Analytic Studies on Pseudo-holomorphic Maps

伪全纯映射的解析研究

• 批准号: 0406454
• 负责人: Thomas Parker
• 金额: --
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-15 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406454&HistoricalAwards=false
• 关键词: Analytic; Studies; Pseudo; holomorphic; Maps

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.

摘要奖：DMS-0406454主要研究者：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枚举不变量的有效方法。 主要目标是与E. Ionel关于六维流形的Gromov-Witten不变量弦论在预测一种特殊类型的六维空间时特别有效，这种空间被称为卡-丘3-折叠空间。弦论最吸引人的预测之一是贡巴-瓦法猜想，它声称Gromov-Witten不变量的过多是由一个更有限的“BPS数”集合决定的。 第一个提出的项目旨在通过采用C. Taubes发展了Seiberg-Witten和Gromov不变量的关系。 这种方法的BPS数是不同的，更多的几何比物理学家的。拟议的工作还包括努力扩大PI的相对GW不变量的不变量相对于“分层辛子空间”的sprevious工作，并给出一个几何和计算有效的描述虚拟fundamentalclass辛流形不一定是半正的。 第四个项目涉及P.I.定义的“修改的GWinvariants”。他的学生李俊昊;这使得研究一类流形（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参数。 最后一个项目是不同类型的几何分析，旨在理解超对称性如何简化流形上热核的表达式。过去十年中数学最显著的发展之一是来自弦理论、代数几何、偏微分方程和辛几何的思想的显著融合。 这个故事的一个主要部分围绕着计算代数流形中全纯曲线的想法。 这个问题研究了100年，但收效甚微。 然后，大约在1990年，人们意识到这个问题可以转化为辛几何，并使用强大的数学规范理论来解决。（规范理论最初是物理学的一部分，在过去的20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-维滕理论，以及弦论）。 这种“Gromov-Witten不变量”的方法很快导致了一些回答原始曲线计数问题的公式。 这门学科现在在许多方面都在迅速发展，并不断受到物理学预言的刺激。 本项目旨在进一步发展这一理论，使用规范理论，这是最接近物理学的方法，并使用辛方法，使概念之间的桥梁代数和物理学的观点。