CAREER: New Methods and Applications for Smooth Rigidity of Algebraic Actions

动力系统领域起源于微分方程式和天体力学。它研究受运动规律约束的系统的长期行为。刚性现象的研究是动力系统的中心主题之一，应用于数论、几何和数学物理。粗略地说，如果系统的各种属性在适当的修改下得到保留，那么一个动作就是刚性的。到目前为止，人们对具有强混沌特性的系统已经有了很好的了解。然而，在许多感兴趣的系统中，只能观察到微弱的混沌行为。这个项目的目标是开发新的工具来研究这类系统，然后将这些结果应用于研究其他数学领域，如数论和表示论。该研究计划还包括教育和推广活动，包括对本科生、研究生和博士后助理的培训，以及促进不同领域的女性研究人员之间的合作。拟议的研究计划由几个连贯的项目组成，范围从动力系统、李群的调和分析、表象理论和数论。主要研究人员计划建立上循环刚性，并利用表示理论研究一大类代数、部分双曲和抛物型作用的(扭曲)上同调方程。抛物作用的研究结果和技巧将被应用于数论，以研究某些半直积群的齐次空间上的某种酉幂等流和映射的有效等价分布。主要研究人员还将结合KAM方法和表示理论来研究一般的部分双曲和抛物型代数作用，以研究局部刚性。将探索一大类几何性质与现有例子截然不同的新例子。这一奖项反映了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 field of dynamical systems originated from differential equations and celestial mechanics. It studies the long run behavior of a system subject to laws of motion. The study of rigidity phenomenon is one of the central themes in dynamical systems, with applications to number theory, geometry, and mathematical physics. Roughly speaking, an action is rigid if various properties of the system are preserved under appropriate modifications. So far, systems with strong chaotic properties have been well understood. In many systems of interest however, only weak chaotic behaviors can be observed. The goal of this project is to develop new tools to study such systems and then apply these results to study other areas of mathematics, such as number theory and representation theory. The research plan is complemented by educational and outreach activities involving the training of undergraduates, graduate students, and postdoctoral associates, and fostering collaborations among female researchers in different areas.The proposed research plan consists of several coherent projects, ranging from dynamical systems, harmonic analysis for Lie groups, representation theory and number theory. The principal investigator plans to establish cocycle rigidity and to study (twisted) cohomological equations for a large class of algebraic partially hyperbolic and parabolic actions by using representation theory. The results and techniques from the study of parabolic actions will be applied to number theory for the study of effective equidistribution for certain unipotent flows and maps on some homogenous spaces of semidirect product groups. The principle investigator will also combine KAM approach and representation theory to general partially hyperbolic and parabolic algebraic actions to study local rigidity. A large class of new examples whose geometric properties are distinctly different from existing examples will be explored.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.