Global Analysis for Pseudo-holomorphic and Harmonic Maps

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AbstractAward: DMS-1011793Principal Investigator: Thomas H. ParkerThis project involves analytic aspects of the theory of pseudo-holomorphic and harmonic maps. The PI's recent work with JunhoLee has established a remarkable link between Gromov-Wittentheory in dimensions 2 and 4. The first project aims to completethis program (with Lee) by explicitly calculating the local GWinvariants that determine the GW invariants of a wide class ofKahler surfaces. The planned technique, a homotopy of operatorstogether with the symplectic sum formula, should be useful inother contexts. GW is especially effective in making predictionsabout Calabi-Yau 3-folds. The second proposed project seeks touse geometric analysis methods to prove the Gompakumar-VafaConjecture, which is a fascinating prediction by physicists thatthe plethora of GW numbers are determined by a more limitedcollection of "BPS numbers". In a third project, the PI andJ. Lee aim to prove that, under certain hypotheses, the space ofstable $J$-holomorphic maps is generically a smooth manifold, along-sought result. Finally, Project 4 is a study of thefinite-time singularities of the harmonic map heat flow. Theidea is to use analysis on path space to show that the mapsdevelop "necks" whose images converge to geodesics in a precisemanner.One of the most notable developments in recent mathematics hasbeen the remarkable confluence of ideas coming from stringtheory, algebraic geometry, and symplectic geometry. A majorpart of the story revolves around the idea of countingholomorphic curves in algebraic manifolds (this is a geometricformulation of one of the most fundamental programs inmathematics: classifying the solutions of systems of polynomialequations). The counting 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 themachinery of mathematical Gauge Theory, a theory that arose frominteractions between mathematics and physics. This"Gromov-Witten invariant" approach quickly led to formulasanswering some of the original curve-counting problems. Thisproject is aimed at further developing Gromov-Witten theory usingthe methods of geometric analysis, and applying similartechniques to related problems in the theory of harmonic maps.