Fourier optimization problems appear naturally within several
different fields of mathematics, particularly in analysis and number
theory. These are problems in which one imposes certain conditions on a
function and its Fourier transform, and then wants to optimize a certain
quantity. A recent example is given in the proof of the optimal sphere
packing in dimensions 8 and 24. In this talk I want to show how certain
optimization problems of this sort appear naturally in connection to the
question of bounding the maximal gap between consecutive primes, under
the Riemann hypothesis. In particular, we improve the best known bounds
for this problem, that dates back to the works of Cramer in the 1920's.
This is a joint work with M. Milinovich (Univ. of Mississippi) and
K. Soundararajan (Stanford Univ.)