轨形的格罗莫夫-威腾理论（Orbifold Gromov-Witten theory）、辛双有理几何是目前辛几何中最重要的研究领域。然而目前关于辛轨形上的辛双有理几何的研究还处于起步当中。这主要是由于目前关于轨形群胚的一般的、基本的微分拓扑和辛拓扑的研究很缺乏。很多时候人们希望微分流形的一些结论能够自然延伸到轨形群胚上，然而却未仔细研究给出证明。本项目就是结合已有的研究结果和申请人的研究积累来对轨形群胚的微分拓扑和辛拓扑进行系统的研究。并以此为基础来研究辛轨形群胚上的辛双有理几何。我们首先将微分流形的一些微分拓扑的性质严格的推广到轨形群胚上。然后利用这些结果来研究辛轨形群胚的基本的辛几何性质，如辛邻域定理等等。由此我们可以研究轨形群胚上的辛手术：辛约化，辛切割，辛粘合等等。最后我们将研究辛轨形群胚的辛双有理不变量，比如：symplectic uniruledness。

Orbifold Gromov-Witten theory and symplectic birational geometry are the most important research areas in symplectic geometry. However, the research on the symplectic birational geometry of symplectic orbifolds has only a few progresses. This is mainly because that the differential topology and symplectic topology of orbifold groupoids are lack of research. Sometimes, people think that some results on manifolds can be extended to the orbifold groupoid case, but without a careful study and a complete proof. Based on the previous studies, this proposal studies the differential topology and symplectic topology of orbifold groupoids and their applications in the symplectic birational geometry on symplectic orbifold groupoids. We first study the differential topology of orbifold gropoids. Then we use these results to study the symplectic topology of symplectic orbifold groupoids, such as the symplectic neighborhood theorem, etc. Then we move on to study the orbifold groupoid version symplectic surgeries: symplectic reduction, symplectic cutting, symplectic gluing, etc. At last, we study the symplectic birational invariants of symplectic orbifold groupoids, such as symplectic uniruledness.