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

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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.