In his monograph “Systèmes Orthogonaux” (Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910), Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of PDEs, where for each unknown function a certain subset of partial derivatives is prescribed and the values of the unknown functions are prescribed along the corresponding transversal coordinate…

One way to effectively show a group is non-trivial is to find a non-trivial representation.  A major open question in low-dimensional topology is whether the fundamental group of a closed three-manifold other than S^3 has a non-trivial SU(2) representation, and this is a strategy for an alternate proof of the three-dimensional Poincare conjecture.  We will…

In this talk I'll explain a surprising relationship between the objects in the title. 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 it into $Q$. The scissors congruence problem asks: when can we do this? what obstructs this? In…

The Penrose triangle, also known as the impossible tribar is an icon for cohomology.  It is literally the icon for Cech cohomology on Wikipedia.  The idea goes back to a paper by Roger Penrose in 1992, but was first reported by Penrose several years earlier.  There, he shows how the impossibility of the figure depends…