Persi Diaconis: Shuffling Cards and Hopf Algebras

Hopf algebras were introduced by topologists, but now they seem to be all over the place. Combinatorialists are using them to study procedures for pulling apart and putting together sets, they occur in chemistry, physics and quantum groups. In work with Amy Pang and Arun Ram we have found that the 'Hopf-square' yields natural Markov chains, such as the Gilbert -Shannon-Reeds method of shuffling cards, a rock breaking model of Kolmogorov, and natural walks on graphs and simplicial complexes. These chains can be explicitly diagonalized using the combinatorics of the free Lie algebra. No previous knowledge of Hopf algebras or shuffling required.