CAREER: Dynamical Rigidity Related to Group Actions and Arithmetics

职业：与群体行动和算术相关的动态刚性

• 批准号: 1753042
• 负责人: Zhiren Wang
• 金额: $ 42.5万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1753042&HistoricalAwards=false
• 关键词: CAREER; Dynamical; Rigidity; Related; Group

The proposed project belongs to the field of dynamical systems, which is the study of long-term trajectories of movements that take place over time in a geometric space and are governed by a fixed set of mathematical rules. The theory of dynamical systems has its roots in celestial mechanics and has become an important tool in many other scientific fields. The proposed research deals with the question of whether a given dynamical system has rigidity properties, namely whether different trajectories must display a lot of common patterns. The principal investigator will conduct research on the topic, collaborate with other researchers, work with graduate and undergraduate students, and disseminate ideas developed in the project. The principal investigator will also organize seminars and conferences. In particular, the principal investigator will organize a series of research workshops for undergraduate students.This research will try to determine properties of certain classes of dynamical systems of group theoretical or number theoretical origins. One of such classes are actions on smooth manifolds by higher-rank commutative groups or by lattices in higher-rank semisimple Lie groups. These actions are often expected to have rigidity, a phenomenon characterized by the scarcity of invariant structures under the action. Another proposed line of research is the study of some special cases of the Mobius disjointness conjecture. These include certain distal topological systems and other systems whose trajectories have strict patterns in sufficiently long term. The approaches in the proposed research will use ideas from ergodic theory, smooth dynamics, representation theory and number theory.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.

拟议的项目属于动力系统领域，即研究在几何空间中随时间发生的长期运动轨迹，并受一套固定的数学规则支配。动力系统理论起源于天体力学，并已成为许多其他科学领域的重要工具。所提出的研究涉及的问题是，给定的动力系统是否具有刚性属性，即不同的轨迹是否必须显示出许多共同的模式。首席研究员将对该主题进行研究，与其他研究人员合作，与研究生和本科生合作，并传播项目中开发的想法。首席研究员还将组织研讨会和会议。特别是，首席研究员将为本科生组织一系列的研究工作坊。这项研究将试图确定某些类别的群论或数论起源的动力系统的属性。其中一类是高阶交换群或高阶半单李群中的格在光滑流形上的作用。这些动作通常被期望具有刚性，这种现象的特征在于动作下的不变结构的稀缺性。 另一个研究方向是研究莫比乌斯不相交猜想的一些特殊情况。这些包括某些远端拓扑系统和其他系统，其轨迹在足够长的时间内具有严格的模式。在拟议的研究方法将使用遍历理论，光滑动力学，表示理论和数论的想法。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

项目成果

Three-stage evolution and fast equilibrium for SGD with non-degenerate critical points

具有非简并临界点的 SGD 的三阶段演化和快速平衡

• 发表时间: 2022
• 期刊: Proceedings of the 39th International Conference on Machine Learning
• 作者: Wang, Yi;Wang, Zhiren
• 通讯作者: Wang, Zhiren

Möbius disjointness for nilsequences along short intervals

短间隔上的零序列的莫比乌斯不相交性

• DOI: 10.1090/tran/8176
• 发表时间: 2021
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: He, Xiaoguang;Wang, Zhiren
• 通讯作者: Wang, Zhiren

Arbitrarily slow decay in the Möbius disjointness conjecture

莫比乌斯不相交猜想中的任意缓慢衰减

• DOI: 10.1017/etds.2022.61
• 发表时间: 2023
• 期刊: Ergodic Theory and Dynamical Systems
• 影响因子: 0.9
• 作者: ALGOM, AMIR;WANG, ZHIREN
• 通讯作者: WANG, ZHIREN

Polynomial effective equidistribution

多项式有效均匀分布

• DOI: 10.5802/crmath.411
• 发表时间: 2023
• 期刊: Comptes Rendus. Mathématique
• 作者: Lindenstrauss, Elon;Mohammadi, Amir;Wang, Zhiren
• 通讯作者: Wang, Zhiren

On \epsilon -escaping trajectories in homogeneous spaces

关于齐次空间中的 epsilon 逃逸轨迹

• DOI: 10.3934/dcds.2020365
• 发表时间: 2021
• 期刊: Discrete & Continuous Dynamical Systems - A
• 作者: Rodriguez Hertz, Federico;Wang, Zhiren
• 通讯作者: Wang, Zhiren