Geometrical Methods in Algebra and Topology

PI建议开展一系列项目，将代数、几何和拓扑技术结合起来，着眼于在这三个领域的应用。更准确地说，PI提出了以下四个思想圈：(1)在空间上适当的几何假设下回答拓扑中的各种经典问题（如诺维科夫猜想，Borel猜想，代数k理论的计算）。(2)研究有限生成群的准等距分类问题。(3)探索固定流形上CAT（-1）度量空间与负弯曲黎曼度量空间的边界。(4)研究局部对称空间的拓扑（上同调、有界上同调、子流形）。这些项目旨在展示代数（主要是群论）、几何（负弯曲空间）和拓扑（几何和代数）这三个领域之间的方法和结果的相互依存关系。群是配备了“乘法”的集合，并且经典地作为底层几何对象的对称而出现。几何学是研究带有一些额外结构的空间的性质，通常是度量性质的（即允许人们测量空间中的距离）。在拓扑学领域，人们忘记了潜在的距离，而只保留空间中点“靠近”的概念。虽然这三个领域一直有一些重叠，但过去二十年的研究已经导致它们之间的思想和应用交流日益丰富。PI的研究旨在进一步发展这三个领域之间的联系。

The PI proposes to work on a series of projects that incorporate techniques from algebra, geometry, and topology, with a view to applications in all three of these fields. More precisely, the PI proposes to work on the following four circles of ideas:(1) Answering various classical questions in topology (such as the Novikov conjecture, the Borel conjecture, computation of the algebraic K-theory) under suitable geometric hypotheses on the spaces.(2) Working on problems related to the quasi-isometric classification of finitely generated groups.(3) Exploring the boundary between the space of CAT(-1) metrics and the space of negatively curved Riemannian metrics on a fixed manifold.(4) Studying the topology (cohomology, bounded cohomology, submanifolds) of locally symmetric spaces.These projects are intended to exhibit the interdependence of methods and results between the three fields of algebra (primarily group theory), geometry (negatively curved spaces) and topology (both geometric and algebraic).Groups are sets equipped with a "multiplication", and classically arose as symmetries of underlying geometric objects. Geometry is the study of properties of spaces equipped with some additional structure, usually of a metric nature (i.e. allowing one to measure distances in the space). In the field of topology, one forgets about the underlying distance, and only retain the notion of points in the space being "close together". While these three fields have always had some overlap, research from the last twenty years have led to an increasingly fertile exchange of ideas and applications amongst them. The PI's research is intended to further develop the links between these three fields.