Coarse Geometry of Topological Groups

分组出现在数学、化学和物理的许多领域，并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物体的对称集合，例如三维空间、分子或晶体。通常，对称集本身，即群，具有自然的拓扑结构。也就是说，我们可以说一种对称接近另一种对称，就像在三维空间的两个旋转中，如果它们相差一个小角度，它们就会很接近。后一种群被称为拓扑变换群，它们通过几何对象（它们是几何对象的对称集合）与几何有平凡的联系。然而，其他拓扑群不那么容易被视为来自几何学。例如，这是许多微分方程的解的系统的情况，它们在加法下形成一个群，叫做巴拿赫空间。然而，本课题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，这种几何结构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掩盖掩盖群体整体或大规模属性的更精细的细节来提供对群体结构的重要洞察。该项目的主要目的是研究拓扑的粗糙几何，特别是波兰群体。在早期的研究中，PI已经建立并研究了每个拓扑群都具有的自然粗结构，这与传统上对紧流形的有限生成或局部紧群、Banach空间甚至同胚群的研究一致。要研究的特别有趣的子类是波兰有界几何群，它们的几何结构理论非常先进。关于这类群的范围的几个有趣的问题仍然是开放的，例如，是否每一个有界几何的波兰群都粗等价于一个局部紧群。这个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，包括功能分析、逻辑、几何和微分拓扑。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.