格罗莫夫-威腾理论和量子上同调是辛几何与辛拓扑近20年来非常热门的研究领域。量子自然性问题是格罗莫夫-威腾理论中的一个基本问题。大量研究表明，量子上同调在双有理映射下具有良好的表现。阮勇斌首先观察到这个现象并提出了著名的K-等价猜想：任意两个K-等价的流形（或Gorenstein 轨形）具有同构的量子上同调环。本项目主要研究与一类广义典则奇点有关的整体辛flop变换，其中典则奇点自然出现在Mori的极小模型列表中。计划用辛切割技巧和辛局部化公式计算出Ruan上同调环和量子上同调环，从而验证Ruan猜想，并期望进一步探索格罗莫夫-威腾理论与双有理几何之间的紧密联系。

Gromov-Witten theory and Quantum cohomology are very hot topics in Symplectic geometry and Symplectic topology during the past twenty years. One of the fundamental problems in Gromov-Witten theory is Quantum Naturality. A lot of studies show that quantum cohomologies behave well under birational maps. Yongbin Ruan first observed this phenomenon and proposed a famous K-equivalence conjecture: any two K-equivalent manifolds (or Gorenstein orbifolds) have isomorphic quantum cohomology rings. This project mainly studies a global singular symplectic flop associated to a class of generalized canonical singularities, where this kind of canonical singularity appears naturally in the list of Mori’s Minimal model program. This project plans to use the methods of symplectic cutting and symplectic virtual localization formula to obtain Ruan cohomology rings and quantum cohomology rings, hence verifies Ruan’s conjecture and hopes to further investigate the intimate relationship between Gromov-Witten theory and birational geometry.