In this proposal Baldridge works on two research strands: The study of the geography problem in 4-dimensions and the study of symplectic 4-manifolds that admit a circle action. The first strand centers on the ``Symplectic Poincare Conjecture,?? a question about whether or not the topological space of the complex projective plane has a unique smooth structure that supports a symplectic form. The second strand centers on a question first posed by C. Taubes: if the product space of a 3-manifold and a circle admits a symplectic form, does the 3-manifold fiber over a circle? Baldridge?s previous work in each strand has formed and influenced his work in the other--the two strands are deeply connected. The research in this proposal would investigate the following questions related to the conjectures above: (1) Which manifolds with small Euler characteristic admit smooth exotic structures? Which of those exotic structures admit a symplectic form? (2) Are there criteria under which a symplectic 4-manifold is (up to diffeomorphism) uniquely defined by its topological data? (3) What interesting phenomena occur in the geography problem as one gets close to the Bogomolov-Miyaoka-Yau line? (4) Which circle bundles over a fibered 3-manifold admit a symplectic form?
A 3-manifold is a space that locally looks like the familiar space in which we live. 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 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. Baldridge will investigate the shapes of symplectic 4-manifolds with circle actions and of 4-manifolds in general. This investigation is done by constructing new examples of smooth and symplectic 4-manifolds and by distinguishing 4-manifolds from each other by relating invariant features of a 4-manifold to the number of solutions of certain systems of nonlinear partial differential equations on that manifold. In addition to his research work in mathematics, Baldridge will strive to help the next generation of U.S. citizens learn mathematics through his work on three projects. The first project is to complete a comprehensive college curriculum for prospective elementary and middle school teachers. The second is to bring outstanding mathematicians to Louisiana State University to be ``Scholars in Residence?? to interact with participants in a new professional Master?s degree program for teachers. The final project is professional development for teachers in a high-need school district that has the potential to create a replicable model for building high-performing mathematics programs in high-need schools.
