Global Analysis for Pseudo-holomorphic and Harmonic Maps

摘要奖：DMS-1011793主要研究员：托马斯·H·帕克这个项目涉及伪全纯映射和调和映射理论的分析方面。PI最近与JunhoLee的工作在2维和4维的Gromov-Witten理论之间建立了显著的联系。第一个项目的目的是通过显式计算确定一大类Kahler曲面的GW不变量的局部GW不变量来完成这个程序(与Lee一起)。所计划的技巧，即算符与辛和公式的同伦，在其他情况下应该是有用的。GW在预测Calabi-Yau的3倍时尤其有效。第二个提议的项目寻求使用几何分析方法来证明Gompakumar-Vafa猜想，这是物理学家提出的一个有趣的预测，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。Lee的目的是证明，在某些假设下，稳定的$J$-全纯映射的空间一般是光滑流形的，这是人们一直在寻找的结果。最后，项目4研究了调和映射热流的有限时间奇异性。其思想是利用对路径空间的分析来显示地图发展的“脖子”，其图像在精确的管理器中收敛到测地线。现代数学中最显著的发展之一是来自弦理论、代数几何和辛几何的思想的显著融合。故事的很大一部分围绕着计算代数流形中的全纯曲线的想法(这是数学中最基本的程序之一的几何表述：对多项式方程组的解进行分类)。这一计数问题研究了100年，但几乎没有结果。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的机制来解决，数学规范理论是数学和物理之间相互作用的理论。这种“Gromov-Witten不变量”方法很快就产生了解答一些原始曲线计数问题的公式。本项目旨在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.