Flexibility and Rigidity in Dynamics and Geometry

动力系统领域旨在了解数学系统如何根据一组规则随时间变化。它是一个活跃和不断增长的数学研究领域，本身就至关重要，并与许多其他领域有着深刻的相互作用。 动力学系统可以模拟各种现象，从人类心脏的运动到疾病在人群中的传播。 由于系统经常表现出混沌行为，通过其潜在的几何结构和动力学不变量的透镜来研究它们是很自然的。 通过研究动力学中的柔性和刚性现象，PI将研究这些不变量与结构特性之间的各种关系，从而揭示光滑动力系统和流形几何的新特征。 因此，动力系统的结构方面将得到澄清，目标是使动力学工具更容易应用于其他数学领域以及其他科学领域。 作为该项目的一部分，PI的目标是让本科生和研究生在柔性和刚性的背景下通过计算机实验探索一些低维系统的行为。该项目的主要目标分为两个不同的方向。首先，PI将开发技术来构建一致双曲系统，例如负弯曲流形上的测地线流和Anosov体积保持的非线性同态，在一个固定的类中具有特定的不变量集合，例如熵和李雅普诺夫指数。同时，PI将确定在一个固定的类中对这些不变量的自然限制，并搜索隐含新的动态和/或几何刚性实例的关系。第二个主要目标是产生对动力学行为进行编码并在新情况下具有良好统计属性的自然度量，重点关注CAT（0）空间上的非共形排斥子和测地线流。 该项目由数学科学部的分析项目和刺激竞争性研究的既定项目（EPSCoR）共同资助。该奖项反映了NSF的法定使命，并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，被认为值得支持。

The field of dynamical systems aims to understand how mathematical systems change over time according to a set of rules. It is an active and growing area of mathematical research that is vital in its own right, and has profound interactions with many other fields. Dynamical systems can model a variety of phenomena, from the motion of the human heart to the spread of disease within a population. Since systems often demonstrate chaotic behavior, it is natural to study them through the lens of their underlying geometric structure and dynamical invariants. By studying flexibility and rigidity phenomena in dynamics, the PI will investigate various relationships between these invariants and structural properties, thereby revealing new features of smooth dynamical systems and the geometry of manifolds. Thus, structural aspects of dynamical systems will be clarified with the goal of making the tools of dynamics more readily applicable in other areas of mathematics, as well as other scientific fields. As part of the proposed project, the PI aims to involve undergraduate and graduate students in exploring the behavior of some low-dimensional systems with computer experiments in the context of flexibility and rigidity.The main goals of the project split into two distinct directions. First, the PI will develop techniques to construct uniformly hyperbolic systems, such as geodesic flows on negatively curved manifolds and Anosov volume-preserving diffeomorphisms, in a fixed class with a particular collection of invariants such as entropies and Lyapunov exponents. Concurrently, the PI will determine the natural restrictions on those invariants in a fixed class and search for relations that imply new instances of dynamical and/or geometric rigidity. The second main goal is to produce natural measures that encode dynamical behavior and have good statistical properties in new cases with a focus on non-conformal repellers and geodesic flows on CAT(0) spaces. This project is jointly funded by the Analysis Program of the Division of Mathematical Sciences and the Established Program to Stimulate Competitive Research (EPSCoR).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.