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

这个差分拓扑项目的共同主题是 研究三维和四维流形的不变量，它们之间的关系 变换群理论，子流形几何， 量子物理学、瞬子模空间及相关问题 离散群的上同调 研究者将继续 他与Sylvain Cappell和E. Y.米勒的理论 广义Casson流，关于Witten-Reshetikhin-Turaev不变量， 3-流形的谱流不变量及其应用 Atiyah-Patodi-Singer指标的辛几何技巧 理论 他打算与达留什·威尔钦斯基一起调查 用极小曲面表示4-流形的同调类 属 一个相关的主题是我的联合工作。汉布尔顿和 4-上光滑变换群理论的研究者 流形，瞬子的等变模空间，以及 等变唐纳森不变量 与史蒂文·温特劳布一起， 他计划继续研究算术的上同调 关于Sp 4的子群及分歧覆盖的一个不变量 空间源自J. - P. Serre。 最后以 艾伦布朗斯坦，研究者计划研究运动组 3-空间中的n-弦，它的上同调，以及相关的模空间。 传统上，差分拓扑提供了一种 分析我们生活的世界的某些问题， 几何学及其物理学，但近年来， 变得更加流畅，物理学经常提供有用的 几何概念，反之亦然。 最新的此类外国 可以说，进口一直是4的唐纳森不变量， 流形和相关的不变量发现，因为，但沿着 同样的线条。 当然，数学家们 参与到物理学的问题中， 两个主题的相互丰富。 开发工具来解决 物理问题，这些工具，然后变成有很多 更大的通用性，并在数学中得到广泛应用。 的 三维流形的新量子不变量的故事是一个变化 在这个主题上。 数学家对问题感兴趣 量子场论的理论被引入了某种几何学 建设 这种构造又导致了一个不变量， 只取决于拓扑特征，而不是完全 基本流形的几何特征。 现在有 原始结构的修改和许多结果 量子不变量 它们的起源与 其他已知的不变量，他们可以预期， 来检测传统不变量无法检测到的东西 现在 拓扑学家理应证明这一点，以及整理出 不同的量子不变量以及它们与 传统不变量 沿着这些路线工作将是 调查员与他的一些军团进行的项目， 合作者

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.