# Hilbert's Generalized Third Problem and the Algebraic K-theory of Fields

In this talk I'll explain how one might attack Hilbert's Generalized Third Problem via homotopy theory. Two n-dimensional polytopes, $P$, $Q$ are said to be scissors congruent if one can cut $P$ along a finite number of hyperplanes, and re-assemble the pieces into $Q$. The scissors congruence problem, aka Hilbert Generalized Third Problem, asks: when can we do this? what obstructs this? In two dimensions, two polygons are scissors congruent iff they have the same area. In three dimensions, there is volume AND another invariant, the Dehn Invariant. In higher dimensions, very little is known. I'll give an introduction to this very classical problem, and explain how homotopy theory can be used to get purchase on it. Prerequisites: The discussion of Hilbert's Third Problem and Dehn's invariant will be widely accessible. No knowledge of algebraic K-theory will be assumed. This is all joint work with Inna Zakharevich.