Geometry of manifolds with large volume

一个薄的橡胶球可以被拉伸、推入或扭曲，从而失去其圆度。例如，可以收缩赤道以将圆形球体转换为沙漏。虽然沙漏的颈部看起来不再像一个圆球，但两个灯泡仍然很圆。著名的高斯-博内定理说，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。数学家将这一结果视为一个例子，其中对象的拓扑结构（即，它是通过扭曲圆形球体而产生的事实）约束了它的几何形状。这个建议的广泛主题是开发类似的结果有关高维抽象几何对象。Perelman在2003年证明的Thurston的几何化猜想指出，每个封闭的三流形都可以被切割成碎片，每个碎片都允许八种类型的齐次度量之一。在这些作品中，只有双曲三流形没有被分类。 佩雷尔曼结果的完整陈述包括了当闭三流形M容许双曲度量时的拓扑条件。Mostow刚性定理意味着这样的M上的双曲度量是唯一的，如果它存在的话，所以很自然地试图从M的拓扑中提取关于度量的具体几何信息。有效几何化（Effective geometrization）是PI及其合作者研究的一个程序，用于从三维流形的拓扑结构中提取关于双曲度量的具体几何信息。这个项目的很大一部分涉及到找到一个描述的几何M只给需要产生的基本组M的元素的数量是有界的，或M有一个拓扑分解与某些特点。其关键技术是通过几何极限来渐近地理解闭双曲三流形序列。这种观点也将适用于双曲几何以外，在研究的增长贝蒂数序列的高秩局部对称空间，在一个程序的灵感来自一个活跃的领域在图论。

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.