Mathematical Sciences: Symplectric Topology and It's Applications to String Theory

9401367阮研究员打算继续他在辛拓扑及其在弦理论中的应用领域的工作。更确切地说，他计划研究拓扑sigma模型-量子上同调理论的数学基础，利用拓扑sigma模型不变量研究Calabi-Yau流形和一般类型的曲面，继续研究辛极值射线和辛异形，特别是精确辛异形的退化不动点。拓扑性质是研究对象在微小变形下保持不变的特殊性质。在过去的一个世纪里，研究数学各个分支的拓扑性质一直是数学研究的中心。本项目研究数学物理，特别是弦理论中产生的一些拓扑性质。但这些拓扑性质并不是以前所知的普通拓扑现象。它们属于一种新的拓扑理论——辛拓扑，研究者认为辛拓扑将为拓扑理论和弦理论提供更好的理解。*＊*

9401367 Ruan The investigator intends to continue his work in the area of symplectic topology and its application to string theory. More precisely, he plans to work on the mathematical foundations of topological sigma model-quantum cohomology theory, to use topological sigma model invariants to study Calabi-Yau manifolds and surfaces of general type, to continue to work on symplectic extremal rays and symplectomophisms, in particular, degenerate fixed points of exact symplectomorphisms. Topological properties are special properties that remain unchanged under the small deformation of objects that the investigator studies. Studying topological properties in all branches of mathematics has been in the center of mathematics for the past century. This project studies some topological properties arising from mathematical physics, in particular, from string theory. But those topological properties are not ordinary topological phenomenon known before. They belong to a new kind of topological theory-Symplectic topology, which the investigator believes will provide much better understanding both for topology theory and for string theory. ***