Structures on combinatorial K-theory: TR, zeta-functions, and motivic measures

There is a classic approach to solving large and complicated problems commonly employed in the field of algebraic topology. The idea is to break down a large problem into smaller ones, solve each of the smaller ones, and then reassemble the answers into a solution to the larger problem. This approach can be very fruitful, but it comes with one important caveat: it must be possible to reassemble the smaller solutions into a larger one, and to know when this reassembly is uniquely determined. A central focus of this project is to use the powerful tool of K-theory to address the question of which different objects can be reconstructed out of the same pieces. This will be done by modifying and extending techniques from algebraic and topological K-theory and applying them to the more recently emerging field of combinatorial K-theory. The outcomes of this project will have wide applications in geometry and combinatorics. In parallel with this research activity, the PI will continue their engagement with student mentoring, through enrichment activities at the K-12 level, and through mentioning at the college and post-grad level, with an overall focus on improving the accessibility of mathematics to a wide audience. The spectrum topological restriction homology (TR) has been useful in classical computations of algebraic K-theory and topological Hochschild Homology (THH). However, the construction of this spectrum relies on having a spectral enrichment, which combinatorial K-theory does not have. The goal of this project is to produce new constructions of TR that do not rely on this enrichment, and to use them to construct TR for examples of combinatorial K-theory, such as varieties. In recent work it has been shown that TR is the codomain of universal zeta-functions in many contexts, and this project hopes that a new construction will allow for a deeper understanding of the structure of zeta-functions and their relationship to combinatorial K-theory, especially in the examples of finite sets and varieties. In addition, the novel constructions of combinatorial K-theory (using categories with covering families or categories with squares) are far more general than previously-understood constructions. Another goal of the project is to study the behavior of TR in these examples and construct new types of zeta-functions for them. 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.