Mathematical Sciences: Localization and Periodicity in Homotopy Theory

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，S在对稳定同伦理论的全球理解方面取得了显著进展，表明一些主要特征是在层次中排列的周期现象的相互作用下产生的。这个项目是发展对不稳定同伦理论的类似理解的努力的一部分。在最近的工作中，Bousfield引入了一个局部化层次，称为正整数n的vn-周期化，它用于将给定的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vn-周期化以及密切相关的同调局部化。他已经在空间的v1-周期化和K-同调局部化之间建立了很强的联系，从而解决了关于迭代循环空间的K-理论的老问题。几何对象或空间在数学中扮演着中心角色，例如，源于方程组的解集。除了它们通常的尺寸之外，这种空间还有更深层次的特性，即使在剧烈变形之后仍然存在。这些特性从独立组件的数量和孔洞的数量开始几乎是微不足道的，但在更高的维度上变得极其丰富和丰富。同伦理论领域致力于利用强大的代数和几何技巧来研究它们。自50年代的S以来，同伦理论中的各种周期现象被发现，而在80年代的S中，发展了一种令人兴奋的“色”方法，可以将复杂的现象分解成简单得多的周期部分。这种方法最初局限于某些更简单的现象，称为稳定现象，但现在它正在扩展到一个完整的、有趣的、重要的不稳定同伦理论现象NA。***

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. ***