I will try to give a glimpse of exciting developments in representation theory over the last two decades. A central focus will be on the representations of symmetric groups over the complex numbers and fields of positive characteristic. Over the complex numbers our understanding is very good, however the case of positive characteristic fields has turned out to be more complicated than (I suspect) the pioneers would have ever imagined. Remarkably, there appears to be a way forward which combines ideas which emerged in the Langlands programme with techniques from mod p algebraic topology (Smith theory).
This video was produced by the Hausdorff Center for Mathematics, and was part of the conference Panorama of Mathematics II.
