Mathematical Sciences: On the Relationship between conjectures about elliptic cohomology, and the cohomology theory E-sub-n

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9306938 Ando One of the principal tools of algebraic topology is the use of algebraic invariants called homology theories to analyze topological spaces. In the course of his work on conformal field theory, Edward Witten made several surprising observations about the relationship between three important homology theories, singular homology, K-theory, and elliptic cohomology. The general principal is that to understand the K-theory of a space, one should understand as much as possible about the singular homology of its free loop space, and similarly for elliptic cohomology and K-theory. At about the same time, Mike Hopkins and various collaborators were showing the fundamental importance to algebraic topology of a sequence of homology theories E-sub-n. Their work involves a very profitable interaction between algebraic topology and number theory. It turns out that E-sub-zero is close to singular homology, E-sub-one is close to K-theory, and E-sub-two is close to elliptic homology. Ando outlines how certain conjectures of Witten and of Hopkins, et al., might be closely related. He intends to investigate these relationships and make them more precise, in the hopes that the advances in these two areas of research can inform each other more directly and fruitfully. The theme of this project, drawing connections between deep algebraic and number theoretic theories on the one hand and deep geometric (topological) theories on the other, is repeated again and again in modern topology. As the theories grow increasingly intricate, it is only through such overall principles of organization that mathematicians keep the structure manageable and within the grasp of human mental powers. The instant project bears on one of the most fruitful recent developments of this nature. ***