Mathematical Sciences: Low Dimensional Manifolds: Their Symmetries and Topological Invariants

三维拓扑学中一个著名的问题是列出作为某闭三维流形的基本群的所有有限群。由于所有已知的例子都来自于自由作用于3球的旋转群的子群，因此所涉及的流形具有正的常曲率，并且该问题被称为球面空间形式问题。罗尼·李和i·汉布尔顿计划用唐纳森的瞬子理论来解决这个问题。与上述密切相关并与S.Cappell和E.Miller合作，Ronnie Lee计划研究Casson不变量的推广，并证明它们属于所谓的“有限型不变量”。首席研究员李卫平计划研究两种类型的花同源性。第一个被称为瞬时花同调，是通过将莫尔斯理论应用于无限维的连接空间而得到的，第二个被称为辛花同调，是基于一个表示变体的拉格朗日交集理论。他们计划证明这些都是相同的理论，从而建立一个关于阿蒂亚的猜想。最后，与Wilczynski一起，首席研究员计划研究寻找表示单连通4流形的给定同调类的最小属局部平面的问题。在光滑的情况下，相应的问题被称为托姆猜想，最近P.Kronheimer和T.Mrowka解决了这个问题，获得了极大的赞誉。简而言之，该项目计划研究三维和四维流形理论中几个众所周知的问题，这些问题涉及这些物体的基本对称性质以及它们内部子流形的几何形状。用于解决这些问题的许多技术都是新的，以前没有。它们是近年来在量子物理和拓扑学的相互作用下发展起来的。在数学方面，它们被称为“对称不变量”，而在物理方面，它们在量子场论中有对应的名称。＊＊＊

9529310 Lee A famous problem in 3-dimensional topology is to list all finite groups which are the fundamental group of some closed 3-manifold. Since all the known examples come from subgroups of the rotation group that act freely on the 3-sphere, the manifolds involved have positive constant curvature, and the problem is known as the spherical space-form problem. Together with I. Hambleton, Ronnie Lee plans to solve this problem using Donaldson's theory of instantons. Closely related to the above and in collaboration with S.Cappell and E.Miller, Ronnie Lee plans to study generalizations of Casson's invariants and show that they belong to the so-called "finite type invariants." With Weiping Li, the principal investigator plans to investigate two types of Floer homologies. The first, known as the instanton Floer homology, is obtained by applying Morse theory to an infinite dimensional space of connections, and the second, known as the symplectic Floer homology, is based on the Lagrangian intersection theory of a representation variety. They plan to show that these are the same theory and hence establish a conjecture of Atiyah. Finally, with Wilczynski, the principal investigator plans to study the problem of finding locally flat surfaces of minimum genus representing a given homology class of a simply connected 4-manifold. In the smooth setting, the corresponding problem is known as the Thom conjecture and has been solved recently by P.Kronheimer and T.Mrowka to great acclaim. In short, the project plans to study several well-known problems in 3- and 4-dimensional manifold theory concerning the basic symmetric nature of these objects and also the geometry of the submanifolds inside them. Many of the techniques to be used in solving these problems are new and were not previously available. They were developed through the interaction between quantum physics and topology in recent years. On the mathematics side they are known under the heading of Sympl ectic Invariants, while on the physics side they have their counterpart in Quantum Field Theory. ***