Gromov-Witten Theory and its Applications

摘要奖：DMS-0803193主要研究人员：阮永斌在过去的二十年里，数学与物理之间发生了大量的相互作用。这种相互作用的一个重要领域涉及Gromov-Wittenant不变量理论。在物理学中，它计算的是Calabi-Yau空间的瞬时修正。在数学上，它为代数几何和辛几何提供了重要的不变量。受数学和物理学的启发，格罗莫夫-威滕理论在各种情况下都得到了发展，例如相对论格罗莫夫-威腾理论和奥比诺夫-威腾理论。它们中的每一个都是大局中不可或缺的一部分，它们对于全面理解格罗莫夫-维滕理论是必不可少的。这一建议旨在向两个方向发展Gromov-Witten理论；(I)将Gromov-Witten理论扩展到Landau-Ginzburg/奇点模型，并将其发展为普通Gromov-Witten理论的有效计算工具；(Ii)应用Gromov-Witten理论研究辛生几何。数学一直是我们理解我们生活的宇宙结构的重要工具和语言。在宇宙的弦理论模型中，现代数学学科，如几何和拓扑学，发挥着越来越大的作用。特别是，Gromov-Witten不变量作为称为瞬时修正的重要物理量进入这些模型。这个项目将提高我们计算瞬子修正的能力，从而增强我们对宇宙可能结构的理解。

AbstractAward: DMS-0803193Principal Investigator: Yongbin RuanDuring the past twenty years, there has been a great deal ofinteraction between mathematics and physics. An important area ofthis interaction involves the theory of Gromov-Witteninvariants. In physics, it counts the instantons corrections ofCalabi-Yau space. In mathematics, it provides importantinvariants for algebraic and symplectic geometry. Inspired byboth mathematics and physics, Gromov-Witten theory has beenestablished in a variety of situations such as relativeGromov-Witten theory and orbifold Gromov-Witten theory. Each ofthem is an integral part of the big picture and they have beenessential for a complete understanding of Gromov-Wittentheory. This proposal seeks to advance Gromov-Witten theory intwo directions; (i) expanding Gromov-Witten theory to theLandau-Ginzburg/singularities model and developing it as aneffective computational tool for ordinary Gromov-Witten theory;(ii) applying Gromov-Witten theory to study symplectic birationalgeometry.Mathematics is always an important tool and language in ourunderstanding of the structure of the universe we live in. Inthe string theoretic model of the universe, modern mathematicalsubjects such as geometry and topology play an ever biggerrole. In particular, Gromov-Witten invariants enter these modelsas important physical quantities called instantoncorrections. This project will improve our ability to calculateinstanton corrections and hence enhance our understanding of thepossible structure of the universe.