Geometry and Dynamics of K3 Surfaces

动力系统研究的目标是了解一个结构的长期行为，它根据某种预定的规律发生变化。动力系统的结构和定律来自于不同的领域，如物理学、经济学、生物学等，因此动力系统渗透到科学及其应用的大多数领域。在动力系统中，面积保持映射是普遍存在的，但人们对它的理解却很少。这个项目研究一类称为K3曲面的空间上的面积保持映射及其几何。这样的动力学系统作为一个基本模型，为广泛的一类情况和international不可预测性（混沌）与驯服，可预测的行为。只表现出不可预测性的系统，以及只表现出驯服行为的系统，到目前为止已经得到了很好的研究，本项目的目标是了解这两个极端的边界和共存。在一个方向上，PI将研究K3曲面模空间中的动力学。模空间将给定类型的所有可能对象参数化，是数学和理论物理学中的基本工具。模空间中的动力学描述了表面的几何形状如何变化，从而导致对表面本身的动力学的理解。研究计划的一部分是基于均匀和Teichmüller动力学的早期发展，遵循K3和Riemann曲面之间的类比。K3曲面的几何形状由Ricci平坦度量控制，这是Monge-Ampère偏微分方程的解。PI将把这些方程与更多的动力学不变量联系起来，比如李雅普诺夫指数和熵。此外，PI将研究这些问题的非阿基米德版本，并将在非阿基米德动力学中开发必要的工具。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

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.