We describe mean value estimates for exponential sums of degree exceeding 2
that approach those conjectured to be best possible. The vehicle for this
recent progress is the efficient congruencing method, which iteratively
exploits the translation invariance of associated systems of Diophantine
equations to derive powerful congruence constraints on the underlying
variables. There are applications to Weyl sums, the distribution of polynomials
modulo 1, and other Diophantine problems such as Waring's problem.Comment: Submitted to Proceedings of the ICM 201