Ergodic Theory and Dynamics over Teichmuller Space

William a . Veech在一般闭黎曼曲面上研究了一类亚纯有限范数二次微分的素测地定理的存在性问题。利用他最近发展的西格尔测度理论，他被引导到一个问题，即在范数1二次微分的模空间中，在一个地层的拓扑分量上，SL（2，R）的相关Teichmuller映射作用是否几乎没有不变向量，在相对于Liouville测度的常数的正补中。Veech正在研究穿孔表面上具有锥奇点的平面度量的模空间。对于固定锥角，在Teichmuller模空间和尺度度量的模空间之间有一个标识，这就产生了在Teichmuller模空间上有趣的积分。Veech试图证明这些表示模空间的自然体积的积分是有限的，使用他最近开发的模空间的“Delaunay划分坐标”。他正在研究截断的二十面体上相对于自然平面（锥）度尺的测地线是否闭合或均匀分布的问题，将其简化为与截断的二十面体相关的SL（2，R）的某个离散子群是否是晶格的问题。William a . Veech正在研究一类动力系统的周期轨迹。这些系统的描述只需要高中几何知识，但其分析需要来自不同领域的深刻概念，包括复分析、遍历理论和李群的表示理论。周期轨迹的定位和枚举是动力系统理论中的核心问题。前者可以解释为根据初始条件的知识预测周期的问题，而后者则是获得所有周期轨迹集合的定量信息的问题。包括一个关于大T时周期小于T的轨迹数目的渐近公式。这两个问题在很大程度上都无法解决的系统的一个基本例子是一个点状粒子的均匀运动，它被包含在一个平面多边形区域中，并根据斯涅尔定律从容器的侧面反弹，即。入射角等于反射角。为了表明情况的复杂性，可以提到正多边形（除等边三角形、正方形和正六边形外）中这种运动的周期轨迹的定位和枚举问题，直到最近才由Veech解决。

William A. Veech is pursuing the question of existence of a prime geodesic theorem for a generic meromorphic finite norm quadratic differential on a generic closed Riemann surface. Using his recently developed theory of Siegel measures, he is led to the question of whether a related Teichmuller map action of SL(2,R) on a topological component of a stratum in the moduli space of norm one quadratic differentials does not almost have invariant vectors, in the orthocomplement of the constants relative to the Liouville measure. Veech is studying moduli spaces of flat metrics with cone singularities on punctured surfaces. For fixed cone angles, there is an identification between the Teichmuller moduli space and the moduli space of metrics up to scale which gives rise to interesting integrals over the Teichmuller moduli space. Veech is attempting to prove that these integrals, which represent natural volumes of the moduli space, are finite, using his recently developed "Delaunay partition coordinates" for moduli space. He is pursuing the question of whether every geodesic relative to the natural flat (cone) metric on the truncated icosahedron is either closed or uniformly distributed, reducing this to a question of whether a certain discrete subgroup of SL(2,R) which is associated to the truncated icosahedron is a lattice.William A. Veech is studying periodic trajectories for a class of dynamical systems. These include systems whose descriptions require little more than high school geometry but whose analysis requires rather deep notions from disparate fields, including complex analysis, ergodic theory and representation theory of Lie groups. Location and enumeration of periodic trajectories are central problems in the theory of dynamical systems. The former may be interpreted as the problem of predicting periodicity from knowledge of intitial conditions while the latter is the problem of obtaining quantitative information about the set of all periodic trajectories, including an asymptotic formula for the number of trajectories whose period is less than T for large T. An elementary example of a system for which both questions are for the most part unresolved is the uniform motion of a pointlike particle which is contained in a planar polygonal area and which rebounds from the sides of the container according to Snell's law, i.e. "angle of incidence equals angle of rebound". To indicate the complexity of the situation it may be mentioned the the problems of location and enumeration of periodic trajectories for such motion in a regular polygon (other than equilateral triangle, square and regular hexagon) were open until only recently when they were solved by Veech.