Analytic Studies on Pseudo-holomorphic Maps

摘要奖：DMS-0406454主要研究员：托马斯·H·帕克这个项目涉及伪全纯曲线理论的分析方面。目的是发展计算辛流形的Gromov-Witten不变量和代数流形的计数不变量的有效方法。主要推力是与E.Ionel关于六维流形的Gromov-Witten不变量的持续项目。弦理论特别有效地预测了一种称为Calabi-Yau 3重的特殊类型的六维空间。弦理论最吸引人的预测之一是Gompakumar-Vafa猜想，它声称Gromov-Witten不变量的过多是由更有限的‘BPS数’集合决定的。第一个提议的项目试图通过采用C.陶布斯发展的将Seiberg-Witten和Gromov不变量联系起来的分析方法来推导和证明Gompakumar-Vafa公式。这种处理BPS数的方法与物理学家的方法不同，而且更具几何性质。我们的工作还包括将PI关于相对GW不变量的工作扩展到相对于“分层辛子空间”的不变量，并给出了不一定是半正的辛流形的虚拟基本类的几何和计算上的有效描述。第四个项目涉及P.I.S的学生李俊浩定义的“修正的GW不变量”；这些不变量使得研究一类流形(具有正几何亏格的Kahler曲面)成为可能，其中存在目前GW方法无法接近的重要猜想。最后一个项目是一种不同类型的几何分析，旨在了解超对称如何简化流形上热核的表达式。在过去的十年里，数学中最显著的发展之一是弦理论、代数几何、偏微分方程式和辛几何思想的显著融合。这个故事的主要部分围绕着计算代数流形中的全纯曲线的想法。这个问题研究了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，并利用数学规范理论的强大机制来解决。(规范理论最初是物理学的一部分，在过去的二十年里，它一直是数学家和物理学家之间许多卓有成效的互动的焦点；它包括杨-米尔斯和塞伯格-威腾理论，以及弦理论)。这种“格罗莫夫-威腾不变量”方法很快就产生了解答某些原始曲线计数问题的公式。这门学科现在正在许多方面迅速发展，并不断受到物理学预测的刺激。这个项目的目的是使用与物理学最接近的规范理论来进一步发展这一理论，并使用辛方法在代数和物理观点之间建立概念桥梁。

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.