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

职业：色同伦理论中的分解、对偶性和皮卡德群

• 批准号: 2239362
• 负责人: Irina Bobkova
• 金额: $ 41.27万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239362&HistoricalAwards=false
• 关键词: CAREER; Decomposition; duality; Picard; groups

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.

这个职业生涯奖支持代数拓扑学方面的研究：一个通过代数的透镜研究几何对象的数学领域。也就是说，它通过分配给它们各种代数不变量（如数字）来对几何形状进行分类。这些工具对于研究我们无法轻易描绘的更大维度空间中的几何对象非常强大，例如，球体的高维类似物。 PI的工作涉及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数学丰富计划，并领导一个研究团队参加Women in Topology研讨会。色同伦理论是一个概念和计算框架，用于通过正式群律变形的Lubin-Tate理论来理解稳定同伦范畴。它是稳定同伦理论的关键工具，无论是计算还是组织对大尺度现象的搜索。PI将研究与色同伦理论中的对偶性和可逆性相关的几个项目。更具体地说，PI将研究Spanier-Whitehead和Gross-Hopkins对偶，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二元性密切相关的可逆性现象。PI将调查K（n）的Picard组-特别感兴趣的地方类别。该奖项反映了NSF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。