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

代数拓扑学的主要工具之一是使用称为同调理论的代数不变量来分析拓扑空间。爱德华·威滕在研究共形场论的过程中，对奇异同调、k -理论和椭圆上同调这三种重要的同调理论之间的关系作了一些惊人的观察。一般的原则是，为了理解一个空间的k理论，人们应该尽可能多地理解它的自由环空间的奇异同调，同样地，对于椭圆上同调和k理论也是如此。大约在同一时间，迈克·霍普金斯和其他合作者正在展示一系列同调理论E-sub-n的代数拓扑的基本重要性。他们的工作涉及代数拓扑学和数论之间非常有益的相互作用。证明了e - 0趋近于奇异同调，e - 1趋近于k理论，e - 2趋近于椭圆同调。安藤概述了Witten和Hopkins等人的某些猜想是如何密切相关的。他打算调查这些关系，并使它们更加精确，希望这两个研究领域的进展能够更直接、更富有成效地相互告知。这个项目的主题，一方面是深入的代数和数论理论，另一方面是深入的几何（拓扑）理论之间的联系，在现代拓扑学中一再重复。当这些理论变得越来越复杂时，数学家们只有通过这些总体的组织原则，才能保持结构的可管理性，并使其处于人类智力的掌握之中。这个即时项目与这一性质的最近最富有成果的发展之一有关。＊＊＊

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. ***