Geometries of topological groups

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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.