Rigidity of abelian actions

提案中有三个主要主题。第一部分研究了阿贝尔高阶抛物作用的上同性；特别地，我们研究了完全没有双曲的作用的上同调方程和环刚性。其次用几何方法研究了由四元数群生成的齐次空间上的Cartan作用的光滑作用刚性；这是对高阶代数作用的光滑刚性研究的补充。最后，研究了Cartan作用和抛物作用的一般光滑作用刚度；本文将著名的Kolmogorov-Arnold-Moser （KAM）方法应用于齐次函数空间。动力系统之所以出现，是因为牛顿发现机械物体的运动可以用常微分方程的解来描述。更一般地说，许多其他自然现象和社会现象，如放射性衰变、化学反应、人口增长或市场上价格的动态变化，都可以用各种动力系统来建模。因此，动力系统这门学科不仅在数论、几何和表示论等数学的其他分支中得到了广泛的应用，而且在物理学、生物学、经济学和其他科学中也得到了广泛的应用。本文研究的重点是代数作用的光滑刚度结果，即与参数变化有关的稳定性。PI提出从表征理论和谐波分析发展动力系统研究的新方法和新技术。PI相信新工具在解决问题和激发动力系统的新方向方面是强大的。PI计划在该奖项期间与一些研究生合作并教授研究生水平的课程。

There are three main topics in the proposal. The first is the study of the cohomology of abelian higher-rank parabolic actions; in particular we study the cohomological equation and cocycle rigidity for actions completely absent of hyperbolicity. The second considers the smooth action rigidity for the Cartan action on homogeneous space generated by quaternion groups by geometric method; this complements the project of studying smooth rigidity for higher-rank algebraic actions. Finally, we study the general smooth action rigidity for Cartan action and parabolic action; here we apply the famous Kolmogorov-Arnold-Moser (KAM) method to homogenous function space. Dynamical systems appeared first because of Newton's discovery that the motions of mechanical objects can be described by the solutions of ordinary differential equations. More generally, many other natural and social phenomena,such as radioactive decay, chemical reaction, population growth, or dynamics of prices on the market, maybe be modeled with various dynamical systems. So the subject of dynamical systems hasfound numerous applications not only in other branches of mathematics, such asnumber theory, geometry and representation theory, but also in physics, biology, economics and other sciences.The proposed research focuses on smooth rigidity results of algebraic actions, that is, the stabilityproperties related to change of parameters. The PI proposes to develop new methods and techniques from representation theory andharmonic analysis in the study of dynamical systems. The PI believes that the new tool is powerful in solving problems and inspiring new directions in dynamical systems. The PI intends to work with some graduate students and teach graduate-level course during the period of this award.