Mathematical Sciences: Ergodic Theory and Topological Dynamics

在数学及其应用中，一个群作用在一个空间上的情况是经常遇到的，例如作用于物理系统状态空间上的实数群。人们感兴趣的是总体上发生了什么，平均而言，从长期来看，这可能很难仅仅从短期内发生的事情来推断。在Veech教授的项目中，一个看似直截了当的例子占据了突出的位置，即台球在三角形区域内来回跳跃所描述的流动。轨迹要么随着时间的推移基本上填满整个三角形，要么是周期性的，在有限次数的反弹后自行恢复。对周期性轨迹进行分类和描述需要一些令人惊讶的复杂数学。特别是，事实证明，将台球流与流和更奇特的几何对象上的其他群操作联系起来是有利的。在技术层面上，这与SL(2，R)在Teichmueller空间上作用的遍历理论和动力学有关。Veech教授正在研究这个空间中的点的普遍性，这些点的作用的各向同性子群是格子。某些这样的分数是由三角台球产生的。在不同的几何背景下，将研究流动的长度谱的困难问题，即可以出现多长的周期轨道。

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.