Geometry and Topology of Counting Curves in Projective Manifolds

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AbstractAward: DMS-0072182Principal Investigator: Kefeng LiuMy project will focus on the understanding of the geometry andtopology of mirror symmetry, in particular the aspect of countingcurves in projective manifolds. More precisely there are threeoutstanding and closely related problems that I would like towork on and hope to solve. The first is to prove the most generalmirror principle for counting rational curves in any projectivemanifold and its Calabi-Yau submanifolds; the second is tounderstand the geometry and algebra of the local mirror symmetry,in particular, of the surprising role played by the ellipticcurves appeared in our computation by applying mirror principle;the third is to develop a mirror principle for counting curves ofhigher genus in projective manifolds. I hope to solve theseproblems by further extending the techniques we developed toprove the mirror principle for balloon manifolds. These threeproblems are the different aspects of a single problem: to proveand understand the mirror principle in its most general form.Superstring theory, one of the most ambitious theory in sciences,is proposed for the grand unification of the laws of the nature.Based on this theory, string theorists have made many remarkablemathematical conjectures. Mirror symmetry is among one of them.The proofs of these conjectures will not only give beautifulmathematical results, but also help gain confidence in thephysical theory. The mirror principle we developed has partiallyverified some of these conjectures. It has many interestingmathematical consequences and is remarkably compatible with thephysical theory.