Geometries of topological groups

这个项目旨在解决关于拓扑群的几何化的基本开放问题。拓扑群，特别是波兰群，是该建议的中心关注对象，出现在整个数学中，并且经常以200年前伽罗瓦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。虽然这些群本身没有明确的距离概念，因此也没有明确的几何，但该项目致力于揭示可以从它们的代数和拓扑结构中定义的隐式几何。提出的数学问题汇集了来自数学逻辑，分析和度量几何的想法，而同时他们的解决方案将开发一套适用于其他领域的工具，如几何拓扑。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，为数学科学社区建设做出贡献。拓扑群很好地携带-以规范的粗糙和均匀结构的形式定义的大尺度和小尺度几何形状，其可以是或可以不是底层的大或小规模的Lipschitz几何。虽然这个Lipschitziation问题是令人满意地解决了大规模的几何，它仍然在很大程度上开放的小规模的几何。该提案的一个方面是通过为允许小规模Lipschitz几何的群体构造适当的Banach-李代数来在这个问题上取得进展，并可能最终将它们作为Banach-李群的闭子群。其他问题的建议涉及提升有界集在波兰团体，非交换类似物的粗适当的行动，局部紧空间和新的建设团体承认粗适当和cocompact说空间。该项目还旨在通过用Wasserstein距离中的邻近性取代L1-norm中的邻近性，从而将舒适性与最佳运输理论中的问题联系起来，更好地掌握舒适性的大规模几何含义。该奖项反映了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。

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.