A Comprehensive Program in Rigidity

这一研究项目的重点主要在双曲动力系统领域，这是数学的一个子领域，涵盖了随着时间发生变化的广泛的现实世界系统。这些系统的复杂性各不相同，从简单机械系统的自由运动到封闭容器中气体的行为。双曲动力学研究的是这类系统的“平均”性质和长期行为。该项目将解决双曲动力学中的基本问题，这些问题可能会潜在地影响物理、化学和工程等领域，在这些领域，混沌动力系统自然会出现。此外，该项目还有一个教育部分，并为研究生的培训提供支持。第一级动力学中的刚性计划旨在从先验的较弱形式的等价(如轨道等价或拓扑共轭)建立更强形式的等价(如光滑共轭)，假设通常与周期轨道有关的障碍(如周期轨道周期或周期点上的特征值数据)消失。最近PI和合作者提出的想法开辟了许多新的领域，特别是对于更高维的系统。PI将探索这些在动力学中实现刚性的新途径。更重要的是，这些想法中的一些还可以在邻近地区取得进展。因此，PI计划将刚性计划扩展到平滑动力学问题之外，在这些问题中，平滑动力学的工具可能会被转移。例如，PI计划在负曲线流形上追求标记和加权的标记长度谱刚性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The focus of this research project mainly lies within the field of hyperbolic dynamical systems, a subfield of mathematics which encompasses a wide range of real-world systems that undergo changes over time. These systems can vary in complexity, ranging from the free motion of a simple mechanical system to the behavior of gas in a closed vessel. Hyperbolic dynamics studies the properties and long-term behavior "on average" of such systems. The project will address fundamental questions in hyperbolic dynamics that could potentially impact areas such as physics, chemistry, and engineering, where chaotic dynamical systems naturally arise. Additionally, the project has an educational component and provides support for the training of graduate students.The rigidity program in rank one dynamics aims to establish a stronger form of equivalence (such as smooth conjugacy) from a priori weaker forms of equivalence (such as orbit equivalence or topological conjugacy), assuming the vanishing of obstructions that are usually associated with periodic orbits (such as periods of periodic orbits or eigenvalue data at periodic points). Recent ideas proposed by the PI and collaborators open up a lot of new ground, especially for higher dimensional systems. The PI will explore these new avenues for rigidity in dynamics. Even more importantly, some of these ideas also enable progress in neighboring areas. Accordingly, the PI plans to expand the rigidity program beyond smooth dynamics questions, where tools of smooth dynamics could potentially be transferred. For example, the PI plans to pursue marked and weighted marked length spectrum rigidity on negatively curved manifolds.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.