Parameterized, Algebraic, and Chromatic Traces

This project uses symmetry in an essential way to understand topological spaces and their invariants. The Euler characteristic is a very illuminating, fundamental, and accessible first example of such an invariant. It can be computed simply in terms of counting certain objects from which the topological space is constructed. Furthermore, symmetries respected by the invariant lead to powerful refinements that detect rich information and have computationally useful properties such as additivity and multiplicativity. This project will develop similar rich generalizations of topological invariants where symmetry plays an essential role. In addition to graduate student mentoring, summer school organization, and service to local and national committees, the PI will continue to organize undergraduate quilting projects at U of Kentucky, designing and making quilts with significant mathematical content, while attracting new students to mathematics. This project centers on three clusters of ideas: 1) invariants for fixed points, periodic points, and flows refining the Euler characteristic, 2) generalizations of algebraic Lefschetz and Riemann-Roch theorems to endomorphisms, and 3) traces in topological settings that enjoy significant algebraic structure. These questions all build on a vast generalization of the trace of a matrix that can be applied in topological and algebraic examples. For this trace, finite dimensionality is replaced by a notion of dualizability and the examples above manifest different forms of this dualizability. The discrepancy between the topological and algebraic examples provides an opportunity to reconcile different perspectives and a major goal of this project is to use this compatibility to prove topological generalizations of algebraic results. This project is jointly funded by the Topology & Geometric Analysis (TGA) Program, and the Established Program to Stimulate Competitive Research (EPSCoR). 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.