Coarse Geometry of Topological Groups

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对象的对称性集合，例如，三维空间，分子或晶体。通常，对称性本身的集合，即，该群具有自然的拓扑结构。也就是说，我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，那么它们就很接近。这些后一组被称为拓扑变换群，并通过几何对象，它们是几何对象的对称性的集合，与几何有着平凡的关系。然而，其他拓扑群不那么容易被视为来自几何。这是，例如，系统的解决方案，以许多微分方程，形成一组下，除了所谓的Banach空间。然而，本项目的主要思想之一是，所有的拓扑群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，这种几何结构可以在许多情况下提供显着的洞察到组的结构，通过涂抹更精细的细节，掩盖了全球或大规模的属性group.The项目的主要目的是研究粗糙的几何拓扑，特别是波兰的群体。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一种自然粗糙结构，这与传统上研究的局部紧群，Banach空间甚至紧流形的同胚群相一致。特别有趣的子类进行研究的波兰群体有界几何的几何结构理论是先进的。几个有趣的问题的程度，这类群体仍然开放，例如，是否每一个波兰组的有界几何是粗略相当于一个局部紧组。该研究计划是自然跨学科。虽然它的起源在于描述集合论，但要研究的群体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。这个奖项反映了NSF的法定使命，并被认为值得通过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来支持。

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.