Anton Zorich, IMJ-PRG
    (joint work with V. Delecroix, E. Goujard, and P. Zograf)

Twenty years ago Vladimir Arnold posed a question on asymptotic
statistics of cycle decomposition of special permutations of large
number of elements; such permutations can be seen as random
integral exchange transformations with a fixed permutation. It was
proved by I. Pak and A. Redlich in 2008 that a random interval
exchange permutation (A,B,C) -> (C,B,A) is transitive with
asymptotic probability $6/pi^2$. However, there was no further
progress in Arnold’s problem ever since.

I will present a solution of Arnold’s problem for an arbitrary
permutation based, as usual, on combination of combinatorics,
geometry, and dynamics of the moduli space of Abelian
differentials. En route I will suggest some further conjectures. I
will try to make the talk accessible to a general audience. The
talk would be in English or in French depending on preferences of
the audience.