Problems in Surface Geometry and Topology

表面几何和拓扑问题

• 批准号: 0104151
• 负责人: Howard Masur
• 金额: $ 10.59万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2005-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104151&HistoricalAwards=false
• 关键词: Problems; Surface; Geometry; Topology

Problems in Surface Geometry and TopologyAbstractHoward MasurThe principal investigator proposes to work on problems in the geometry and topology of surfaces. The geometric problems consist of studying the dynamics of flows on flat surfaces with cone angle singularities. A particular example is the study of billiards in polygons with angles that are rational multiples of pi. The particular aspect of this study will be to attempt to find the asymptotic growth rate for the number of periodic orbits. For some billiards such as the square, the growth rate is known. Other more interesting examples are certain right triangles. The principal investigator will study additional examples such as thesquare with a slit or barrier. The idea is to find asymptotic quadraticestimates. The topological problems concern the mapping class group of a surface. This group is one of the main objects of study in surface topology. One of the recent developments in group theory is to study the large scaleor asymptotic geometry of a group. The principal investigator proposes tostudy the large scale geometry of the mapping class group with a goal ofshowing that the mapping class group is quasi-isometrically rigid. In the field of dynamical systems one studies the motion of objects. The field had its origins in the study of the motion of theplanets. Billiards are another closely related example. There one has apoint mass that moves in straight lines so that when it encountersthe boundary of a region, it rebounds so that the angle of reflectionequals the angle of incidence. Billiards have been studied for over 100years. In order to understand such a dynamical system oneneeds to understand the long term behavior of the orbits. One particularexample of this long term behavior are the study of the periodicorbits. These are the orbits that repeat themselves. The study ofperiodic orbits for billiards in polygons in the plane is the main subjectof this proposal.

表面几何和拓扑问题摘要：霍华德·马苏尔主要研究表面的几何和拓扑问题。几何问题包括研究具有锥角奇点的平面上的流动动力学。一个特别的例子是用角为π的有理倍数的多边形研究台球。本研究的特别之处在于试图找出周期轨道数目的渐近增长率。对于一些台球，如方块，增长率是已知的。其他更有趣的例子是某些直角三角形。首席研究员将研究额外的例子，如带有狭缝或屏障的正方形。目的是找到渐近的二次估计。拓扑问题涉及曲面的映射类群。这一群体是表面拓扑学的主要研究对象之一。群论的最新发展之一是研究群的大尺度或渐近几何。主要研究者提出研究映射类群的大比例尺几何，目的是证明映射类群是准等距刚性的。在动力系统领域，人们研究物体的运动。这个领域起源于对行星运动的研究。台球是另一个密切相关的例子。有一个质点沿直线运动，当它碰到一个区域的边界时，它会反弹，使反射角等于入射角。人们研究台球已有100多年了。为了理解这样一个动力系统，我们需要了解轨道的长期行为。这种长期行为的一个特别的例子是对周期轨道的研究。这些是重复的轨道。研究平面上多边形台球的周期轨道是本文的主要课题。