Award: DMS 1708310, Principal Investigator: Clifford H. Taubes
The immediate goal for the project is to develop new tools and methods to study the structure of three and four dimensional spaces. Questions about these spaces are central to much of current research in both geometry and topology (not to mention physics and astrophysics/cosmology - four dimensions is relevant because space with time is four dimensional). There exist powerful mathematical tools for studying these spaces, but more tools are needed. This is especially true for four dimensional spaces because the ignorance here is broad and deep. This lack of information motivates the project to develop new tools and sharpen the old ones while keeping in mind the long term goal of understanding more about the spaces involved. The research to be carried out could construct bridges between the fields of topology, differential geometry, symplectic geometry and mathematical physics. Concepts from the mathematical fields of the project are now used extensively (and centrally) in condensed matter physics, high energy physics, and theories of gravity and cosmology.
The long term goal of this project is to understand the structure of smooth, low dimensional manifolds through the following primary areas of study: (1) The behavior of sequences of solutions to certain generalizations of Seiberg-Witten equations will be studied on four dimensional manifolds. The ultimate goal in this regard is to determine whether non-convergent sequences have pathologies that obstruct solution counting definitions of invariants that can distinguish different 4-manifolds or, when the four dimensional manifold has a boundary, distinguish different knots or links on the boundary. (2) The structure and topological significance of the zero locus of a harmonic section of a spin bundle that is defined only where the section is not zero will be investigated. This second research focus has implications for the first because harmonic sections of the sort described arise in the study of sequences of solutions to certain equations of interest. (3) The third research focus will look for invariants that could distinguish path components of the space of Riemannian metrics on a 4-manifold with anti-self dual Weyl curvature. Of particular interest would be an invariant that distinguishes the products of a circle with hyperbolic three-dimensional manifolds because an invariant that does this might distinguish certain pairs of homeomorphic, but not necessarily diffeomorphic, four dimensional manifolds that can not be separated with existing tools for smooth structures. All of these topics use methods from differential geometry and analysis to construct new tools for studying the differential topology of low dimensional manifolds.
