Problems in Surface Geometry and Topology

表面几何和拓扑问题AbstractHoward Masur主要研究者建议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。一个特别的例子是研究台球在多边形的角度是合理的倍数π。 本研究的特别之处是试图找到周期轨道数的渐近增长率。对于一些台球，如广场，增长率是已知的。其他更有趣的例子是某些直角三角形。主要研究者将研究其他示例，例如带狭缝或屏障的正方形。我们的想法是找到渐近二次估计。拓扑问题涉及曲面的映射类群。这一群体是曲面拓扑学研究的主要对象之一。 群论最近的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，目标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，它沿直线运动，当它越过一个区域的边界时，它会反弹，反射角等于入射角。台球已经被研究了100多年。为了理解这样一个动力学系统，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轨道的研究。 这些是重复的轨道。本文主要研究平面多边形台球的周期轨道。

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.