Gromov-Witten theory

摘要奖：DMS-1103368主要研究人员：阮永斌在过去的二十年里，数学和物理之间发生了大量的相互作用。物理学中产生的各种对应或二元性对数学产生了巨大的影响。这些通信之一是Landau-Ginzburg/Calabi-Yau通信。这一建议提供了一个全面的方案，以建立数学中的这种严格的数学对应。自牛顿时代以来，数学一直是帮助我们理解宇宙的重要工具。爱因斯坦广义相对论的微分几何就是一个例子。在过去的二十年里，人们在建立所谓的弦理论宇宙模型方面进行了大量的工作。宇宙的弦论模型是一维的，我们通常的4维时空与一个非常小的6维空间一起被称为Calabi-Yau流形。如此小的内部空间将通过某些数学变量影响我们的时空，如Gromov-Wittenant不变量。这些不变量的计算一直是几何学和物理学中的核心问题。这是数学工具非常有用的领域。目前的方案设想了一个理论框架以及开发计算Gromov-Witten不变量的技术工具

AbstractAward: DMS-1103368Principal Investigator: Yongbin RuanDuring the past twenty years, there has been a great deal ofinteraction between mathematic and physics. Variouscorrespondences or dualities arising from physics have had agreat impact on mathematics. One of these correspondences is theLandau-Ginzburg/Calabi-Yau correspondence. This proposal offers acomprehensive program to establish this correspondence inmathematics with mathematical rigor. There is a wide range ofapplications such as global mirror symmetry, the computation ofGromov-Witten invariants of compact Calabi-Yau manifolds, and tomodular forms.Since the era of Newton, mathematics has been a key tool inhelping us to comprehend the universe. An example isdifferential geometry via Einstein's theory of generalrelativity. During the last twenty years there has been a greatdeal of activity in building so-called string-theoretic models ofthe universe. The string-theoretic model of the universe isten-dimensional, our usual 4-dimensional space-time togeher witha very small 6-dimensional space called a Calabi-Yau manifold.Such a small internal space will affect our space-time throughcertain mathematical quantities such as Gromov-Witteninvariants. The computation of these invariants has been acentral problem in geometry and physics. This is the area inwhich mathematical tools can be very useful. The currentproposal envisages a theoretical framework as well as developingtechnical tools to compute Gromov-Witten invariants