Geometry and topology of hyperbolic manifolds

这一提议的主要目标是将低维双曲几何最近取得的惊人进展中获得的一些技术和直觉转移到更高的维度。目前的强大工具包括佩雷尔曼关于利玛奇流的工作，以及三十多年来从发展瑟斯顿的双曲三维流形理论中获得的洞察力。虽然这些技术通常不会从字面上转移到更高的维度，但PI认为现在是研究更高维双曲流形的几何和拓扑的好时机，并受益于更低维的直觉和洞察力。这项工作将建立在与Steven Kerckhoff现有的合作基础上。由于理解我们周围的3(或4)维世界的自然动机，大多数几何研究集中在“低维”，通常意味着不到5维。然而，在现代世界，更高维的几何对象是丰富的：现代计算机科学使用非常高维的抽象“单纯复合体”来模拟具体的系统。研究金融市场往往需要估计高维空间的积分，而互联网搜索引擎依赖于非常大的向量空间的高效线性代数算法。这项建议将重点研究一种特定类型的高维几何，即双曲几何。传统上，如果没有一台功能强大的计算机，研究这一主题是很困难的。这不再是一个主要障碍，理解具体而复杂的例子是可能的。我们的目标是建立在我们的低维直觉之上，以更好地理解高维对象。

The main goal of this proposal is to transfer into higher dimensions some of the techniques and intuition gained from recent spectacular progress in low dimensional hyperbolic geometry. Current powerful tools include Perelman's work on the Ricci flow, and over thirty years of insight gained from developing Thurston's theory of hyperbolic 3-manifolds. While these techniques usually do not transfer literally into higher dimensions, the PI thinks it is an opportune moment to study the geometry and topology of higher-dimensional hyperbolic manifolds, and benefit from lower-dimensional intuition and insights.This work will build on an existing collaboration with Steven Kerckhoff.With the natural motivation of understanding the 3 (or 4) dimensional world around us, most research in geometry focuses on "low dimensions", usually meaning less than 5. Nonetheless, in the modern world, higher dimensional geometric objects are abundant: contemporary computer science employs abstract "simplicial complexes" of very high dimension to model concrete systems, studying financial markets often requires estimating integrals over high dimensional spaces, and internet search engines rely on efficient linear algebra algorithms for very large vector spaces. This proposal will focus on studying a specific type of geometry in high dimensions, namely hyperbolic geometry. Traditionally, approaching this subject was difficult without a powerful computer. This is no longer a major obstacle, and it is possible to understand concrete, yet complex, examples. The goal is to build on our low dimensional intuition to better understand high dimensional objects.