Geometric Structures

最近在Kleinian群和3流形几何领域有了巨大的进展。几个长期存在的猜想已经得到了解决，最引人注目的是佩雷尔曼对几何化猜想的证明，但也包括结束层压猜想（Minsky， Brock和Canary）和驯服猜想（Agol, Calegari-Gabai）。本文的目标是更精确地理解3流形上的几何结构是如何反映在它们的拓扑性质上的。由于闭流形上的双曲结构是唯一的，通过族来研究它们通常需要允许奇点或通过粘接有边界的柔性流形来获得奇点。要获得关于几何结构的明确的、定量的信息，需要对从中获得这些信息的柔性结构有精细的理解。这些变分技术也可以应用于理解其他类型的几何结构，例如没有不变度规的投影和洛伦兹结构。我们还将考虑其他维度几何结构的有趣问题（高维的双曲结构，曲面上的投影结构）。在这个建议中，我们希望研究三维几何结构，以及其他维度，使用分析，几何和拓扑工具的组合。这是一个非常活跃和有趣的领域，最近有新的想法和技术涌入。三维几何特别吸引人，因为它对很多人来说都是直观的，包括初学数学的学生和那些没有太多技术背景的人。它还应用于物理学，并导致了许多图形界面的创建，这些界面已被广泛使用。

There has recently been a tremendous amount of progress in the areas of Kleinian groups and the geometry of 3-manifolds. Several long-standing conjectures have been solved, the most spectacular being Perelman's proof of the Geometrization Conjecture, but also including the Ending Lamination Conjecture (Minsky, Brock and Canary) and the Tameness Conjecture (Agol, Calegari-Gabai). The goal of this proposal is to get a more exact understanding of how geometric structures on 3-manifolds are reflected in their topological properties. Since hyperbolic structures on closed manifolds are unique, studying them through families typically requires allowing singularities or obtaining them by gluing flexible manifolds with boundary. Getting explicit, quantitative information about a geometric structure requires a refined understanding of the flexible structures from which they are obtained. These variational techniques can also be applied to understand other types of geometric structures, such as projective and Lorentzian structures, that don't have an invariant metric. We will also consider interesting questions about geometric structures in other dimensions (hyperbolic structures in high dimensions, projective structures on surfaces).In this proposal we wish to study geometric structures in dimension 3, as well as other dimensions, using a combination of analytic, geometric, and topological tools. This is a very active and interesting area that has recently had an influx of new ideas and techniques. Geometry in dimension 3 is particularly appealing because it is visually accessible to many people, including beginning mathematics students and those with less technical backgrounds. It also has applications to physics and has led to the creation of a number of graphical interfaces that have been widely utilized.