# Scissors congruence and algebraic K-theory

# Scissors congruence and algebraic K-theory

Hilbert's third problem asks the following question: given two polyhedra with the same volume, can we decompose them into finitely many pairwise congruence pieces? The answer, provided by Dehn in 1901 is no; there is a second invariant on polyhedra, now called the Dehn invariant. Classical scissors congruence asks this question in other dimensions and geometries. In this talk we construct an abstract framework for discussing scissors congruence problems using algebraic K-theory. By discarding much of the geometric underpinning of scissors congruence problems we are able to construct decomposition invariants in much more general settings, including Grothendieck rings of arbitrary model's. As an application of this framework we construct a "derived Grothendieck ring of varieties" and construct a spectral sequence converging to it's homotopy groups.