FRG: Rational billards and geometry and dynamics on Teichmuller Space

主要研究人员将致力于几何分析领域的两个基础项目。第一个项目涉及有理台球、非常微分模空间和TeichMuller测地线流的动力学的相关分析研究。最近的一个主题是均匀空间上的流动动力学与有理台球和平坦表面上的流动动力学之间的相互作用。这种协同效应导致了最近这两个学科的突破。一般说来，主要研究者建议应用齐次空间中发展的方法和条件来研究TeichMuller空间上的有理台球和动力学。第二个项目是给出研究TeichMuller空间及其几何的组合方法。主要目标之一是在这一理论和经典的模曲线几何理论之间建立一个完全的类比。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。另一个有数百年历史的例子是台球理论，即研究分子碰撞的理论。三角台球是在研究两个质量在一个区间内的弹性碰撞时出现的。动力系统的研究可以有多种形式。一个例子是研究周期轨道(或重复行为)；另一个是研究合唱行为。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平面多边形台球，以及一种被称为平面的推广。这些表面还可以组合成一个被称为阿莫杜利空间的家族，主要研究人员计划研究模空间上的动力学。该项目的主要组成部分将是教育性的。主要研究人员计划为本科生和研究生开设研讨会，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.