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

点击翻译按钮获取中文摘要

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.