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

研究者将继续他对椭圆亏格和椭圆上同调的研究。1986年与Douglas Ravenel和Robert Stong合作发现了一族新的周期上同调理论，其目的是根据M.Kreck，S.Stolz和M.Hovey最近关于构造椭圆上同调的工作，研究确定这些上同调理论的内在几何性质的中心问题。进一步的问题涉及通常的2级椭圆亏格的变体，产生与Thom谱MO8有关的1级模形式，以及与Jacobi多项式有关的N(N 1)级模形式。此外，还希望能够建立连接纽结不变量和量子群到椭圆上同调的联系。这个项目致力于将几何信息归结为一个计算主题。涉及的几何信息的性质是问题的症结所在。虽然关于长度、面积、角度、体积等的问题实际上需要归结为计算，但它与几何对象的拓扑属性有很大的不同。这些属性包括连通性(完好无损)、多节、无洞等。所有对这些性质的系统研究，例如，如何区分两个几何物体在这些性质中的一个是否真的不同，或者仅仅是表面上的不同，或者如何对可能发生的各种不同进行分类，所有这些只有在归结为计算的问题时才被真正理解和掌握。现代代数提供了许多工具和对所需工具的态度，代数和拓扑学之间的相互作用仍然是一个令人着迷的主题。由这位研究人员和几位同事发明的椭圆上同调，有望成为代数拓扑学家可用的重要工具之一。

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.