FRG: Rational billards and geometry and dynamics on Teichmuller Space

主要研究人员将从事几何分析领域的两个基本项目。第一个项目涉及理性台球，阿贝尔微分的模空间和teichmullergedesic流的动力学的相互关联的分析研究。最近的一个主题是均匀空间上的流动动力学与理性台球和平面上的流动动力学之间的相互作用。这种协同作用导致了这两个学科最近的突破。从广义上讲，主要研究者建议将齐次空间的方法和思想应用于研究Teichmuller空间上的有理台球和动力学。第二个项目是给出一种组合方法来研究泰希穆勒空间及其几何。其中一个主要的目标是发展一个完全类似的理论之间的这一理论和经典理论的几何模曲线。许多自然现象都是通过被称为动力系统的数学分支来研究的。一个重要的例子是行星运动。另一个古老的例子是台球理论，或分子碰撞的研究。三角台球是在研究两个物体在一段时间内的弹性碰撞时出现的。对动力系统的研究可以采取多种形式。一个例子是研究周期轨道（或重复行为）；另一种是研究choatic行为。主要研究人员计划研究数学中出现的各种动力系统。其中一个突出的主题将是在平面上的多边形台球，以及被称为平面的泛化。这些表面也可以组合成一个称为模空间的家族，主要研究人员计划研究模空间上的动力学。这个项目的一个主要组成部分将是教育。主要研究人员计划为本科生和研究生开设讲习班，以便向他们介绍这些数学领域。

Abstract The principal investigators will work on two fundamentalprojects in the field of geometric analysis. The firstproject concerns the interrelated analytic study ofrational billiards, moduli spaces of abeliandifferentials, and the dynamics of the Teichmullergeodesic flow. A recent theme has been the interplaybetween the dynamics on flows on homogeneous spaces on theone hand, and the dynamics of flows on rational billiardsand flat surfaces on the other. This synergy has led torecent breakthroughs in both subjects. Broadly speaking, theprincipal investigators propose to apply the methods andideas developed in the case of homogeneous spaces to studyrational billiards and dynamics on Teichmuller space. The second project is to give a combinatorial approachto the study of Teichmuller Space and its geometry. Oneof the major goals is to develop a complete analoguebetween this theory and the classical theory of thegeometry of the modular curve.Many natural phenomena are studied via the branch ofmathematics known as dynamical systems. One importantexample is planetary motion. Another centuries old exampleis the theory of billiards, or the study of collisions ofmolecules. Triangular billiards occur in the study ofelastic collisions of two masses in an interval. Thestudy of a dynamical system can take many forms. Oneexample is to study periodic orbits (or repeatingbehavior); another is to study choatic behavior.The principal investigators plan to study a variety ofdynamical systems that occur in mathematics. One of theprominent topics will be billiards in polygons in theplane, and a generalization known as flat surfaces. These surfaces also fit together into a family known as amoduli space, and the principal investigators plan tostudy dynamics on moduli spaces. A major component of theproject will be educational. The principal investigatorsplan to give workshops for students, both at theundergraduate and graduate levels, in order to introducethem to these fields of mathematics.