This NSF funded research project is in topology, a central field in modern mathematics. The main goal is to use ideas from theoretical particle physics (gauge theory)to develop new tools for studying three-dimensional geometric shapes. The principal investigator also plans to apply these tools to study the problem of which high dimensional shapes can be triangulated, that is, decomposed into simple pieces (similar to a decomposition of a surface into triangles). Triangulations have applications beyond pure mathematics, for example in computer graphics. The project also aims to support graduate education and to disseminate mathematical ideas to the general public.
In more technical terms, the project concerns Floer theories associated to three-manifolds, and their topological applications. In particular, the principal investigator will investigate Seiberg-Witten Floer stable homotopy types, the associated Pin(2)-equivariant Seiberg-Witten Floer homology, and involutive Heegaard Floer homology. These theories can be used to get information about the homology cobordism group in three dimensions. In turn, homology cobordism gives insight into the classification of triangulations for manifolds of dimension at least five. The principal investigator also proposes to understand the behavior of Heegaard Floer homology for coverings of 3-manifolds, and to develop 3-manifold invariants from the moduli spaces of SL(2,C) flat connections.