  (This text is a survey written for the Bourbaki seminar on the work of F.
Manners.)
  Gowers uniformity norms are the central objects of higher order Fourier
analysis, one of the cornerstones of additive combinatorics, and play an
important role in both Gowers' proof of Szemer\'{e}di's theorem and the
Green-Tao theorem. The inverse theorem states that if a function has a large
uniformity norm, which is a robust combinatorial measure of structure, then it
must correlate with a nilsequence, which is a highly structured algebraic
object. This was proved in a qualitative sense by Green, Tao, and Ziegler, but
with a proof that was incapable of providing reasonable bounds. In 2018 Manners
achieved a breakthrough by giving a new proof of the inverse theorem. Not only
does this new proof contain a wealth of new insights but it also, for the first
time, provides quantitative bounds, that are at worst only doubly exponential.
This talk will give a high-level overview of what the inverse theorem says, why
it is important, and the new proof of Manners.
