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

这个微分拓扑学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、子流形的几何、量子物理、瞬子的模空间以及离散群上同调的相关问题的关系。研究人员将继续与Sylvain Cappell和E.Y.Miller一起研究广义Casson流理论、Witten-Reshetikhin-Turaev不变量、三维流形的谱流不变量以及辛几何技术在Atiyah-Patodi-Singer指数理论中的应用。与Dariusz Wilczynski合作，他打算研究用极小亏格的曲面表示4-流形的同调类。一个相关的主题是I.Hambleton和研究者在4-流形上的光滑变换群理论、瞬子的等变模空间和等变Donaldson不变量方面的共同工作。他计划与Steven WeIntrub一起继续研究SP4的算术子群的上同调，以及源于J.-P.Serre工作的分支覆盖空间的一个不变量。最后，与Alan Brownstein一起，研究人员计划研究3-空间中n-弦的运动群、它的上同调以及相关的模空间。传统上，微分拓扑学提供了一种分析关于我们生活的世界的某些问题的方法，它的几何和物理，但近年来，情况变得更加不稳定，物理学经常提供有用的几何概念，反之亦然。最近引进的这类所谓的外国流形是唐纳森不变量的4流形和相关的不变量发现，但沿着相同的路线。当然，数学家们投身于物理问题的历史由来已久，往往是为了这两个学科的相互充实。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普遍性，并在数学中得到广泛应用。三维流形的新量子不变量的故事就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。这种构造又导致了一个不变量，该不变量只依赖于基本流形的拓扑特征，而不依赖于基本流形的全部几何特征。现在有了对原始结构的修改和许多由此产生的量子不变量。它们的起源与以前已知的其他不变量的起源有很大的不同，因此可以预期它们可以检测到传统不变量无法检测的东西。现在，拓扑学家应该证明这一点，并理清不同的量子不变量以及它们与更传统的不变量的关系。这些工作将是调查员和他的一些合作者军团承担的项目之一。

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.