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

平移表面上的流动动力学、Teichmuller 空间和 3-流形的组合学和几何

• 批准号: 0603980
• 负责人: Howard Masur
• 金额: $ 17.85万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603980&HistoricalAwards=false
• 关键词: Dynamics; Flows; Translation; Surfaces; Combinatorics

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.

首席研究员将致力于曲面的几何和拓扑问题以及三个流形。该提案的第一部分是关于翻译表面上的流动动力学的理解。 这个主题起源于多边形中台球的动力学。 具体来说，该计划是研究条件的翻译表面，这将保证有方向的流量是最小的，但不是唯一的遍历。第二部分的目标是研究Teichmuller空间上的Weil-Petersson度量。很少有人知道在这个度量的测地线。主要研究者将使用组合和几何方法来研究这些测地线。第三个主要的研究领域是通过组合学和几何学的盘复杂的研究Heegaard分裂的三个流形。本研究的一个初始目标是证明圆盘复形是一个Gromov双曲空间。人们希望利用这一事实来建立模型流形的Heegard splittings.The首席研究员工作在数学领域的几何和动力系统的二维表面。 对表面的研究可以追溯到几百年前，与我们对周围世界的理解密切相关。 令人惊讶的是，关于表面还有很多东西要学。几何学关注的是它们的形状，如角和直线，而在动力系统中，人们研究的是 物体是如何随时间演化的 动力系统的研究可以追溯到几千年前研究行星运动的时候。该基金的主要目标之一是研究物体在表面上作直线运动的动力学性质。预计这项工作的发现将使人们更好地了解这些基本对象。 主要研究者计划通过发表结果和公开演讲来传播通过该赠款活动获得的知识。 主要研究者积极参与研究生的教育和培训，无论是在他自己的机构，伊利诺伊大学芝加哥分校，并在芝加哥大学。