Mathematical Sciences: Low Dimensional Manifolds, Transformation Groups, and Cohomology of Discrete Groups

这个微分拓扑项目的共同主题是研究三维和四维流形的不变量，它们与变换群理论的关系，子流形几何，量子物理，实例的模空间，以及离散群上同调的相关问题。研究者将继续与Sylvain Cappell和e.y. Miller一起研究广义Casson流理论、Witten-Reshetikhin-Turaev不变量、3流形的谱流不变量以及辛几何技术在Atiyah-Patodi-Singer指标理论中的应用。他与Dariusz Wilczynski共同研究了用最小属曲面表示4流形的同调类。一个相关的课题是I. Hambleton和研究者关于4-流形上的光滑变换群理论、瞬子的等变模空间和等变Donaldson不变量的联合工作。与Steven Weintraub一起，他计划继续他们对Sp4的算术子群的上同性的研究，以及对源自j.p的分支覆盖空间的不变量的研究。Serre。最后，研究者计划与Alan Brownstein一起研究三维空间中的n弦运动群及其上同调和相关模空间。传统上，微分拓扑学提供了一种方法来分析我们所生活的世界的某些问题，它的几何和物理，但近年来，情况变得更加多变，物理学经常提供有用的几何概念，反之亦然。可以这么说，最近从国外引进的是4流形的Donaldson不变量以及自那以后发现的相关不变量。当然，数学家研究物理问题的历史悠久，常常使这两门学科相互丰富。工具是为了解决物理问题而开发的，这些工具后来被证明具有更大的通用性，并被广泛应用于数学。关于3流形的新量子不变量的故事是这个主题的一个变体。对量子场论问题感兴趣的数学家们得到了某种几何结构。这种构造反过来又导致了一个不变量，它只依赖于底层流形的拓扑特征，而不是完全的几何特征。现在有原始结构的修改和许多由此产生的量子不变量。它们的起源与其他已知的不变量有很大的不同，因此它们可以探测到传统不变量无法探测到的东西。现在拓扑学家应该证明这一点，并整理出不同的量子不变量以及它们与更传统的不变量的关系。这方面的工作将是研究员与他的一些合作者进行的项目之一。

The common theme of this differential topology project is to study invariants of 3- and 4-dimensional manifolds, their relations to the theory of transformation groups, geometry of submanifolds, quantum physics, moduli spaces of instantons, and related questions on cohomology of discrete groups. The investigator will continue his work with Sylvain Cappell and E. Y. Miller on the theory of generalized Casson flows, on Witten-Reshetikhin-Turaev invariants, on spectral flow invariants of 3-manifolds, and on the application of symplectic geometry techniques in Atiyah-Patodi-Singer index theory. Jointly with Dariusz Wilczynski, he intends to investigate representing homology classes of 4-manifolds by surfaces of minimal genus. A related topic is the joint work of I. Hambleton and the investigator on the theory of smooth transformation groups on 4- manifolds, equivariant moduli spaces of instantons, and the equivariant Donaldson invariant. Together with Steven Weintraub, he plans to continue their work on the cohomology of arithmetic subgroups of Sp4 and also on an invariant of ramified covering spaces originating from the work of J.-P. Serre. Finally, with Alan Brownstein, the investigator plans to study the motion group of n-strings in 3-space, its cohomology, and related moduli spaces. Traditionally, differential topology has provided a means of analyzing certain questions about the world we live in, its geometry and its physics, but in recent years the situation has become much more fluid, with physics often providing useful geometric notions as much as vice versa. The latest such foreign imports so to speak have been the Donaldson invariant of 4- manifolds and related invariants discovered since but along the same lines. There is, of course, a long history of mathematicians involving themselves in the problems of physics, often to the mutual enrichment of both subjects. Tools are developed to solve physical problems, and these tools then turn out to have much greater generality and become widely used in mathematics. The story of the new quantum invariants of 3-manifolds is a variation on this theme. Mathematicians interesting themselves in problems of quantum field theory were led to a certain geometric construction. The construction led in turn to an invariant that depended only on the topological character and not the full geometric character of the underlying manifold. There are now modifications of the original construction and numerous resulting quantum invariants. Their origin is sufficiently different from that of other previously known invariants that they can be expected to detect things the traditional invariants cannot. It now behooves topologists to demonstrate this as well as to sort out the different quantum invariants and their relationships to more traditional invariants. Work along these lines will be among the projects undertaken by the investigator with some of his legion of collaborators.