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

职业：代数动作的平滑刚性的新方法和应用

• 批准号: 1845416
• 负责人: Zhenqi Wang
• 金额: $ 40万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1845416&HistoricalAwards=false
• 关键词: CAREER; New; Methods; Applications; Smooth

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.

动力系统领域起源于微分方程和天体力学。它研究受运动定律影响的系统的长期行为。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以保留，则动作是刚性的。到目前为止，具有强混沌特性的系统已经被很好地理解了。然而，在许多感兴趣的系统中，只能观察到微弱的混沌行为。该项目的目标是开发新工具来研究此类系统，然后应用这些结果来研究数学的其他领域，例如数论和表示论。该研究计划还辅以教育和外展活动，包括对本科生、研究生和博士后的培训，以及促进不同领域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，包括动力系统、调和分析等。对于李群、表示论和数论。首席研究员计划利用表示论建立余循环刚性并研究一大类代数部分双曲和抛物线作用的（扭曲）上同调方程。抛物线作用研究的结果和技术将应用于数论，以研究某些单能流的有效均匀分布以及半直积群的一些齐次空间上的映射。主要研究者还将结合 KAM 方法和表示理论到一般的部分双曲和抛物线代数作用来研究局部刚性。将探索一大批几何特性与现有示例明显不同的新示例。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。