Speakers, their affiliations, and topics

Detailed syllabus
Since the pioneering work of Diaconis and Shahshahani on generating random permutations using transposi- tions in 1981, representation theory has become an important tool in understanding random walks on finite groups. We will study certain natural random walk models on the symmetric group in increasing order of difficulty. In these models, one prescribes a probabilistic rule that describes what the possible permutations are at time n + 1 if one has a certain permutation at time n (time is taken to be discrete). For each model, we will understand how a random permutation looks like after a long time and how long it takes for the distribution to reach stationarity. The first three models are well-studied; we will develop our understanding of the theory by working through them. In the end, we will discuss the fourth model, where some of the same questions remain open. Very recently (on 29th September 2015) a preprint has been posted to the arXiv announcing proofs of the eigenvalue conjectures in this model.

1. Random Transpositions
1. This is the original model of Diaconis and Shahshahani, where one obtains a new permutation at each step by performing a random transposition. The study of this model requires a basic understanding of Markov 
   chains and character theory of the symmetric group.
2. Top to random shuffle
1. In this model, one obtains a new permutation at each step by making the transposition (1, j) where j is chosen at random. That is, the first symbol is interchanged with a uniformly randomly chosen symbol. The study of this model requires an understanding of the so-called Jucys-Murphy elements and the more recent Vershik-Okounkov theory of the representation of the symmetric group.
3. Tsetlin library
1. This is a simplistic model of a library with a single shelf! One obtains a new permutation at each step by multiplying the existing permutation with the cycle (1, 2, ..., j) where again j is chosen at random. This model is qualitatively different from the previous two because it requires the study of the representation theory of semigroups. This is the simplest such model and the ideas involved in understanding this model  follow naturally from ideas in the theory of group representations.
4. Random to random shuffle
1. This model is a natural generalisation of the top-to-random shuffle, but several conjectures about this model remain wide open. One chooses two symbols i and j uniformly at random, and then multiplies the existing permutation by the cycle (i, i + 1, ..., j − 1, j) if i ≤ j or (j, j + 1, ..., i − 1, i) if i > j. One of the main aims of this workshop will be to explore this model, understand recent progress, and make some headway towards
   the open questions.
5. Reference
1. Persi Diaconis (1988): Group representations in probability and statistics, Institute of Mathematical Statistics

Timetable