Mathematical Sciences: Periodic Homotopy and Unstable Localization

9401166 Langsetmo该职业进阶奖支持拓扑学领域的数学研究。继续在同伦理论中研究周期同伦与更高Morava k理论之间的关系。特别地，工作将在扩展Dyer-Lashof运算的构造和计算无限循环空间的Morava k理论方面进行。计算了各种有限环路空间的Morava k理论，并确定了cobar （base - to - fiber）频谱序列收敛的时间。计划利用不稳定局域化和映射望远镜进一步研究非连接同调理论下稳定局域化与不稳定局域化之间的关系，以及不稳定周期同伦与周期同调理论之间的关系。最后，将考虑在相关领域的应用，如表示理论。拓扑学是对空间的研究，其中形状和距离变得不那么重要，而连通性和可变形性的质量。特别是同伦，关注的是一种数学结构的发展，它测量了当曲线被限制在特定表面或给定空间中时，曲线可能或可能不会变形成另一个曲线的程度。＊＊＊

9401166 Langsetmo This career advancement award supports mathematical research in the field of topology. The work continues studied in homotopy theory involving the relationship between periodic homotopy and higher Morava K-theories. In particular, work will be done in extending the construction of Dyer-Lashof operations and calculating the Morava K-theory of infinite loop spaces. The calculation of the Morava K-theory of various finite loop spaces and determining the when the cobar (base to fiber) spectral sequence converges will also be carried out. Further study of the relationship between stable and unstable localization in the case of non-connective homology theories and the relationship between unstable periodic homotopy and periodic homology theories via unstable localization and mapping telescopes are planned. Finally, applications to related fields such as representation theory will be considered. Topology is the study of spaces in which shape and distance become less important while qualities of connectedness and deformability. Homotopy, in particular, focuses on the development of a mathematical structure which measures the extent to which curves may or may not be deformed into one another while constrained to stay on a particular surface or within a given space. ***