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

Lefschetz不动点定理及其逆定理的标准方法集中在整数值不变量：Lefschetz数和Nielsen数。这些不变量很容易定义，但很难推广。PI开发了一种方法，该方法产生不变量，这些不变量是球体和扭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与经典情况下的整数值不变量一致。在这个项目中，PI提出了对巧合点和周期点的进一步推广。她还计划探索将不动点不变量的重要经典性质扩展到诸如等变不动点不变量和纤维不动点不变量的推广。PI打算指导研究生和研究生院的本科生，扩大研究生对拓扑学研讨会的参与，并促进学生与外部访客之间的更多互动。不动点出现在数学的许多不同领域，有各种各样的应用。在某些情况下，固定的点被很好地理解，但也有许多重要的问题没有得到解决。本项目的目标是使用代数拓扑中的工具来定义新的不变量，以检测不动点及其推广，并提供更好地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的一个经典领域的新方法，它应该适合于做计算。不同层次的学生都将参与这个项目。

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.