The Heisenberg group ( 3 by 3 upper-triangular matrices with entries in a ring) is a venerable mathematical object. Simple random walk picks one of the bottom two rows at random and adds or substracts it from the row above.
I will use Fourier analysis to get sharp results about the long term behavior. For matrix entries in integers mod n, the walk converges to uniform after order n squared steps. The Fourier arguments
connect to Harper's operator, Hofstadter's butterfly, and the Fast Fourier Transform. For integer entries there is a kind of local CLT with connections to Gowers' higher Fourier analysis, and Levy's Brownian area.
I will explain all this for a non-specialist audience. This is joint work with Dan Bump, Bob Hough and Harold Widom.