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This special colloquium will host a faculty candidate.

Speaker: Megan Bernstein, Visiting Assistant Professor, Georgia Institute of Technology, School of Mathematics

Title: Progress in showing cutoff for random walks on the symmetric group

Abstract: Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff with window for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group. Results include joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.