Topology and non-positive curvature

主要研究者（PI）建议围绕非正曲率的概念开展一系列项目。提案中的大多数项目本质上是跨学科的，可能需要涉及各种不同领域（微分和度量几何，代数和微分拓扑等）技术的解决方案。PI的项目大致分为三大类：1. CAT（0）-流形的拓扑（更一般地说，非球面流形的拓扑）：PI计划专注于流形的黎曼非正曲率和度量非正曲率之间的区别。PI还提出了一些寻找格子群的同调准则。另一个项目涉及研究一些新的非球面流形的例子。2.黎曼几何：PI计划从黎曼几何的观点探索非正曲率的流形。这包括研究测地线长度的分布（算术级数），在谱理论中构造新现象（人们能听到Hauptvermutung吗？），研究几乎非正曲流形的结构。3.代数K-理论：PI计划继续研究非正曲率下代数K-理论的计算方面。一些具体的目标包括建立经典Kleinian群的Farrell-Jones同构猜想，构造映射类群的分类空间的“小”模型，以及找到计算3-orbifold群的低代数K-理论的有效算法。拓扑学专注于定性地理解对象。例如，一个球体和一个椭球体在数量上是不同的：球体包含一个点（中心），通过该点所有的“切片”都是相同的，而椭球体不包含这样的点。但在性质上，它们是相同的：人们可以“压扁”一个椭球体，并把它变成一个球体（没有穿刺或撕裂）。相比之下，球体和甜甜圈在性质上是不同的;甜甜圈有一个“洞”，你不能把一个球体变成一个甜甜圈而不穿刺。本提案中的项目研究几何和拓扑之间的关系。换句话说，空间的定性性质如何约束空间的定量性质（反过来又被空间的定量性质约束）。

The Principal Investigator (PI) proposes to work on a series of projects that are centered around the notion of non-positive curvature. Most of the projects in the proposal are interdisciplinary in nature, and will likely require solutions that involve a mix of techniques from various different fields (differential and metric geometry, algebraic and differential topology, etc.). The PI's projects loosely fall into three broad categories: 1. Topology of CAT(0)-manifolds (and more generally, of aspherical manifolds): the PI plans on focusing on the distinction, for manifolds, between Riemannian non-positive curvature and metric non-positive curvature. The PI also proposes some homological criterion for finding lattice subgroups. Another project involves studying some new examples of aspherical manifolds. 2. Riemannian geometry: the PI plans on exploring manifolds of non-positive curvature from the viewpoint of Riemannian geometry. This includes studying the distribution of lengths of geodesics (arithmetic progressions), constructing new phenomena in spectral theory (can one hear the Hauptvermutung?), and studying the structure of almost non-positively curved manifolds. 3. Algebraic K-theory: the PI plans on continuing his research on computational aspects of algebraic K-theory in the presence of non-positive curvature. Some concrete goals include establishing the Farrell-Jones Isomorphism conjecture for classical Kleinian groups, constructing "small" models for classifying spaces for mapping class groups, and finding efficient algorithms for computing the lower algebraic K-theory of 3-orbifold groups.Geometry is usually focused on understanding objects quantitatively. Topology is focused on understanding objects qualitatively. For example, a sphere and an ellipsoid are quantitatively different: the sphere contains a point (the center) through which all "slices" are identical, while an ellipsoid contains no such point. But qualitatively they are the same: one can "squish" an ellipsoid and turn it into a sphere (without puncturing or tearing). In contrast, a sphere and a donut are qualitatively different; the donut has a "hole", and you cannot turn a sphere into a donut without puncturing. The projects in this proposal study the relationship between geometry and topology. In other words, how do qualitative properties of a space constrain (and in turn, are constrained by) the quantitative properties of the space.