Gromov--Witten theory and its relations with moduli of curves, birational geometry, K-theory and orbifold mirror symmetry

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The proposed research deals with the interaction between Gromov--Witten theoryand other subjects in mathematics and physics, including birational geometry,moduli of curves, K-theory, integrable systems, and mirror symmetry.Some of the main problems investigated are:the structure of the tautological rings,quantum cohomology under birational transformations,Virasoro symmetries,and orbifold mirror symmetry.Gromov--Witten theory lies in the intersection of many exciting researchareas in mathematics and physics.On the one hand, the theory itself has some remarkable conjectural structures.Proof of these conjectures and discovery of new ones will require somenew insights into the theory and input from other areas.On the other hand, it has also provided many powerful ideas and deepconnections in many directions from string theory to classical subjects inmathematics.Some of these ideas and connections will be explored in this proposal.