Probing Near-Symplectic 4-Manifolds and Contact 3-Manifolds with Seiberg-Witten Theory

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Gauge theory describes and exploits the symmetries of nature; among other things it forms the foundation of classical electrodynamics and quantum physics. Symplectic geometry describes the dynamics of nature; among other things it forms the foundation of classical mechanics. Our universe seemingly has four dimensions, three spatial and one time, and it has some curvature, but we do not know exactly what the global picture is. Motions of particles in our universe can be cyclical, but we do not always know how many such periodic trajectories these particles can have. The goal of this project is to use both gauge theory and symplectic geometry in tandem to detect the possible smooth shapes of our universe and to probe the evolution of systems that can arise in said universe, and to develop these mathematical tools further in order to be more efficient in our calculations. Especially, the PI will check whether certain dynamical systems have infinitely many periodic orbits, using the gauge-theoretic “Seiberg-Witten monopoles” and the symplectic-style “Reeb orbits”. The broader impacts of this pursuit involve mentoring students and outreach.The classification of smooth 4-manifolds and dynamics of Reeb vector fields on 3-manifolds have been long-sought out goals in the community, with much progress. This project will be able to contribute with new and refined methods, by extending and exploiting the relations between Seiberg-Witten solutions and pseudoholomorphic curves and Reeb orbits. The Seiberg-Witten invariants of 4-manifolds may be recovered by suitable counts of said curves and orbits, using 2-forms that are symplectic almost everywhere. The PI intends to use this transcription and these near-symplectic 2-forms to probe the structure of the SW invariants, and to search for diffeomorphisms between two homeomorphic symplectic 4-manifolds that are known to become diffeomorphic after a blow-up. In 3 dimensions the PI intends to give quantitative refinements to the Weinstein conjecture that asserts the existence of periodic Reeb orbits on every closed contact 3-manifold, by re-analyzing and extending the proof of said conjecture to the case of certain non-closed contact 3-manifolds using a newly developed SW-Floer homology 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.