Higher Order Fourier Uniformity of Bounded Multiplicative Functions in Short Intervals

The Higher order Fourier uniformity conjecture asserts that on most short intervals, the Mobius function is asymptotically uniform in the sense of Gowers; in particular, its normalized Fourier coefficients decay to zero.  This conjecture is known to be equivalent (after a "logarithmic" averaging) to Sarnak's conjecture on the disjointness of the Mobius function from zero entropy sequences.  In this talk we survey the known progress on this problem, and the main remaining barrier to its resolution, namely to break the logarithmic scale barrier.