Analytic Studies on Pseudo-holomorphic Maps

Michigan State University, East Lansing MI

Investigators

Abstract

Abstract Award: DMS-0406454 Principal Investigator: Thomas H. Parker This project involves analytic aspects of the theory of pseudo-holomorphic curves. The aim is to develop effective methods for computing Gromov-Witten invariants of symplectic manifolds and enumerative invariants of algebraic manifolds. The main thrust is a continuing project with E. Ionel on the Gromov-Witten invariants of six-dimensional manifolds. String Theory is especially effective in making predictions about a special type of six-dimensional spaces called Calabi-Yau 3-folds. One of the most fascinating predictions of String Theory is the Gompakumar-Vafa conjecture, which claims that the plethora of Gromov-Witten invariants are determined by a more limited collection of `BPS numbers'. The first proposed project seeks to derive and prove the Gompakumar-Vafa formulas by adapting the analytic methods that C. Taubes developed to relate the Seiberg-Witten and Gromov invariants. This approach to the BPS numbers is different and more geometric than the physicists'. The proposed work also includes efforts to extend the PI's previous work on Relative GW invariants to invariants relative to ``a stratified symplectic subspace'', and to give a geometric and computationally effective description of the virtual fundamental class for symplectic manifolds which are not necessary semipositive. A fourth project concerns the ``modified GW invariants'' defined by the P.I.'s student Junho Lee; those make it possible to study a class of manifolds (Kahler surfaces with positive geometric genus) where there are important conjectures that are not currently approachable by GW methods. The last project is geometric analysis of a different sort aimed at understanding how supersymmetry simplifies expressions for heat kernels on manifolds. One of the most notable developments in mathematics in the past decade has been the remarkable confluence of ideas coming from String Theory, Algebraic Geometry, Partial Differential Equations, and Symplectic Geometry. A major part of that story revolves around the idea of counting holomorphic curves in algebraic manifolds. That problem was studied for 100 years with few results. Then, around 1990, it was realized that the problem can be translated into symplectic geometry and tackled using the powerful machinery of mathematical gauge theory. (Gauge theory, originally part of physics, has been the focus of many very fruitful interactions between mathematicians and physicists over the past twenty years; it includes Yang-Mills and Seiberg-Witten Theory, and String Theory). This `Gromov-Witten invariant' approach led quickly to formulas answering some of the original curve-counting problems. The subject is now advancing rapidly on many fronts, and is being continually stimulated by the predictions of physics. This project is aimed toward further developing this theory using the gauge theory that is approach most closely aligned with physics, and to use the symplectic approach to make conceptual bridges between the algebraic and physics viewpoints.