roth famously used fourier analysis to upper bound the size of a set of integers without 3-term arithmetic progressions. one might hope that similar techniques can be used for 4 or more term progressions, and some simple examples demonstrate otherwise. however, gowers (2001) introduced a ``higher order'' fourier analysis which generalizes roth's proof to longer arithmetic progressions. in this talk, we will give a combinatorial sketch of the methods of higher order fourier analysis culminating in the key problem of the field, the inverse conjecture for the gowers norms.