Abstract: Integrals with rapidly oscillating factors and their cousins, exponential sums, arise in many mathematical contexts. For instance, Plancherel's inequality for the Fourier transform, one of the most fundamental inequalities of all, can be viewed as a bilinear oscillatory integral inequality.
The theory of upper bounds for multilinear oscillatory integral forms of degree three and higher is far less developed. Some inequalities for the trilinear case will be presented, and the relatively elementary) leading ideas underlying them will be exposed in broad terms. A dichotomy between structure and pseudorandomness is a key theme. Additive combinatorics Szemeredi's theorem), the theory of weak limits of solutions of nonlinear partial differential equations, web geometry, and multiscale analysis will make appearances in the story.