Coarse Geometry of Topological Groups

小组出现在数学、化学和物理的许多领域，并作为数学中的一种组织工具产生了巨大的影响。它们通常表现为几何对象的一组对称，例如，三维空间、分子或晶体的对称性。通常，对称组本身，即群，具有自然的拓扑结构。也就是说，人们可以说一个对称性接近另一个对称性，就像三维空间的两个旋转如果相差一个小角度就接近了。这些后一类群被称为拓扑变换群，它们通过它们是其对称集的几何对象与几何关系平凡地相关。然而，其他拓扑组并不那么容易被视为来自几何图形。例如，这就是许多微分方程组的解的情况，这些微分方程组在加法下形成一个群，称为Banach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，这是由它们的拓扑结构和代数结构共同定义的。此外，这种几何结构在许多情况下可以通过遮盖掩盖群的全局或大规模性质的更精细的细节来提供对群的结构的重要洞察。这个项目的主要目的是研究拓扑群的粗略几何，特别是波兰群。在早期的研究中，PI已经建立并研究了每个拓扑群都配备的一个自然粗结构，它与传统上研究的有限生成或局部紧群、Banach空间甚至紧流形的同胚群的结构一致。要研究的特别有趣的子类是有界几何的波兰群，其几何结构理论是非常先进的。关于这类群的范围的几个有趣的问题仍然悬而未决，例如，是否每个有界几何的波兰群粗略地等价于一个局部紧群。该研究项目本质上是跨学科的。虽然它的起源是描述集合论，但要研究的群体的主要例子出现在不同的数学学科，包括泛函分析、逻辑、几何和微分拓扑学。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.