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

数学科学：论椭圆上同调猜想与上同调理论的关系 E-sub-n

• 批准号: 9306938
• 负责人: Matthew Ando
• 金额: $ 5.55万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-09-01 至 1996-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9306938&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Relationship; between; conjectures

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. ***

小行星9306938 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。 在过程中，他的工作共形场理论，爱德华维滕了一些令人惊讶的意见之间的关系，三个重要的同源性理论，奇异的同源性，K理论，椭圆上同调。 一般的原则是，为了理解空间的K-理论，人们应该尽可能多地理解它的自由圈空间的奇异同调，椭圆上同调和K-理论也是如此。 大约在同一时间，迈克霍普金斯和各种合作者显示了根本的重要性，代数拓扑的一系列同源理论E分n。 他们的工作涉及一个非常有益的互动代数拓扑和数论。 证明了E-sub-0接近于奇异同调，E-sub-1接近于K-理论，E-sub-2接近于椭圆同调。 安藤概述了维滕和霍普金斯等人的某些观点，可能是密切相关的。 他打算研究这些关系，使它们更加精确，希望这两个研究领域的进展可以更直接和更有成效地相互通报。 这个项目的主题，绘制深代数和数论理论之间的联系，一方面和深几何（拓扑）理论，是一次又一次地在现代拓扑。 随着理论变得越来越复杂，数学家们只有通过这样的总体组织原则，才能使结构易于管理，并在人类智力的掌握范围内。 目前的项目关系到这一性质的最富有成果的最近发展之一。 ***