A random sorting network (RSN) is a uniformly chosen shortest path between the identity and the reverse permutation in the Cayley graph of the symmetric group generated by adjacent transpositions. Based on empirical evidence some remarkable conjectures have been made on the geometric behaviour of RSN. For example, the half way permutation appears to be the projection of spherical measure onto the disk.

I will explain these conjectures and recent joint work with Balint Virag and Mate Vizer.