Tuesday, November 5, 2019 - 6:00pm
Abstract: Two polygons in the plane are called scissor congruent (or equipartite) if one can cut one into polygonal pieces and reassemble them to get the other. The Wallace-Bolyai-Gerwien theorem, first proved in the early 19th century, says that two polygons are scissor congruent if and only if they have the same area. The same question makes sense for polytopes in three-space. In 1900, Hilbert presented a list of problems that he thought would be important in the 20th century. Number three in the list was the question of whether an analog of that theorem holds in three dimensions. This problem didn’t stand up for very long: shortly, Dehn, found a counterexample: a cube and a regular tetrahedron of the same volume are not scissor congruent.
For this, Dehn produced an invariant of scissor congruence that is different for the cube and the tetrahedron. A modern formulation presents Dehn’s invariant as an element of a tensor product of two infinite dimensional vector spaces over the rationals, giving a proof of Dehn’s theorem that is only a few pages long. In this talk, I will explain what the tensor product is and construct Dehn’s invariant.