Georgia Institute of Technology College of Sciences

The scissors congruence group of polytopes in $\mathbb{R}^n$ is defined tobe the free abelian group on polytopes in $\mathbb{R}^n$ modulo tworelations: $[P] = [Q]$ if $P\cong Q$, and $[P \cup P'] = [P] + [P']$ if$P\cap P'$ has measure $0$. This group, and various generalizations of it,has been studied extensively through the lens of homology of groups byDupont and Sah. However, this approach has many limitations, the chief ofwhich is that the computations of the group quickly become so complicatedthat they obfuscate the geometry and intuition of the original problementirely. We present an alternate approach which keeps the geometry of theproblem central by rephrasing the problem using the tools of algebraic$K$-theory. Although this approach does not yield any new computations asyet (algebraic $K$-theory being notoriously difficult to compute) it hasseveral advantages. Firstly, it presents a spectrum, rather than just agroup, invariant of the problem. Secondly, it allows us to construct suchspectra for all scissors congruence problems of a particular flavor, thusgiving spectrum analogs of groups such as the Grothendieck ring ofvarieties and scissors congruence groups of definable sets. And lastly, itallows us to construct filtrations by filtering the set of generators ofthe groups, rather than the group itself. This last observation allows usto construct a filtration on the Grothendieck spectrum of varieties that does not (necessarily) exist on the ring.