RUI: Interactions between Number Theory and Ergodic Theory

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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.