Localization and Periodicity in Unstable Homotopy Theory

9803601 Bousfield研究员将继续他在代数拓扑学方面的研究，使用局部化方法揭示和分析空间不稳定同伦理论中的周期现象。在过去的二十年里，在对稳定同伦理论的全球理解方面取得了显著的进展，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。这位研究人员引入了一种称为周期定位的局部化层次结构，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分期以及与Morava K理论密切相关的局部化。他最近在同伦理论的稳定和不稳定单色层之间发现了令人惊讶的新函子。使用这些函数器和其他工具，他将探索单色层的结构。在经典的K-理论水平上，这项工作导致了关于周期同伦群和空间的K-理论的旧问题的解决。例如，拓扑或几何空间作为方程系统的解的集合而产生，它们在数学中发挥着核心作用。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。这些特性从低维空间中独立组件的数量和“洞”的数量开始非常简单，但在高维空间中变得极其丰富和丰富。同伦理论领域致力于它们的研究，使用了强大的代数和几何技术。近年来，出现了一种令人兴奋的色度方法，可以将复杂的同伦现象分解成更简单的周期部分。然而，这种色度方法最初只是停留在某些稳定的现象上。在本项目中，研究人员正致力于将其扩展到不稳定同伦理论现象的全部范围。***

9803601 Bousfield The investigator will continue his research in algebraic topology, using localization methods to expose and analyze periodic phenomena in the unstable homotopy theory of spaces. During the past two decades, remarkable progress has been 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. The investigator has introduced a hierarchy of localizations called periodizations, which serve to resolve a given space into a chromatic tower with monochromatic fibers. He will work to understand these periodizations as well as the closely related localizations with respect to Morava K-theories. He recently discovered surprising new functors between the stable and unstable monochromatic layers of homotopy theory. Using these functors and other tools, he will explore the structure of the monochromatic layers. At the classical K-theoretic level, this work is leading to solutions of old problems concerning periodic homotopy groups and the K-theory of spaces. Topological or geometric spaces arise, for instance, as sets of solutions to systems of equations, and they play a central role in mathematics. Beyond their usual measurements, such spaces have deeper properties that persist even after drastic deformations. These properties begin very simply with the number of separate components and the number of "holes" in a low dimensional space 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. In recent years, an exciting chromatic approach to the subject has emerged, allowing decompositions of complicated homotopical phenomena into much simpler periodic parts. However, this chromatic approach was originally rest ricted to certain stable phenomena. In the present project, the investigator is working to extend it to a full range of unstable homotopy theoretic phenomena. ***