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

拟研究Gromov-维滕理论与其他数学和物理学科之间的相互作用，包括双有理几何、曲线模、K-理论、可积系统和镜像对称。重言式环的结构，双有理变换下的量子上同调，Virasoro对称和orbifold镜像对称。Gromov-维滕理论位于数学和物理学中许多令人兴奋的研究领域的交叉点。一方面，该理论本身有一些显著的结构。2这些结构的证明和新结构的发现需要对该理论的新见解和其他领域的投入。3另一方面，它还提供了许多强有力的思想和深刻的联系，从弦理论到数学中的经典学科。这些思想和联系中的一些将在本提案中进行探讨。

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.