Scissors congruence for manifolds via K-theory

The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German "schneiden und kleben," cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of  Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic. This is joint work with Hoekzema, Semikina, Rovi, and Wells.