Consider a list of n particles labelled in increasing order. A sorting
network is a way of sorting this list into decreasing order by swapping adjacent
particles, using as few swaps as possible. Simulations of large-n uniform random
sorting networks reveal a surprising and beautiful global structure involving
sinusoidal particle trajectories, a semicircle law, and a theorem of Archimedes.
Based on these simulations, Angel, Holroyd, Romik, and Virag made a series of
conjectures about the limiting behaviour of sorting networks. In this talk, I will
discuss how to use the local structure of random sorting networks to prove these
conjectures.