Rigid Structures and Statistical Properties of Smooth Systems

这个项目的目的是发现动力系统领域的新现象。动力系统(简称“动力学”)是对运动的研究，特别是由一套不变的规则所规定的运动，例如控制行星运动的力。动力学中众所周知的实验现象，如混沌轨迹与稳定运动相结合，已经在实验上观察到了，但从理论上还远未完全理解。该项目将涉及几个主题。第一个是动力系统的稳定性--即初始条件甚至规则本身的微小扰动如何影响进化的结果。理解稳健的稳定机制是一项基本的追求，研究人员已经发现了几种新的此类机制。第二个主题是概括性，或者说粗略地理解一个典型系统中存在哪些动力学特征。第三个主题是刚性，研究动力系统的对称性和那些具有最佳对称性的系统，即所谓的动力学理想晶体。该项目的一个重要方面是进一步促进数学和邻近科学界之间的互动，例如物理学。PI已经在围绕粒子加速器设计的问题上进行了合作，目前正在与一名物理学家合作研究黑洞出现背后的量子动力学。此外，国际数学家协会还举办了几次关于动力学的公开讲座，并在大众媒体上发表了有关数学家工作的文章。国际和平协会将在未来几年扩大这些活动。这个项目为本科生和研究生提供了研究培训的机会。这个项目从一般的角度来考虑光滑动力系统中的问题，特别是关于某些叶状动力学的一般性的问题，到局部的问题，集中在特定的群体作用族的刚性上。这些问题的动机是众所周知的猜想，也是发现和探索新的动力学现象的愿望。第一个问题围绕着玻尔兹曼最初的遍历假设，以及Pugh和Shub关于稳定遍历的现代和相关猜想。它们解决的基本问题是，什么时候人们可能会期望一个动力系统是遍历的。一个仍然悬而未决的重要问题是C1Pugh-Shub猜想的辛版本，研究人员将对其进行攻击。证明这一猜想的策略涉及双曲和部分双曲动力学以及扩展叶的有趣和及时的研究方面。第二个项目研究了部分双曲系统的不稳定叶的拓扑和统计性质，特别是具有部分双曲分裂的Anosov微分同胚。第三个项目涉及部分双曲阿贝尔作用量的刚性性质。在这里，超完整力学小组的作用起着重要作用：在所考虑的作用中，环境、部分双曲动力学和超完整力学小组的联合作用受到某些可解群的约束，上述已知刚性结果适用于这些可解群。该奖项反映了NSF的法定使命，并通过使用基金会的智力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 aim of this project is to discover new phenomena in the area of dynamical systems. Dynamical systems ("dynamics," for short) is the study of motion, and in particular motion that is dictated by an unchanging set of rules, such as the forces controlling planetary motion. Well-known experimental phenomena in dynamics such as chaotic trajectories combined with stable motion have been observed experimentally but are far from being fully understood from a theoretical perspective. The project will address several themes. The first is the stability of dynamical systems - that is, how small perturbations in initial conditions and even in the rules themselves affect the outcome of the evolution. Understanding robust mechanisms for stability is a fundamental pursuit, and the investigator has already discovered several novel such mechanisms. The second theme is genericity, or loosely to understand what dynamical features are present in a typical system. The third theme is rigidity, the study of symmetries of dynamical systems and those systems with optimal symmetries, the so-called ideal crystals of dynamics. An important aspect of the project is to further interaction between mathematical and adjacent scientific communities, such as physics. The PI has already collaborated on questions surrounding the design of particle accelerators and is currently collaborating with a physicist on studying the quantum dynamics behind the emergence of black holes. Furthermore, the PI has given several public lectures on dynamics and has written in the popular press about the work of mathematicians. The PI will expand these activities in the coming years. The project provides research training opportunities for undergraduate and graduate students.This project considers questions in smooth dynamical systems all the way from a general perspective, in particular those about genericity of certain foliation dynamics, to a local one, focused on the rigidity of specific families of group actions. These questions are motivated by well-known conjectures, but also by the desire to discover and explore new dynamical phenomena. The first circle of questions centers around Boltzmann’s original ergodic hypothesis as well as the modern and related conjectures of Pugh and Shub about stable ergodicity. The basic question they address is when one might expect a dynamical system to be ergodic. An important question that remains open is the symplectic version of the C1 Pugh-Shub conjecture, which the investigator will attack. The strategy to prove this conjecture involves interesting and timely aspects of the study of hyperbolic and partially hyperbolic dynamics and expanding foliations. A second project investigates the topological and statistical properties of the unstable foliations of partially hyperbolic systems, and in particular Anosov diffeomorphisms with a partially hyperbolic splitting. A third project concerns the rigidity properties of partially hyperbolic abelian actions. Here the action of the su-holonomy group plays an important role: in the actions considered, the joint action of the ambient, partially hyperbolic dynamics and the su-holonomy group are constrained by certain solvable groups for which known rigidity results described above hold.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.