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

点击翻译按钮获取中文摘要

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. ***