Cutoff profiles, from transpositions to more general conjugacy classes

Diaconis and Shahshahani used Fourier analysis on groups (i.e. representation theory) when they proved in 1981 that there is a cutoff for random transpositions. Their technique has since then been very fruitful. In this talk we will introduce some ideas about the representation theory of the symmetric group, and explain how it can be used to derive cutoff profiles. We will also discuss some difficulties that arise when considering more general conjugacy classes, and announce upcoming results in this direction.