Gromov-Witten theory

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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