Dynamics and Kleinian Groups

三维双曲空间可以用三维欧几里得空间的上半部分空间来建模，其中两点之间的最短路径要么沿着垂直线，要么沿着垂直半圆。这个空间的边界是平面和无穷远处的点。大多数情况下，该空间中对称集合的轨迹聚集在平面上的分形上，其研究与双曲流形上的流动动力学有关，由对称群给出双曲三维空间的商。这种分形的一个美丽的例子是阿波罗垫圈，它是由古希腊几何学家阿波罗尼乌斯证明的定理构造的。分形在自然界中是丰富的，可以在海岸线、海浪、罗马花椰菜等中看到。酱油中的一滴油会形成一个类似阿波罗垫圈的圆形填料。该项目有望在解释自然界中出现的分形结构方面具有价值。大多数双曲三流形具有无限体积，而双曲三流形的动态刚度特性研究大多是在有限体积情况下进行的。本课题研究的是无限体积双曲流形的动力学问题，这是一个非常丰富的理论，有几个深刻而引人注目的应用。该项目涉及双曲几何、Kleinian群、遍历理论和数论之间的新相互作用。研究这个新领域需要几何学和动力学的许多新想法，并在这些领域之间开辟了新的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The three-dimensional hyperbolic space can be modeled by the upper half space of the three-dimensional Euclidean space, where the shortest path between two points is either along a vertical line or a vertical semi-circle. The boundary of this space is the plane together with the point at infinity. Most often the trajectory of a collection of symmetries in this space accumulates on a fractal on the plane, the study of which is related to dynamics of flows on the hyperbolic manifold, given by the quotient of the hyperbolic three space by the symmetry group. One beautiful example of such fractals is the Apollonian gasket which is constructed by a theorem proven by the ancient Greek geometer Apollonius. Fractals are abundant in nature as can be seen in the coastlines, ocean waves, Romanesco cauliflowers, etc. A drop of oil in soy sauce creates a circle packing akin to the Apollonian gasket. This project is expected to be of value in interpreting the structure of fractals arising in nature.Whereas most of hyperbolic three manifolds have infinite volume, the dynamical rigidity properties of hyperbolic manifolds have been investigated mostly for finite volume cases. The project concerns with dynamics on infinite volume hyperbolic manifolds, which turn out to be quite a rich theory and has several deep and striking applications. This project involves new interactions between hyperbolic geometry, Kleinian groups, ergodic theory and number theory. Investigating this new territory requires many new ideas both from geometry and dynamics and opens up new interactions between these fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.