Mixing times and the cutoff phenomenon
Markov chains are random processes that retain no memory of the past. The mixing time of a Markov chain is the time it takes for it to reach equilibrium. During the last three decades, there has been a lot of progress in developing various techniques to estimate mixing times for various chains and to understand the cutoff phenomenon which means that the Markov chain has an abrupt convergence to equilibrium. We will present recent work establishing cutoff for the random to random card shuffle which confirms a 2001 conjecture of Diaconis. We will also present a proof of uniform lower bounds for Glauber dynamics for the Ising model, extending a result of Ding and Peres. The proofs employ both probabilistic and algebraic techniques.