Geometric Cut-and-Paste Problems
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 the 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.
Inna Zakharevich, Mathematics, is investigating 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, Zakharevich is shedding light on longstanding conjectures relating to geometry and algebra. The research 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 real and complex numbers. Using Rognes' stable rank filtration, Zakharevich and collaborator Jonathan Campbell at Vanderbilt University are constructing maps between the filtered parts of Rognes' filtration and the derived scissors congruence of polytopes.
The second part of this project investigates the construction of derived motivic measures on the Grothendieck spectrum of varieties. The third part of this project is an investigation of "squares K-theory," which uses four-term (instead of three-term) relations. Zakharevich is extending the definition of K-theory, thus allowing the construction of higher derived invariants for these problems.