RUI: Interactions between Number Theory and Ergodic Theory

主要研究者将致力于数论和遍历理论的接口问题。 该提案的第一部分是了解李群的有限体积商空间上的齐次流的动力学。 具体来说，该计划是研究行为的发散轨迹的半简单流，并给出了一些应用的问题，数论有关奇异向量，如确定其Hausdorff维数。 建议的第二部分涉及研究平移表面上的流动。 该计划是一个双管齐下的方法来研究条件的翻译表面上，保证存在的最小的，但不唯一遍历方向的一方面，和条件，确保他们的不存在的另一方面。 其中一个目标是确定第一个非平凡的Masur-Smillie常数。首席研究员在动力系统的数学领域工作。 动力系统研究的最终目标是能够控制和预测其通常复杂和混沌的行为。 主要研究者的研究主题是寻找混沌系统中可能表现出的规律，例如自相似性。这些规律通常可以用数字的微妙属性来描述，而遍历理论方法专注于系统轨道的统计属性，不断提供新的见解，提高了我们对混沌系统的理解。

The principal investigator will work on problems at the interface of number theory and ergodic theory. The first part of the proposal is to understand the dynamics of homogeneous flows on finite volume quotient spaces of Lie groups. Specifically, the plan is to study the behavior of divergent trajectories of semisimple flows and give a number of applications to problems in number theory concerning singular vectors, such as the determination of their Hausdorff dimension. The second part of the proposal concerns the study of flows on translation surfaces. The plan is a two-pronged approach to study conditions on a translation surface that guarantee the existence of minimal but not uniquely ergodic directions on the one hand, and conditions that ensure their non-existence on the other. One of the objectives is to determine the first nontrivial Masur-Smillie constant.The principal investigator works in the mathematical field of dynamical systems. The ultimate goal in the study of dynamical systems is to be able to control and predict their often complex and chaotic behavior. The overall theme behind the principal investigator's research is to search for regular patterns that may be exhibited in a chaotic system, such as self-similarity.These regular patterns are often describable in terms of subtle properties of numbers, and the ergodic theoretic approach, which focuses on the statistical properties of the orbit of a system, has continued to provide new insights that has improved our understanding of chaotic systems.