Gromov-Witten Theory and its Applications

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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.