Coarse Geometry of Topological Groups

拓扑群的粗略几何

基本信息

批准号：
2204849
负责人：
Christian Rosendal
金额：
$ 23.99万
依托单位：
University of Maryland, College Park
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-11-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204849&HistoricalAwards=false
关键词：
Coarse Geometry Topological Groups

项目摘要

Groups appear in numerous areas of mathematics, chemistry and physics and have had tremendous impact as an organizational tool within mathematics. They often appear as the set of symmetries of a geometric object, for example, of 3-dimensional space, a molecule or a crystal. Oftentimes, the set of symmetries themselves, i.e., the group, has a natural topological structure. That is, one can speak of one symmetry being close to another, as in the case of two rotations of 3-dimensional space being close if they differ by a small angle. These latter groups are called topological transformation groups and are trivially related to geometry via the geometric object of which they are the set of symmetries. However, other topological groups are not so easily viewed as coming from geometry. This is, for example, the case for the systems of solutions to many differential equations which form a group under addition called a Banach space. However, one of the principal ideas of the present project is that all topological groups have natural intrinsic geometric structure which is defined jointly by their topological and algebraic structure. Moreover, this geometric structure can in many cases provide significant insight into the structure of the group by blotting out finer details that obscure the global or large scale properties of the group.The primary aim of this project is to investigate the coarse geometry of topological and, in particular, Polish groups. In earlier research, the PI has established and investigated a natural coarse structure that every topological group is equipped with and which coincides with that traditionally studied on finitely generated or locally compact groups, Banach spaces or even homeomorphism groups of compact manifolds. Particularly interesting subclasses to be studied are the Polish groups of bounded geometry for which the geometric structure theory is well-advanced. Several interesting questions on the extent of this class of groups remain open, for example, whether every Polish group of bounded geometry is coarsely equivalent to a locally compact group. The research program is by nature interdisciplinary. While its origins lie in descriptive set theory, the main examples of groups to be studied arise in various disciplines of mathematics, including functional analysis, logic, and geometric and differential topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

小组出现在数学、化学和物理的许多领域，并且作为数学中的组织工具产生了巨大的影响。它们通常表现为几何对象的对称性集合，例如 3 维空间、分子或晶体。通常，对称性集本身（即群）具有自然的拓扑结构。也就是说，我们可以说一种对称性接近另一种对称性，就像 3 维空间的两次旋转如果相差很小的角度则接近的情况一样。后面的这些群被称为拓扑变换群，并且通过作为对称集的几何对象与几何无关。然而，其他拓扑群并不那么容易被视为来自几何。例如，许多微分方程的解系统就是这种情况，这些微分方程在加法下形成一个群，称为巴纳赫空间。然而，本项目的主要思想之一是所有拓扑群都具有自然的内在几何结构，该结构由它们的拓扑和代数结构共同定义。此外，这种几何结构在许多情况下可以通过消除掩盖群的全局或大尺度特性的更精细的细节来提供对群结构的重要洞察。该项目的主要目的是研究拓扑和拓扑的粗略几何形状。，特别是波兰团体。在早期的研究中，PI建立并研究了每个拓扑群都配备的自然粗结构，该结构与传统上对有限生成或局部紧群、Banach空间甚至紧流形同胚群的研究相一致。要研究的特别有趣的子类是有界几何的波兰群，其几何结构理论非常先进。关于此类群的范围的几个有趣的问题仍然悬而未决，例如，有界几何的每个波兰群是否粗略地等效于局部紧群。该研究项目本质上是跨学科的。虽然它起源于描述性集合论，但要研究的群的主要例子出现在数学的各个学科中，包括泛函分析、逻辑、几何和微分拓扑。该奖项反映了 NSF 的法定使命，并被认为值得通过以下方式获得支持：使用基金会的智力价值和更广泛的影响审查标准进行评估。