Mathematical Sciences: Ergodic Theory and Topological Dynamics

群作用于空间的情况，如实数群随时间演化作用于物理系统的状态空间的情况，在数学及其应用中经常遇到。人们感兴趣的是总体上，平均而言，在长期内发生的事情，这可能很难仅仅从短时间间隔内发生的事情来推断。在Veech教授的项目中，有一个看似简单的例子，即台球在三角形区域内倾斜的流动。随着时间的推移，轨迹要么填满整个三角形，要么是周期性的，在有限次数的反弹后回到自己身上。分类和描述周期轨迹需要一些令人惊讶的复杂数学。特别是，将台球流与更奇特的几何物体上的流和其他群体行为联系起来是有利的。在技术层面上，这与SL（2，R）作用于Teichmueller空间的遍历理论和动力学有关。Veech教授正在研究这个空间中各向同性子群为格的点的普遍性。某些这样的点来自三角台球。流的长度谱的难题，即周期轨道的长度可以出现，将在各种几何背景下进行研究。

The situation of a group acting on a space, for instance the group of real numbers acting by time evolution on the state space of a physical system, is encountered quite often in mathematics and its applications. One is interested in what happens overall, on average, in the long run, which may be quite difficult to infer solely from what happens over short intervals of time. A seemingly straightforward example figures prominently in the project of Professor Veech, namely the flow described by billiard balls careening about inside a triangular region. Trajectories will either essentially fill out the whole triangle over time or be periodic, coming back on themselves after a finite number of bounces. Classifying and describing the periodic trajectories requires some surprisingly sophisticated mathematics. In particular, it turns out to be advantageous to relate billiard flows to flows and other group actions on more exotic geometric objects. At a technical level, this has to do with the ergodic theory and dynamics of the action of SL(2,R) on Teichmueller space. Professor Veech is studying the prevalence of points in this space whose isotropy subgroups for the action are lattices. Certain such points arise from triangular billiards. The difficult problem of length spectrum for a flow, i.e. of what lengths of periodic orbits can occur, will be studied in various geometric contexts.