Topology, Geometry and Physics

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The immediate goal for the project is to develop new tools and methods to study three and four dimensional spaces. Questions about these spaces are central to much of current research in both geometry and topology. There exist powerful tools for studying these spaces, but more is needed, especially to study four dimensional spaces where ignorance is broad and deep. This ignorance motivates the focus here to develop new tools and sharpen the old ones keeping in mind the long term hope of understanding more about the spaces involved. The research described in this proposal could construct bridges between the fields of topology, differential geometry, symplectic geometry and mathematical physics. These fields are central not just to mathematics, but to theoretical physics as well. For example, notions from these fields are used in condensed matter physics, high energy physics, string theory and theories of gravity. The foreseeable future research of C. H. Taubes will focus on differential topology questions with the following three specific areas of concentration. The first area of concentration will be directed towards understanding the behavior of sequences of solutions to certain generalizations of the equations for flat connections on 3 manifolds and for self dual connections on 4-manifolds with a goal to determine if divergent sequences doom solution counting constructions of differential topology invariants. The second area of concentration will investigate the differential topology of certain natural generalizations to dimensions 3 and 4 of the 2-dimensional notion of a holomorphic, quadratic differential. The third area of concentration will use the Seiberg-Witten equations to analyze the dynamics of vector fields on manifolds of dimension three and higher. Of particular interest are conditions that guarantee the existence of closed orbits and other sorts of small dimensional invariant sets. In all of these areas, the research will use methods from differential geometry and analysis to construct new tools for studying the differential topology of low dimensional spaces. Specific topics and questions notwithstanding, the long term goal is to understand the structure of smooth, low dimensional manifolds.