Finitary Analysis in Homogeneous Dynamics and Applications

动力系统研究预测空间中一点的长期行为的规则。动力系统的起源可以追溯到牛顿力学。动力系统数学模型的例子包括流体动力学、气流动力学和许多其他方面的研究。例如，考虑气体粒子按照一定的规则在空间中运动。人们感兴趣的是了解系统在长时间间隔内的各种状态。一个奇怪的现象是：如果一个人假设最初所有的气体粒子都在空间的一半，那么在未来无限地经常，气体分子将聚集在空间的初始部分。这一结论似乎自相矛盾，这是由该系统的数学方面预测的。然而，如果一个人观察到这个系统的自由度非常大，那么这个谜团就可能被揭开，因此为了恢复到原始状态，他需要观察这个系统很长的时间间隔(实际上是不可能的)。这一建议旨在研究动力系统的有限方面，其中一个人感兴趣的是在误差范围内逼近系统的各种状态。这个项目还包括一名研究生的培训。首席研究员寻求对动力学及其在数论和几何中的应用的某些刚性结果的扩展和加强。将特别注意分析的有限和有效方面。以下是主要目标：(I)动力系统已成为现代数学的主要参与者。然而，依赖于遍历理论和动力学技术的论证往往是非定量的。提供这些论点的有限版本是具有挑战性的，也是非常受欢迎的，特别是考虑到各种应用。主要研究者寻求这一方向的结果有两个主要目标：提供产生强刚性多项式率的有限自变量，在一些具有数论和几何学有趣应用的特定例子中产生齐次动力学结果；提供这些著名结果的有限版本，即使在一般情况下可能得不到多项式率。(Ii)从不同角度研究了三维双曲流形中的测地线平面。这些研究表明，测地平面的行为与环境流形的几何、拓扑和算术性质密切相关。这项建议寻求使用来自同质动态的工具来改进和加强这一方向的结果。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。

Dynamical systems study rules which predict the long-term behavior of a point in a space. The origin of dynamical systems may be traced back to Newtonian mechanics. Examples of mathematical models of dynamical systems include the study of fluid dynamics, airflow dynamics, and many others. Consider, for instance, gas particles moving in a space according to a certain rule. One is interested in understanding various states of the system over long-time intervals. A curious phenomenon is the following: if one assumes that initially all gas particles are in one half of the space, then infinitely often in the future, the gas molecules will collect in the initial portion of the space. This conclusion, which is predicted by mathematical aspects of the system, seems paradoxical. However, the mystery may be revealed if one observes that the number of the degree of freedom in this system is very large, thus in order to return to the original state, one needs to observe the system for very long (practically impossible) time intervals. This proposal aims at the study of finitary aspects of dynamical systems where one is interested in approximating various states of the system within an error. This project also include the training of a graduate student.The Principal Investigator seeks extensions and strengthening of certain rigidity results in dynamics and their applications in number theory and geometry. Special attention will be given to finitary and effective aspects of the analysis. The following will be the main objectives: (i) Dynamical systems have become a major player in modern mathematics. However, arguments relying on techniques from ergodic theory and dynamics are often non-quantitative. Providing finitary versions of these arguments are challenging and much sought after, especially in view of various applications. The principal investigator seeks results in this direction with two main goals in mind: provide finitary arguments which yield polynomial rates for strong rigidity results in homogeneous dynamics in some specific examples with interesting applications to number theory and geometry; provide finitary versions of these celebrated results in great generality, even though one may not obtain a polynomial rate in general. (ii) Geodesic planes in hyperbolic 3-manifolds have been studied from different angles. These investigations have made it clear that the behavior of geodesic planes is intimately related to the geometric, topological, and arithmetic properties of the ambient manifold. This proposal seeks to refine and strengthen results in this direction using tools from homogeneous dynamics.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.