Coarse Geometry of Topological Groups

拓扑群的粗略几何

基本信息

批准号：
1764247
负责人：
Christian Rosendal
金额：
$ 23.99万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-06-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764247&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，化学和物理学的许多领域中，并且作为数学中的组织工具产生了巨大影响。它们通常以几何对象的对称性（例如3维空间，分子或晶体）的形式出现。通常，一组对称性本身，即群体具有自然拓扑结构。也就是说，一个人可以说一个对称性靠近另一个对称性，例如，如果三维空间的两个旋转，则如果它们以小角度的不同，则它们是接近的。这些后一组称为拓扑转换组，并通过几何对象与几何对象相关。但是，其他拓扑组并不容易被视为来自几何形状。例如，这是针对许多微分方程的解决方案系统的情况，这些方程形成了一个在添加的班级空间下的组。但是，本项目的主要思想之一是所有拓扑组都有自然的内在几何结构，这是由它们的拓扑结构和代数结构共同定义的。此外，在许多情况下，这种几何结构可以通过掩盖掩盖该组的全球或大规模特性的更细节来对组的结构进行重大见解。该项目的主要目的是研究拓扑，尤其是波兰群体的粗糙几何形状。在较早的研究中，PI建立并研究了每个拓扑组都配备的自然粗糙结构，并与传统上研究有限产生或局部紧凑的群体，Banach空间甚至同型紧凑型歧管的群体相吻合。要研究的特别有趣的子类是几何结构理论的界限几何形状群。例如，关于这类群体范围的几个有趣的问题仍然是开放的，例如，每一个界限的几何形状是否都与本地紧凑的群体相当。该研究计划本质上是跨学科的。尽管它的起源在于描述性集理论，但要研究的群体的主要例子是在数学的各种学科中出现的，包括功能分析，逻辑，几何和差异拓扑。该奖项反映了NSF的法定任务，并通过使用该基金会的智力功能和广泛的影响来评估CRITERIA CRITERIA。