Principal Investigator: Kefeng Liu
My project will focus on the understanding of the geometry and topology of mirror symmetry, in particular the aspect of counting curves in projective manifolds. More precisely there are three outstanding and closely related problems that I would like to work on and hope to solve. The first is to prove the most general mirror principle for counting rational curves in any projective manifold and its Calabi-Yau submanifolds; the second is to understand the geometry and algebra of the local mirror symmetry, in particular, of the surprising role played by the elliptic curves appeared in our computation by applying mirror principle; the third is to develop a mirror principle for counting curves of higher genus in projective manifolds. I hope to solve these problems by further extending the techniques we developed to prove the mirror principle for balloon manifolds. These three problems are the different aspects of a single problem: to prove and 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 remarkable mathematical conjectures. Mirror symmetry is among one of them. The proofs of these conjectures will not only give beautiful mathematical results, but also help gain confidence in the physical theory. The mirror principle we developed has partially verified some of these conjectures. It has many interesting mathematical consequences and is remarkably compatible with the physical theory.
