Mathematical Sciences: Three-Dimensional Hyperbolic Geometry

该项目的目的是分析 3 维流形上双曲结构的各个方面。 Kerckhoff 教授将使用相关 Hodge 理论的微分几何分析，研究双曲锥流形、具有 Dehn 手术奇点的流形和无限体积几何有限流形的变形理论。 Canary 教授希望研究双曲 3-流形末端的解析和拓扑结构及其与克莱因群变形理论的关系。 三流形是非常自然的研究对象。 毕竟我们住的是一处。 因此，令人惊讶的是我们对它们的了解仍然存在巨大差距。

The purpose of this project is to analyze various aspects of hyperbolic structures on 3-dimensional manifolds. Professor Kerckhoff will study the deformation theory of hyperbolic cone manifolds, manifolds with Dehn surgery singularities, and infinite-volume geometrically finite manifolds, using a differential geometric analysis of the relevant Hodge theory. Professor Canary wishes to study the analytic and topological structure of the ends of hyperbolic 3-manifolds and their relation to the deformation theory of Kleinian groups. Three-manifolds are very natural objects of study. We live in one, after all. It is therefore surprising what large gaps remain in our knowledge of them.