Global Analysis for Pseudo-holomorphic and Harmonic Maps

摘要奖：DMS-1011793主要研究者：托马斯H.这个项目涉及伪全纯和调和映射理论的分析方面。 PI最近与JunhoLee的工作建立了2维和4维Gromov-Witten理论之间的显着联系。 第一个项目的目的是完成这个程序（与李）明确计算本地GWinvariants，确定GW不变的广泛类的Kahler曲面。 计划的技术，一个同伦算子与辛求和公式，应该是有用的，在其他情况下。 GW在预测Calabi-Yau三重褶皱方面特别有效。 第二个项目的目的是利用几何分析方法来证明Gompaintar-Vafa猜想，这是一个有趣的预测，由物理学家，过多的GW数是由一个更有限的收集“BPS数”。 在第三个项目中，PI和J。Lee的目的是证明在一定的假设下，稳定的$J$-全纯映射空间一般是光滑流形，这是他一直寻求的结果。 最后，项目4是研究调和映射热流的有限时间奇异性。 这一思想是利用对路径空间的分析来表明，地图发展的“脖子”，其图像收敛到测地线在一个精确的manner。在最近的数学最显着的发展之一，一直是显着的汇合的想法来自弦理论，代数几何，辛几何。 故事的大部分围绕着代数流形中计算全纯曲线的想法（这是数学中最基本的程序之一的几何公式：对多项式方程组的解进行分类）。 计数问题研究了100年，但成果寥寥。然后，大约在1990年，人们意识到这个问题可以转化为辛几何，并使用数学规范理论（Gauge Theory）来解决，这是一种产生于数学和物理之间相互作用的理论。 这种“Gromov-Witten不变量”的方法很快就导致了一些原始曲线计数问题的公式。 本项目旨在利用几何分析的方法进一步发展Gromov-Witten理论，并将类似的技术应用于调和映射理论中的相关问题。

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.