Geometry and topology of hyperbolic manifolds

这个建议的主要目标是转移到高维的一些技术和直觉获得最近壮观的进展，在低维双曲几何。 目前强大的工具包括佩雷尔曼的工作里奇流，并超过三十年的洞察力从发展瑟斯顿的理论双曲3流形。 虽然这些技术通常不会从字面上转移到更高的维度，PI认为这是一个研究高维双曲流形的几何和拓扑的合适时机，并受益于低维的直觉和见解。这项工作将建立在与Steven Kerckhoff现有的合作基础上。随着理解我们周围的3（或4）维世界的自然动机，几何学的大多数研究集中在“低维”上，通常意味着小于5。 尽管如此，在现代世界中，更高维的几何对象是丰富的：当代计算机科学使用非常高维的抽象“单纯复形”来建模具体系统，研究金融市场通常需要估计高维空间上的积分，互联网搜索引擎依赖于非常大的向量空间的有效线性代数算法。 这个建议将集中在研究一个特定类型的几何在高维，即双曲几何。 传统上，如果没有强大的计算机，处理这个问题是困难的。 这不再是一个主要障碍，可以理解具体但复杂的例子。 我们的目标是建立在我们的低维直觉上，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.