaces with new tools coming from o-minimality and transcendence theory. These tools will be used to shed light on classification problems that are central to the subject. In addition, existing finiteness results will be made effective and algorithms will be devised to efficiently compute orbit closures. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.