Geometry and Dynamics of K3 Surfaces

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The goal of research in dynamical systems is to understand the long-term behavior of a structure that changes according to some predetermined law. The structures and the laws come from diverse fields such as physics, economics, biology, to name a few, and as a consequence dynamical systems pervade most areas of science and its applications. Among dynamical systems, area-preserving maps are both ubiquitous and poorly understood. This project investigates area-preserving maps and their geometry on a class of spaces called K3 surfaces. Such dynamical systems serve as basic models for a broad class of situations and intertwine unpredictability (chaos) with tame, predictable behavior. Systems exhibiting only unpredictability, as well as systems exhibiting only tame behavior, are by now well-studied and the goal of this project is to understand the boundary and coexistence of these two extremes.In one direction, the PI will study the dynamics in moduli spaces of K3 surfaces. Moduli spaces parametrize all possible objects of a given type and are fundamental tools in mathematics and theoretical physics. Dynamics in moduli spaces describes how the geometry of the surface changes and, consequently, leads to an understanding of the dynamics on the surface itself. Part of the research program is based on earlier developments in homogeneous and Teichmüller dynamics, following analogies between K3 and Riemann surfaces. The geometry of K3 surfaces is controlled by Ricci-flat metrics, which are solutions to Monge-Ampère partial differential equations. The PI will relate these equations to more dynamical invariants, such as Lyapunov exponents and entropy. Additionally, the PI will study non-Archimedean versions of these questions and will develop the necessary tools in non-Archimedean dynamics.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.