Ergodic Theory of Billiards and Quadratic Differentials

首席研究员将研究台球中的问题，这些问题模拟了科尔莫戈洛夫-阿诺德-莫泽理论、二次微分和泰希穆勒理论中的重要问题。他将研究，如果一个人知道泰希穆勒测地线的轨道，是否可以估计测量的叶面的遍历平均值的收敛。本课题研究的另一个重要问题是TeichMuller空间的极长几何。具体地说，是否存在极值长度决定黎曼曲面的有限数量的曲线？不变环带出现在Kolmogorov-Arnold-Moser湍流理论中。函数在这些频带上的迭代可以通过圆桌上台球反弹的角度和位置来建模。出于这些原因，首席研究人员可能会通过回答台球游戏中描述的问题来阐明相关的湍流问题(以及船体的形状)。例如，一个人可能击中台球的可能方向中有多少个会使球处于周期性的轨迹中，从而使球最终返回到完全相同的位置，沿着完全相同的方向运行？球的位置或桌子的形状会有区别吗？

The principal investigator will study problems in billiards which model important questions from Kolmogorov-Arnold-Moser theory, as well as quadratic differentials and Teichmuller theory. He will investigate whether convergence of ergodic averages for measured foliations can be estimated if one knows the orbit of the Teichmuller geodesic it determines. Another important question addressed in this project concerns the extremal length geometry of Teichmuller space. Specifically, are there a finite number of curves whose extremal lengths determine the Riemann surface? Invariant annular bands arise in the Kolmogorov-Arnold-Moser theory of turbulence. Iterations of functions on these bands can be modeled by angles and locations of bounces of a billiard ball on a circular table. For these reasons, the principal investigator might shed light on relevant questions of turbulence (and the shape of ships' hulls) by answering questions described by the game of billiards. For example, how many of the possible directions that one might hit a billiard ball carry the ball in a periodic trajectory, so that the ball eventually returns to the exact same spot travelling in the exact same direction? Does the location of the ball or shape of the table make a difference?