In this talk we study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave an elegant solution which shows that for a standard deck of 52 cards, seven shuffles is enough. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us for random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate. This is joint work with Persi Diaconis and K. Soundararajan.

A good background for interested students would be to read the very entertaining and accessible paper by Bayer and Diaconis titled 'Trailing the dovetail shuffle to its lair'. This material was the precursor to the work I will be presenting and gives a very accessible and fun introduction to the problem.