How many steps does it take to shuffle a deck of n cards, if at each step we pick two cards uniformly at random and swap them? Diaconis and Shahshahani proved that 1/2 n log n steps are necessary and sufficient to mix the deck. Using the representation theory of the symmetric group, they proved that this random transpositions card shuffle exhibits a sharp transition from being unshuffled to being very well shuffled. This is called the cutoff phenomenon. In this talk, I will explain how to use the spectral information of a Markov chain to study cutoff. As an application, I will briefly discuss the random to random card shuffle (joint with M. Bernstein) and the non-backtracking random walk on Ramanujan graphs (joint with P. Sarnak). Speaker(s): Evita Nestoridi (Princeton University)