Global Analysis for Pseudo-holomorphic and Harmonic Maps

项目编号：dms -1011793项目负责人：Thomas H. parker本项目涉及伪全纯和调和映射理论的解析方面。PI最近与JunhoLee的合作已经在2维和4维的gromovo - wittentheory之间建立了一个显著的联系。第一个项目旨在通过显式计算局部gw不变量来完成这个程序（与Lee一起），这些局部gw不变量决定了广泛的kahler曲面的gw不变量。计划的技巧，算子的同伦和辛和公式，在其他情况下应该是有用的。GW在预测Calabi-Yau 3倍时特别有效。第二个提议的项目寻求使用几何分析方法来证明gompakumar - vafecture，这是物理学家的一个迷人的预测，即GW数的过剩是由更有限的“BPS数”集合决定的。在第三个项目中，PI和j。李的目的是证明，在某些假设下，稳定J -全纯映射的空间一般是光滑流形，这是一个长期寻求的结果。最后，课题4是谐波图热流的有限时间奇异性研究。他们的想法是通过对路径空间的分析来显示地图形成的“颈部”，其图像以精确的方式收敛于测地线。近代数学最显著的发展之一是弦论、代数几何和辛几何思想的显著融合。故事的主要部分围绕着在代数流形中计数全纯曲线的思想（这是数学中最基本的程序之一的几何公式：对多项式方程系统的解进行分类）。计数问题被研究了100年，却没有什么结果。然后，在1990年左右，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（一种由数学和物理相互作用产生的理论）的机制来解决。这种“Gromov-Witten不变量”的方法很快引出了一些公式，可以回答一些原始的曲线计数问题。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谐波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.