Monopole Floer homology and the torus of flat connections

A fundamental problem in science is to understand the relation between the shape of a given object and its physical properties: for example, how the shape of a drum influences the frequencies at which it vibrates. To address this question from a mathematical perspective, one needs to study how geometry and topology interact with solutions to partial differential equations. While the study of the Seiberg-Witten equations has provided fundamental tools to tackle problems in low-dimensional topology, their relation with geometry is still very mysterious. The main goal of the project is to tackle this question using spectral theory (the study of the frequencies at which objects vibrate) as a stepping stone. Towards this end, this project will also create many research opportunities both at the undergraduate and graduate level, and support the preparation of an introductory graduate level textbook on the topic. The PI will work on several projects on monopole Floer homology all having as protagonist (for a given torsion spin^c structure on the three-manifold) the torus of flat spin^c connections and the geometry of the locus corresponding to Dirac operators with non-trivial kernel. In particular, the PI will: develop tools to explicitly compute Floer theory purely in terms of hyperbolic geometry for manifolds with large first Betti number; study interactions with the spectral and complex geometry of surfaces; explore the geometric meaning of non-vanishing Floer homology groups in terms of vector fields and plane distributions. 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.