Traces in higher categories, stable homotopy theory, and applications to fixed point theory

高范畴中的迹、稳定同伦理论以及不动点理论的应用

• 批准号: 1207670
• 负责人: Kathleen Ponto
• 金额: $ 10.67万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2015-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207670&HistoricalAwards=false
• 关键词: Traces; higher; categories; stable; homotopy

The standard approaches to the Lefschetz fixed point theorem and its converse have focused on integer valued invariants: the Lefschetz number and Nielsen number. These invariants are easy to define but very difficult to generalize. The PI has developed an approach that produces invariants that are elements of stable homotopy groups of spheres and twisted loop spaces, as well as more algebraic targets. These invariants readily generalize and agree with the integer valued invariants in the classical cases. In this project the PI proposes further generalizations to coincidences and periodic points. She also plans to explore extensions of important classical properties of fixed point invariants to generalizations such as equivariant and fiberwise fixed point invariants. The PI intends to mentor graduate students and graduate school bound undergraduates, expand graduate student participation in the topology seminar and facilitate greater interaction between students and external visitors.Fixed points arises in many different areas of mathematics and have a variety of applications. In some cases fixed points are very well understood, but there are also many important questions that have not been resolved. The goal of this project is to use tools from algebraic topology to define new invariants that detect fixed points and their generalizations and to provide a better understanding of the structure of an important collection of fixed point invariants. This project involves new approaches to a classical area of algebraic topology that should be amenable to doing computations. Students at different levels will be involved in the project.

莱夫谢茨不动点定理及其匡威定理的标准方法集中在整数值不变量：莱夫谢茨数和尼尔森数。这些不变量很容易定义，但很难推广。PI已经开发出一种方法，该方法产生的不变量是球面和扭曲环空间的稳定同伦群的元素，以及更多的代数目标。这些不变量很容易推广，并同意在经典情况下的整数值不变量。在这个项目中，PI提出了对巧合和周期点的进一步推广。她还计划探索扩展的重要经典性质的不动点不变量的推广，如equivariant和fiberwise不动点不变量。PI旨在指导研究生和研究生院的本科生，扩大研究生参与拓扑研讨会，并促进学生和外部访问者之间更大的互动。不动点出现在许多不同的数学领域，并有各种各样的应用。在某些情况下，固定点是很好理解的，但也有许多重要的问题尚未解决。该项目的目标是使用代数拓扑工具来定义新的不变量，以检测不动点及其泛化，并提供更好的理解不动点不变量的重要集合的结构。这个项目涉及到代数拓扑学的经典领域的新方法，应该服从于做计算。不同层次的学生将参与该项目。