Dynamics of Flows on Translation Surfaces, and the Combinatorics and Geometry of Teichmuller Space and 3-Manifolds

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The principal investigator will work on problems in the geometry and topology of surfaces, and three manifolds. The first part of the proposal is concerned with the understanding of the dynamics of flows on translation surfaces. This subject has its origins in the dynamics of billiards in polygons. Specifically, the plan is to study conditions on a translation surface which will guarantee that there are directions in which the flow is minimal, but not uniquely ergodic. The objective of the second part of the proposal is to study the Weil-Petersson metric on Teichmuller space. Little is known about geodesics in this metric. The principal investigator will use combinatorial and geometric methods to study these geodesics. The third major area of research is the study of Heegaard splittings of a three manifold, via the combinatorics and geometry of the disc complex associated to a handlebody. An initial goal of this study is to show that the disc complex is a Gromov hyperbolic space. One hopes to use this fact to build model manifolds for Heegard splittings.The principal investigator works in the mathematical fields of geometry and dynamical systems on two dimensional surfaces. The study of surfaces dates back hundreds of years, and is intimately connected with our understanding of the world around us. Surprisingly, there is still much to be learned about surfaces. Geometry is concerned with their shapes, such as angles and straight lines, while in dynamical systems one studies how objects can evolve in time. The subject of dynamical systems dates back thousands of years to the study of the motion of the planets. One of the main goals of this grant is to study the dynamical properites of objects moving in straight lines on surfaces. It is expected that the findings of this work will lead to greater understanding of these fundamental objects. The principal investigator plans to disseminate the knowledge gained by the activities of this grant by publishing the results and giving public lectures. The principal investigator is actively involved in the education and training of graduate students, both at his own institution, the University of Illinois at Chicago, and at the University of Chicago.