Problems in Surface Geometry and Topology

曲面几何和拓扑学中的问题摘要霍华德·马苏尔首席研究员建议研究曲面几何和拓扑学中的问题。几何问题包括研究具有锥角奇点的平板上的流动动力学。一个特别的例子是研究多边形中的台球，其角度是圆周率的有理倍数。这项研究的特殊方面将是试图找到周期轨道数的渐近增长率。对于一些台球，如正方形，增长率是已知的。其他更有趣的例子是某些直角三角形。首席研究员将研究更多的例子，例如带有狭缝或障碍物的正方形。这个想法是为了找到渐近的二次估计。拓扑问题涉及到曲面的映射类群。这一组是曲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。主要研究者建议研究映射类群的大尺度几何，目的是证明映射类群是拟等距刚性的。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。在那里，有一个点质量沿直线移动，当它遇到一个区域的边界时，它会反弹，这样反射角就等于入射角。对台球的研究已经有100多年的历史了。为了理解这样的动力系统，需要了解轨道的长期行为。这种长期行为的一个特别例子是对周期轨道的研究。这些是重复运行的轨道。研究平面内多边形台球的周期轨道是本方案的主要内容。

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.