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

首席研究员将研究曲面和三个流形的几何和拓扑问题。该提案的第一部分涉及对平移表面上流动动态的理解。这门学科起源于多边形中台球的动力学。具体地说，该计划是研究平移曲面上的条件，以确保存在流动最小但不是唯一遍历的方向。第二部分的目的是研究TeichMuller空间上的Weil-Petersson度量。关于这一度量中的测地线，人们知之甚少。首席研究员将使用组合和几何方法来研究这些测地线。第三个主要研究领域是通过与车把有关的圆盘复合体的组合学和几何来研究三个流形的Heegaard分裂。这项研究的最初目标是证明圆盘复合体是格罗莫夫双曲空间。人们希望利用这一事实来建立Heegard分裂的模型流形。主要研究人员在二维曲面上的几何和动力学系统的数学领域工作。对表面的研究可以追溯到几百年前，它与我们对周围世界的理解密切相关。令人惊讶的是，关于曲面还有很多需要学习的地方。几何学关注的是它们的形状，如角度和直线，而在动力学系统中，人们研究对象如何随时间演变。动力系统这门学科可以追溯到几千年前对行星运动的研究。这项资助的主要目的之一是研究物体在表面上沿直线运动的动力学性质。预计这项工作的结果将有助于更好地理解这些基本目标。首席调查员计划通过公布结果和发表公开演讲来传播从这笔赠款活动中获得的知识。这位首席研究员在他自己的机构--芝加哥伊利诺伊大学和芝加哥大学--积极参与研究生的教育和培训。

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.