Ergodic Theory of Billiards and Quadratic Differentials

首席研究员将研究台球中的问题 它模拟了Kolmogorov-Arnold-Moser 理论，以及二次微分和Teichmuller 理论 他将调查是否收敛遍历 如果人们知道， 它所确定的泰希穆勒测地线的轨道。 另一 本项目涉及的重要问题是 Teichmuller空间的极值长度几何 具体来说， 有有限条曲线，其极值长度决定 黎曼曲面？ 不变的环形带出现在Kolmogorov-Arnold-Moser方程中 湍流理论 这些带上的函数迭代可以 由台球反弹的角度和位置建模 在一张圆桌上。 由于这些原因，校长 研究人员可能会揭示湍流相关问题 (and船体的形状）通过回答描述的问题 被台球游戏所吸引 例如，有多少可能的 击球方向 一个周期性的轨迹，使球最终返回到 在同一个地方朝同一个方向移动 是否 球的位置或桌子的形状会有什么不同吗？

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?