Traces in Algebraic K-theory and Topological Fixed Point Invariants

Overview: The additivity of the Euler characteristic is one of its most important properties of this very important invariant. This is reflected in the observation that the Euler characteristic can be understood as a class in algebraic K-theory. Classically, the generalizations of the Euler characteristic that relate to topological fixed point theory ignored the significance of additivity. The work in this proposal seeks to rectify this omission. Previous results suggest two to approaches to this goal. The first recognizes that the Reidemeister trace, a refinement of the Euler characteristic that gives a converse to the Lefschetz fixed point theorem, takes values in topological Hochschild homology. Topological Hochschild homology receives a map from topological restriction homology and it seems likely that the Reidemeister trace will lift through this map to a topologically meaningful class in topological restriction homology. An alternative approach is to start with the question of understanding K-theory of endomorphisms of modules over E-infinity ring spectra. From here the goal is to describe connections between the cyclotomic trace and trace in bicategories and symmetric monodial categories. In all of this work it also important to ground the results in topological meaning - to verify that classes constructed in these various groups are invariants associated to interesting questions. For example, these constructions should start by giving interesting invariants for periodic points and possibly extend to invariants for dynamical systems. Intellectual Merit: The projects in this proposal are based on several observations: * Methods from stable homotopy theory provide access to fixed point invariants that are richer and more refined than those produced by classical methods. * Trace methods from algebraic K-theory provide a powerful tool for understanding additive invariants. * These historically disparate areas are connected by fixed point invariants lying in topological Hochschild homology - one of the targets for traces in algebraic K-theory. This perspective also fits within a longer term goal of use fixed point theory as a test case for new methods of producing additive invariants that arise from wrong-way-maps or transfers. Broader Impacts: The graduate student population at the University of Kentucky is socioeconomically diverse and primarily composed of alumni of regional state universities and smaller colleges. For this population some of the most significant challenges to success in math arise from a lack of engagement with the mathematically community and an unfamiliarity and distance from mathematical culture. For example, these students may not know how to have a mathematical conversation or not understand the unspoken norms that faculty members assume. These students often struggle to identify their challenges and so over the last five years the PI has worked to build a community where students can identify difficulties, feel comfortable asking questions, and learn to effectively advocate for themselves. This has taken many forms. She and colleagues have revived a weekly topology seminar with external, local faculty, and graduate student speakers. She has also established an expectation that graduate students interact with visitors through meals. She has helped the graduate students establish a biweekly tea that is very well attended and provides opportunities to resolve many forms of graduate student confusion. Through active involvement with the local chapter of the AWM, she has helped the leadership fulfill its mission of supporting traditionally underrepresented groups by providing opportunities for them to fully participate in the community. She also mentors many graduate students, post-docs, and visiting faculty as they move into positions where they become mentors themselves.