Mathematical Sciences: Localization and Periodicity in Homotopy Theory

数学科学：同伦理论中的定域性和周期性

• 批准号: 9504497
• 负责人: Aldridge Bousfield
• 金额: $ 7.5万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504497&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Localization; Periodicity; Homotopy

9504497 Bousfield Bousfield will continue his research in homotopy theory, using localization methods to expose and analyze unstable periodic phenomena. During the 1980's, remarkable progress was made toward a global understanding of stable homotopy theory, showing that some major features arise chromatically from an interplay of periodic phenomena arranged in a hierarchy. This project is part of an effort to develop a similar understanding in unstable homotopy theory. In recent work, Bousfield introduced a hierarchy of localizations, called the vn-periodizations for positive integers n, which serve to resolve a given space into a chromatic tower with monochromatic fibres. He will work to understand these vn-periodizations of spaces as well as the closely related homological localizations. He has already established strong links between v1-periodizations and K-homology localizations of spaces, leading to solutions of old problems concerning the K-theory of iterated loop spaces. Geometric objects or spaces play a central role in mathematics, arising for instance from the solution sets of systems of equations. Beyond their usual measurements, such spaces have deeper properties which persist even after drastic deformations. These properties begin almost trivially with the number of separate components and the number of holes but become tremendously rich and informative in higher dimensions. The field of homotopy theory is devoted to their study using powerful algebraic and geometric techniques. Various periodic phenomena in homotopy theory have been discovered since the 1950's, and an exciting "chromatic" approach to the subject was developed in the 1980's, allowing resolutions of confusingly complicated phenomena into much simpler periodic parts. This approach was originally restricted to certain simpler phenomena known as stable phenomena, but it is now being extended to a full, fascinating, important range of unstable homotopy theoretic phenome na. ***

9504497 Bousfield Bousfield将继续他在同伦理论方面的研究，使用局部化方法来揭示和分析不稳定的周期现象。在20世纪80年代，对稳定同伦理论的整体理解取得了显著进展，表明一些主要特征是由排列在层次结构中的周期性现象的相互作用产生的。这个项目是在不稳定同伦理论中发展类似理解的努力的一部分。在最近的工作中，Bousfield引入了一种定位层次，称为正整数n的n-周期化，用于将给定空间分解为具有单色纤维的色塔。他将致力于理解这些空间的n-周期化以及密切相关的同调局域化。他已经建立了v1-周期化和空间的k -同调局部化之间的紧密联系，从而解决了迭代循环空间的k理论中的老问题。几何对象或空间在数学中起着中心作用，例如从方程组的解集中产生。除了通常的测量之外，这样的空间具有更深层的特性，即使在剧烈变形之后也会持续存在。这些性质开始于几乎微不足道的独立组件的数量和孔的数量，但在更高的维度上变得非常丰富和信息丰富。同伦理论领域致力于使用强大的代数和几何技术来研究它们。自20世纪50年代以来，同伦理论中的各种周期现象已经被发现，并且在20世纪80年代发展了一种令人兴奋的“色”方法，可以将令人困惑的复杂现象分解成更简单的周期部分。这种方法最初局限于某些被称为稳定现象的简单现象，但现在它被扩展到一个完整的、迷人的、重要的不稳定同伦理论现象范围。＊＊＊