CAREER: Dynamical Rigidity Related to Group Actions and Arithmetics

拟议的项目属于动力系统领域，该领域研究在几何空间中随着时间发生的运动的长期轨迹，并受一套固定的数学规则支配。动力系统理论源于天体力学，并已成为许多其他科学领域的重要工具。所提出的研究涉及一个给定的动力系统是否具有刚性性质的问题，即不同的轨迹是否必须显示许多共同的模式。首席研究员将对这一主题进行研究，与其他研究人员合作，与研究生和本科生合作，并传播在该项目中形成的想法。首席调查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研究工作坊。这项研究将试图确定某些群论或数论起源的动力系统的性质。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。这些动作通常被认为具有刚性，这种现象的特征是在动作下缺乏不变的结构。另一条建议的研究路线是研究莫比乌斯不相交猜想的一些特殊情况。这些系统包括某些远端拓扑系统和其他系统，它们的轨迹在足够长的时间内具有严格的模式。建议研究中的方法将使用遍历理论、平滑动力学、表征理论和数论的思想。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 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.