Hyperbolic Geometry and the Mapping Class Group

二维和三维空间是数学家研究的一些最基本的对象。研究它们的一种方法是将它们实现为几何对象，然后使用几何来研究空间。三种最常见的几何首先是经典的欧几里德几何或平坦几何，然后是非欧几里德几何：球面几何和双曲几何。自19世纪末以来，人们就知道大多数二维空间，或者换句话说：曲面，都表现出双曲几何。瑟斯顿在20世纪70年代的开创性工作表明，“大多数”三维空间，称为三流形，也有双曲几何。这是这个项目的重点与双曲几何空间的动机。它包括许多适合培养研究生在这一领域工作的子项目，本项目将在PI先前工作的基础上研究双曲三流形几何的几个问题。其中一个中心课题是对“重整化体积”的研究。PI和他的合作者最近发起了一项研究的梯度流的重整体积函数已被证明是一个非常有用的工具，了解几何的双曲三流形。在另一个方向上，PI和他的合作者将继续他们在几何群论方面的工作，以了解CAT（0）立方体复合体上映射类组的作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Two and three-dimensional spaces are some of the most fundamental objects studied by mathematicians. One approach to study them is to realize them as geometric objects and then use the geometry to study the spaces. The three most common geometries are first, the classical Euclidean or flat geometry, and then the non-Euclidean: spherical and hyperbolic geometries. It has been known since the late 1800s that most two-dimensional spaces, or in other words: surfaces, exhibit hyperbolic geometry. Thurston's groundbreaking work in the 1970s showed that "most" three dimensional spaces, called three-manifolds, also have hyperbolic geometry. This is the motivation for this project's focus on spaces with hyperbolic geometry. It includes many sub-projects suitable for training graduate students to work in this area.This project will study several questions on the geometry of hyperbolic three-manifolds building on PI's prior work. One of the central topics is the study of "renormalized volume". The PI and his collaborators recently initiated a study of the gradient flow of the renormalized volume function which has shown to be an extremely useful tool for understanding the geometry of hyperbolic three-manifolds. In another direction, the PI and his collaborators will continue their work in geometric group theory to understand actions of the mapping class group on CAT(0) cube complexes.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.