Geometry of manifolds with large volume

一个薄薄的橡胶球体可以被拉伸、推入或以其他方式扭曲，从而失去其圆度。例如，人们可以收缩赤道，将圆形球体变成沙漏。虽然沙漏的颈部看起来不再像一个圆形球体，但两个灯泡仍然很圆。著名的Gauss-Bonnet定理说，在某种精确的意义上，无论球体如何扭曲，结果的平均几何形状仍然是圆的。数学家认为这一结果是一个例子，一个物体的拓扑结构(即它是通过扭曲一个圆形球体产生的)限制了它的几何形状。这项提议的主要主题是发展与高维抽象几何对象有关的类似结果。瑟斯顿的几何化猜想，由佩雷尔曼在2003年证明，每一个封闭的三流形都可以切成碎片，每一块都允许八种齐次度量中的一种。在这些作品中，只有双曲三流形没有被归类。Perelman结果的完整陈述包括刻画闭三维流形M何时接纳双曲度量的拓扑条件。Mostow刚性定理意味着这样一个M上的双曲度量是唯一的，如果它存在的话，那么从M的拓扑中提取关于该度量的具体几何信息是很自然的。由PI和他的合作者研究的有效几何化是一个程序，它从三维流形上的双曲度量的拓扑中提取关于该度量的具体几何信息。这个项目的很大一部分涉及到寻找M的几何的描述，只要给定生成M的基本群所需的元素的数目是有界的，或者M具有具有某些特征的拓扑分解。其关键技术是通过传递到几何极限来渐近地理解闭双曲三维流形序列。这一观点也将在双曲几何之外应用于研究高阶局部对称空间序列中Betti数的增长，该程序受图论中的一个活跃领域的启发。

A thin rubber sphere can be stretched, pushed in, and otherwise distorted so that its roundness is lost. For instance, one can contract the equator to transform the round sphere into an hourglass. While the neck of the hourglass does not look much like a round sphere anymore, the two bulbs are still pretty round. The celebrated Gauss-Bonnet theorem says that in some precise sense, no matter how the sphere is distorted, the average geometry of the result will still be round. Mathematicians view this result as an example where the topology of an object (i.e., the fact that it was produced by distorting a round sphere) constrains its geometry. The broad theme of this proposal is to develop similar results pertaining to higher dimensional abstract geometric objects. Thurston's Geometrization Conjecture, proved by Perelman in 2003, states that every closed three-manifold can be cut into pieces, each of which admits one of eight types of homogenous metrics. Of these pieces, only the hyperbolic three-manifolds have not been classified. The full statement of the Perelman's result includes topological conditions that characterize when a closed three-manifold M admits a hyperbolic metric. Mostow's Rigidity Theorem implies that a hyperbolic metric on such an M is unique, if it exists, so it is natural to try to extract concrete geometric information about the metric from the topology of M. Effective geometrization, studied by the PI and his collaborators, is a program to extract concrete geometric information about a hyperbolic metric on a three-dimensional manifold from its topology. A large part of this project involves finding a description of the geometry of M given only that the number of elements needed to generate the fundamental group of M is bounded, or that M has a topological decomposition with certain characteristics. The key technique is to understand a sequence of closed hyperbolic three-manifolds asymptotically by passing to geometric limits. This perspective will also be applied outside of hyperbolic geometry, in the study of the growth of Betti numbers in sequences of higher rank locally symmetric spaces, in a program inspired by an active field in graph theory.