Scissors congruence is about cutting objects up and rearranging the pieces into other objects. For example, any two polygons with the same area are always scissors congruent. Hilbert's third problem asked if there exist polyhedra with the same volume which are not scissors congruent. In 1901 Dehn showed that yes, there do: a cube and a regular tetrahedron with the same volume are not scissors congruent. Analogous questions can be asked in other geometries and other dimensions, and the answers discovered have amazing connections to geometry, algebra and $K$-theory.

We can also consider this question in the context of algebraic varieties. Determining scissors congruence classes of varieties is generally prohibitively difficult, but analyzing algebraic scissors congruence invariants (such as point counting or Euler characteristic) can give interesting non-trivial information about the class of a variety.

In this talk we introduce a general framework for analyzing scissors congruence problems and discuss the new information that this framework reveals. We will also introduce algebraic $K$-theory and discuss why it might be useful for analyzing such problems, and conclude with some theorems which arise from this perspective.