Aldous' conjecture asserted that, for every graph, the random walk and the interchange process have the same spectral gap. Recently, in collaboration with T.M. Liggett and T. Richthammer, we proved the conjecture using a recursive approach. I will review some background material together with previously known results, and will present the main steps involved in the proof. In particular, I will discuss the electric network reduction idea, which allows one to reduce the problem to the proof of certain new comparison inequalities between weighted graphs. Possible extensions of the result together with some open problems will also be discussed.
