Geometric Structures

最近有一个巨大的数额的进展领域的Kleinian群和几何的3-流形。几个长期存在的问题已经解决，最引人注目的是佩雷尔曼的几何化猜想的证明，但也包括结束层压猜想（明斯基，布罗克和金丝雀）和驯服猜想（阿戈尔，Calegari-Gabai）。 这个提议的目标是更准确地理解三维流形上的几何结构如何反映在它们的拓扑性质中。 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将柔性流形与边界粘合来获得它们。 要获得关于几何结构的明确的定量信息，需要对获得这些信息的柔性结构有更好的理解。这些变分技术也可以应用于理解其他类型的几何结构，如投影和洛伦兹结构，没有一个不变的度量。 我们也将考虑其他维度的几何结构（高维的双曲结构，曲面上的射影结构），在这个建议中，我们希望结合分析、几何和拓扑工具来研究三维和其他维度的几何结构。 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。 三维几何特别有吸引力，因为它对许多人来说都是视觉上可访问的，包括数学初学者和那些技术背景较低的人。 它也有物理应用程序，并导致创建了许多已被广泛使用的图形界面。

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.