Localization and Periodicity in Unstable Homotopy Theory

小行星9803601 研究人员将继续他的研究在代数拓扑结构，使用本地化的方法来揭露和分析周期性现象的不稳定同伦理论的空间。 在过去的二十年里，稳定同伦理论在全球范围内的理解取得了显着的进展，表明一些主要特征是由层次结构中的周期性现象的相互作用产生的。 这个项目是努力在不稳定同伦理论中发展类似理解的一部分。 研究者引入了一个层次的定位称为periodizations，这有助于解决一个给定的空间到一个色塔单色纤维。 他将努力理解这些分期以及与摩拉瓦K理论密切相关的局部化。 他最近发现了令人惊讶的新函子之间的稳定和不稳定的单色层同伦理论。 使用这些函子和其他工具，他将探索单色层的结构。 在经典的K-理论水平，这项工作是导致解决老问题的周期同伦群和K-理论的空间。 例如，拓扑空间或几何空间作为方程组的解的集合而出现，它们在数学中起着核心作用。 除了它们通常的测量之外，这样的空间具有更深层次的性质，即使在剧烈变形之后也会持续存在。 这些性质开始非常简单，在低维空间中是由分离的组分的数量和“孔”的数量决定的，但在高维空间中却变得非常丰富和信息丰富。 同伦理论领域致力于他们的研究，使用强大的代数和几何技术。 近年来，出现了一种令人兴奋的色性方法，可以将复杂的同伦现象分解为更简单的周期部分。 然而，这种半色调的方法最初只适用于某些稳定的现象。 在本项目中，研究人员正在努力将其扩展到全方位的不稳定同伦理论现象。 ***

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. ***