Geometries of topological groups

该项目旨在解决有关拓扑群几何化的基本开放问题。拓扑群，特别是作为该提案关注的中心对象的波兰群，出现在整个数学中，并且经常以伽罗瓦在 200 年前最初设想的群的形式出现，即作为各种数学对象的对称性的集合。尽管群本身没有明确的距离概念，因此也没有明确的几何形状，但该项目致力于揭示可以从其代数和拓扑结构定义的隐式几何形状。提出的数学问题汇集了源自数理逻辑、分析和度量几何的思想，同时它们的解决方案将开发出适用于几何拓扑等其他领域的工具集。该项目还将通过培训学生和博士后学者，为美国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，为数学科学领域的社区建设做出贡献。除了拓扑和代数结构之外，拓扑群还以规范的粗糙和均匀结构的形式承载着明确定义的大尺度和小尺度几何，这些结构可能是也可能不是群中潜在的大尺度或小尺度 Lipschitz 几何的实例。尽管这个利普希齐化问题对于大尺度几何体来说已经得到了令人满意的解决，但对于小尺度几何体来说，它仍然很大程度上是开放的。该提案的一个方面是通过为承认小规模 Lipschitz 几何的群构造适当的 Banach-Lie 代数，并最终将它们描述为 Banach-Lie 群的封闭子群，从而在这个问题上取得进展。该提案中的其他问题涉及波兰群中有界集的提升、局部紧空间上粗略适当动作的非交换类比以及在所述空间上承认粗略适当和协紧的群的新构造。该项目还旨在通过用 Wasserstein 距离中的接近度取代 L1 范数中的邻近度，从而更好地掌握舒适度的大规模几何含义，从而将舒适度与最佳交通理论中的问题联系起来。该奖项反映了 NSF 的法定使命，并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，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.