In this proposal Baldridge investigates the topology of smooth 4-manifolds using gauge theory techniques. The goals of the proposer are to find easier ways to calculate the Seiberg-Witten invariants of 4-manifolds, to provide new examples of symplectic 4-manifolds, and to better understand the topology of 4-manifolds which admit a circle action and symplectic form. The first project involves studying the moduli space of solutions to the Seiberg-Witten equations of a smooth 4-manifold when the Dirac operator is conjugated by the exponential of a Morse function. In a collaborative project with Li, the proposer also plans to extend the geography problem for symplectic 4-manifolds to nonsimply connected manifolds and fill out this geography for symplectic manifolds with Kodaira dimension 1. In the third and fourth projects the proposer explores ideas related to Taubes's question, that is: If a 4-manifold is symplectic and diffeomorphic to the product of a three manifold with a circle, does that three manifold fiber over a circle?
A 3-manifold is a space that locally looks like the familiar space we live in. If one imagines replacing every point in the 3-manifold with a circle of points in a nice way, one gets an example of a 4-manifold with a circle action. Here `action' means a rotation of the space along each circle. Four manifolds with circle actions have many nice periodic and symmetry properties, which makes them particularly suitable for modeling and testing physical theories. (For example, the space-time universe we live in is an example of a general 4-manifold which may have a circle action.) These manifolds are especially useful for modeling if they also have a symplectic structure --- a key ingredient in almost all the equations of classical and quantum physics. Work in this proposal investigates what shapes symplectic 4-manifolds with circle actions can have, and the shapes of 4-manifolds in general. This is done by relating invariant features of a 4-manifold to the number of solutions of certain systems of nonlinear partial differential equations on that manifold.
