Gromov-Witten不变量是目前最重要的几何拓扑不变量之一，它在几何拓扑的研究中发挥了至关重要的作用，目前已发展成为几何与物理的研究热点问题。本项目计划开展关于辛流形的辛有理连通性、Donaldson-Thomas理论和量子上同调环的自然性等研究。预期证明Del Pezzo 簇，Fano簇及射影平面的点的Hilbert概型的辛有理连通性；给出曲面上P1-丛的Donaldson-Thomas理论与曲面的嵌套Hilbert概型的Gromov-Witten不变量之间的关系；给出量子上同调环在Blowup下的变化。

Gromov-Witten invariant is one of the most important invariants in geometry and topology, and played very important role in the study of goemetry and topology of manifolds. Now Gromov-Witten theory has become very popular in geometry and Physics. In this project, one will study the symplectic rationally connectedness of symplectic manifolds, Donaldson-Thomas theory and the naturality of quantum cohomology ring. We expect to prove the symplectic rationally connectedness of Del Pezzo varieties and the Hilbert schemes of points of the projective plane; to find the relation between the Donaldson-Thomas theory of the P1-bundle over a surface and the nested Hilbert scheme of points of the surface; to describe the change of quantum cohomology ring under Blowup.