A common approach to problem-solving is to split a problem into smaller sub-problems, solve each of the smaller problems, and assemble the answers into a solution to the original problem. This last step is often very difficult, as there are multiple ways of gluing the pieces of the solution together. The mathematical area of K-theory studies different ways of putting such solutions back together, as well as the relations behind differently-assembled pieces. The invariants constructed by K-theory can be found in many fields, from number theory to algebraic geometry to topology. This National Science Foundation award investigates novel connections between fields through a K-theoretic perspective, by studying how geometric objects, known as polytopes, varieties and manifolds, can be cut apart and reassembled. By analyzing these geometric cut-and-paste problems using traditional K-theoretic techniques found in algebra, the project hopes to shed light on longstanding conjectures relating geometry and algebra. The PI also intends to establish a K-12 math circle available to all local children, and the organization of a district-wide math club for 3-5 graders. Currently, math clubs at schools are heavily dependent on parental involvement and are therefore only available when interested parents have students at the school; organizing such clubs under a central organization would help with institutional memory and make it available to more students.
The research project consists of three parts. The first is an in-depth exploration of the scissors congruence of polytopes as it relates to the algebraic K-theory of the real and complex numbers. Using Rognes' stable rank filtration the PI and collaborator Jonathan Campbell intend to construct maps between the filtered parts of Rognes' filtration and the derived scissors congruence of polytopes; the long-term goal of this project is to investigate the connections between scissors congruence groups and the Beilinson--Soule conjecture. The second part of this project investigates the construction of derived motivic measures on the Grothendieck spectrum of varieties. There are two main flavors of motivic measure: the cohomological measures, which involve structures on the cohomology of the variety, and the Hermitian measures, which involve enriching the Euler number. These measures have proven extremely fruitful in studying the Grothendieck ring of varieties; the project proposes to lift these to the Grothendieck spectrum of varieties and use them to study the higher invariants of cut-and-paste problems on varieties. The cohomological project is joint work with Jonathan Campbell and Jesse Wolfson; the Hermitian project is joint with Kirsten Wickelgren. The third part of this project is an investigation of "squares K-theory", which uses four-term (instead of three-term) relations. Such relations---for example, the principle of inclusion-exclusion [P] + [Q] = [P u Q] + [P n Q]---appear frequently in geometric cut-and-paste problems such as SK-invariants, Bittner's presentation of the Grothendieck ring of varieties, and the definitions of McMullen's polytope algebra. The project proposes to extend the definition of K-theory to such relations, thus allowing the construction of higher derived invariants for these (and similar) problems.
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.
