Analysis, Geometry, and Topology: Episode 1

To a topologist, the Heisenberg Group is indistinguishable from Euclidean space. Nonetheless, I will show how an interesting topological construction, the Hopf Fibration, was used by Hajłasz et. al. to construct an interesting invariant of Lipschitz homotopies between maps into the Heisenberg group. I will mention my current joint work with Hajłasz and Schikorra to extend this to Hölder homotopies. The obstruction is a difficult problem in PDEs. On the other hand, I will show some pictures and data from my computer-assisted work to construct a Hölder homotopy from a circle to a point in the Heisenberg Group - a feat conjectured to be impossible by Gromov.