Analytic Studies on Pseudo-holomorphic Maps

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AbstractAward: DMS-0406454Principal Investigator: Thomas H. ParkerThis project involves analytic aspects of the theory ofpseudo-holomorphic curves. The aim is to develop effectivemethods for computing Gromov-Witten invariants of symplecticmanifolds and enumerative invariants of algebraic manifolds. Themain thrust is a continuing project with E. Ionel on theGromov-Witten invariants of six-dimensional manifolds. StringTheory is especially effective in making predictions about aspecial type of six-dimensional spaces called Calabi-Yau 3-folds.One of the most fascinating predictions of String Theory is theGompakumar-Vafa conjecture, which claims that the plethora ofGromov-Witten invariants are determined by a more limitedcollection of `BPS numbers'. The first proposed project seeks toderive and prove the Gompakumar-Vafa formulas by adapting theanalytic methods that C. Taubes developed to relate theSeiberg-Witten and Gromov invariants. This approach to the BPSnumbers is different and more geometric than the physicists'.The proposed work also includes efforts to extend the PI'sprevious work on Relative GW invariants to invariants relative to``a stratified symplectic subspace'', and to give a geometric andcomputationally effective description of the virtual fundamentalclass for symplectic manifolds which are not necessarysemipositive. A fourth project concerns the ``modified GWinvariants'' defined by the P.I.'s student Junho Lee; those makeit possible to study a class of manifolds (Kahler surfaces withpositive geometric genus) where there are important conjecturesthat are not currently approachable by GW methods. The lastproject is geometric analysis of a different sort aimed atunderstanding how supersymmetry simplifies expressions for heatkernels on manifolds.One of the most notable developments in mathematics in the pastdecade has been the remarkable confluence of ideas coming fromString Theory, Algebraic Geometry, Partial DifferentialEquations, and Symplectic Geometry. A major part of that storyrevolves around the idea of counting holomorphic curves inalgebraic manifolds. That problem was studied for 100 years withfew results. Then, around 1990, it was realized that the problemcan be translated into symplectic geometry and tackled using thepowerful machinery of mathematical gauge theory. (Gauge theory,originally part of physics, has been the focus of many veryfruitful interactions between mathematicians and physicists overthe past twenty years; it includes Yang-Mills and Seiberg-WittenTheory, and String Theory). This `Gromov-Witten invariant'approach led quickly to formulas answering some of the originalcurve-counting problems. The subject is now advancing rapidly onmany fronts, and is being continually stimulated by thepredictions of physics. This project is aimed toward furtherdeveloping this theory using the gauge theory that is approachmost closely aligned with physics, and to use the symplecticapproach to make conceptual bridges between the algebraic andphysics viewpoints.