CAREER: Decomposition, duality and Picard groups in chromatic homotopy theory

该职业奖支持代数拓扑学的研究：代数拓扑学是通过代数的透镜研究几何物体的数学领域。也就是说，它通过赋予几何形状各种代数不变量（如数字）来对几何形状进行分类。这些工具对于研究更大维度空间中的几何物体是非常强大的，我们不能轻易描绘，例如，高维球体的类似物。PI的工作包括使用和开发代数拓扑工具来描述当我们考虑大维度球体之间的连续映射时出现的现象。该项目的教育部分围绕着创建一系列年度会议，目标是在中南部地区建立一个拓扑社区，并为该地区的初级数学家提供指导和交流机会。PI还将为高中生开发和运行一个数学丰富项目，并在“拓扑学中的女性”研讨会上领导一个研究小组。色同伦理论是通过形式群律变形的Lubin-Tate理论来理解稳定同伦范畴的一个概念和计算框架。它是稳定同伦理论的关键工具，无论是计算还是组织对大规模现象的研究。PI将研究与色同伦理论中的对偶性和可逆性有关的几个项目。更具体地说，PI将在这种情况下研究西班牙-怀特黑德和格罗斯-霍普金斯对偶性，特别是应用于拓扑模形式的谱和K(2)-局部范畴中的其他重要谱。该项目的第二部分侧重于与对偶密切相关的可逆性现象。PI将研究K(n)的Picard群——特别感兴趣的局部范畴。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This CAREER award supports research in algebraic topology: an area of mathematics which studies geometric objects through the lens of algebra. Namely, it classifies geometric shapes by assigning to them various algebraic invariants, such as numbers. These tools are very powerful for studying geometric objects in spaces of larger dimensions that we can not easily picture, for example, the high dimensional analogues of spheres. The PI's work involves using and developing tools from algebraic topology for describing phenomena that arise when we consider continuous maps between large dimensional spheres. The educational component of the project is centered around creating a series of annual conferences with the goal of building a topology community in the South Central region and providing mentoring and networking opportunities for junior mathematicians from this region. The PI will also develop and run an enrichment program in mathematics for high school students and lead a research team in a Women in Topology workshop.Chromatic homotopy theory is a conceptual and computational framework for understanding the stable homotopy category through the Lubin–Tate theory of deformations of formal group laws. It is a key tool for stable homotopy theory, both for calculations and for organizing the search for large scale phenomena. The PI will investigate several projects related to duality and invertibility in chromatic homotopy theory. More specifically, the PI will study Spanier-Whitehead and Gross-Hopkins duality in this setting, particularly as applied to the spectrum of topological modular forms and other important spectra in the K(2)-local category. The second part of the project focuses on invertibility phenomena which are closely intertwined with duality. The PI will investigate the Picard groups of K(n)-local categories of particular interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.