Geometries of topological groups

该项目旨在解决关于拓扑群几何化的基本开放问题。拓扑群，特别是波兰群，是该提议关注的中心对象，出现在整个数学中，并且经常以200年前伽罗瓦最初构想的群的形式出现，即作为各种数学对象的对称性的集合。鉴于这些群体本身没有明确的距离概念，因此也没有明确的几何形状，该项目致力于揭示可以从其代数和拓扑结构中定义的隐含几何形状。提出的数学问题汇集了源自数学逻辑、分析和度量几何的思想，同时它们的解决方案将开发一套适用于其他领域（如几何拓扑）的工具集。该项目还将通过培训学生和博士后学者，为美国的劳动力发展做出贡献，并通过有针对性的会议和研讨会组织，为数学科学社区建设做出贡献。除了它们的拓扑和代数结构，拓扑群还以规范的粗糙和均匀结构的形式携带定义良好的大规模和小规模几何，这些结构可能是或可能不是群上潜在的大规模或小规模Lipschitz几何的实例。虽然这种lipschitzi化问题在大尺度几何中得到了令人满意的解决，但在小尺度几何中仍有很大的开放性。这个提议的一个方面就是通过为承认小尺度Lipschitz几何的群构造一个适当的Banach-Lie代数，并最终将它们表征为Banach-Lie群的封闭子群，从而在这个问题上取得进展。其他问题涉及波兰群中有界集的扬弃，局部紧空间上粗固有作用的非交换类似，以及在这些空间上允许粗固有和紧的群的新构造。该项目还旨在通过用Wasserstein距离的接近性代替l1范数的接近性来更好地掌握适应性的大规模几何含义，从而将适应性与最优运输理论中的问题联系起来。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。

This project aims to solve basic open questions regarding the geometrisation of topological groups. Topological groups and, in particular, the Polish groups that are the central object of attention of the proposal, appear throughout mathematics and very often in the form that groups were initially conceived by Galois 200 years ago, namely, as collections of symmetries of various mathematical objects. Whereas the groups themselves have no explicit concept of distance and therefore also no explicit geometry, the project is devoted to unveiling the implicit geometry that can be defined from their algebraic and topological structure. The mathematical problems proposed bring together ideas stemming from mathematical logic, analysis, and metric geometry, while at the same time their solution will develop a tool set applicable to other areas such as geometric topology. The project will also contribute to US workforce development, through the training of students and post-doctoral scholars, and to community building in the mathematical sciences through targeted conference and workshop organisation.Apart from their topological and algebraic structure, topological groups carry well-defined large scale and small scale geometries in the form of canonical coarse and uniform structures that may or may not be instances of underlying large or small scale Lipschitz geometries on the group. Although this Lipschitziation problem is satisfyingly solved for large scale geometry, it remains largely open for small scale geometry. One facet of the proposal is exactly to make progress on this problem by constructing an appropriate Banach–Lie algebra for groups admitting small scale Lipschitz geometry and perhaps ultimately characterise them as closed subgroups of Banach–Lie groups. Other problems in the proposal concern liftings of bounded sets in Polish groups, non-commutative analogs of coarsely proper actions on locally compact spaces and new constructions of groups admitting coarsely proper and cocompact on said spaces. The project also aims to provide a better grasp of the large scale geometric implications of amenability by substituting proximity in L1-norm by proximity in Wasserstein distance and thereby connecting amenability with issues in optimal transport theory.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.