Mathematical Sciences: Research on Elliptic Genera, Elliptic Cohomology and Modules over the Steenrod Algebra

这位研究者将继续他对椭圆形属的研究 椭圆上同调 一族新的周期上同调 理论是在1986年与道格拉斯拉文埃尔的联合工作中发现的， 罗伯特Stong;这是意图研究的中心问题， 确定这些上同调的内在几何性质 理论，根据最近的工作，建设椭圆 同源性M。Kreck，S. Stolz和M.霍维 进一步的问题 处理通常的2级椭圆亏格的变体，产生 与Thom谱MO 8相关的一级模块形式，以及 N（N 1）与雅可比多项式相关的模形式，将是 研究了 此外，还希望能够 建立连接结不变量和量子群椭圆 上同调 这个项目关注的是减少几何信息 一个计算对象。 几何的本质 问题的关键在于资讯。 虽然问题 关于长度、面积、角度、体积等等， 如果要减少计算，它与什么是完全不同的 被称为几何对象的拓扑性质。 这些 是诸如连通性（所有的都是一体的）之类的属性， 结，没有洞，等等。 所有系统研究 例如，如何判断两个几何 对象在这些属性之一方面确实不同，或者 只是表面上的不同，或者如何分类的品种， 可能发生的差异，所有这些都只是真正的 当它们被简化为 计算. 现代代数包含了许多工具， 对所需工具的态度，以及 代数和拓扑学之间的联系仍然是一个令人着迷的课题。 椭圆上同调，由研究者和几个共同发明的， 工人，投标公平地成为一个重要的工具，在阿森纳 代数拓扑学家可用。

The investigator will continue his study of elliptic genera and elliptic cohomology. A new family of periodic cohomology theories was found in 1986 in joint work with Douglas Ravenel and Robert Stong; it is the intention to study the central problem of determining the intrinsic geometric nature of these cohomology theories, in light of the recent work on constructing elliptic homology by M. Kreck, S. Stolz, and M. Hovey. Further problems dealing with variants of the usual level-2 elliptic genus, yielding level-1 modular forms related to the Thom spectrum MO8, and level N (N 1) modular forms related to Jacobi polynomials, will be studied. In addition, it is hoped that connections can be established linking knot invariants and quantum groups to elliptic cohomology. This project is concerned with reducing geometric information to a subject for calculation. The nature of the geometric information involved is the crux of the problem. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation. Modern algebra furnishes many of the tools and the attitudes toward the tools that are needed, and the interplay between the algebra and the topology remains a fascinating subject. Elliptic cohomology, invented by the investigator and several co- workers, bids fair to be one of the important tools in the arsenal available to algebraic topologists.