A characteristic feature of dispersive partial differential equations (such as Schrödinger or wave equations) is that solutions spread out in space and decay in time while preserving the spatial L^2 norm. This effect can be quantified by so-called Strichartz estimates and such estimates play a fundamental role in the perturbative analysis of nonlinear dispersive PDEs.

Strichartz estimates are dual to estimates for the restriction of the Fourier transform of the characteristic surface corresponding to the differential operator. From this perspective, the curvature of the characteristic surface determines the decay. Fourier restriction theory is a classical topic in harmonic analysis which started in the 1960s with first results and conjectures of Elias Stein and his students.

Both in the analysis of nonlinear dispersive PDEs and in Fourier restriction theory more information can be obtained by passing to a bilinear setting. More precisely, the product of two wave packets traveling in transversal directions enjoys better space-time decay. In this talk, I will describe recent progress on bilinear Fourier restriction estimates. Then, I will outline how these can be used to prove small data global well-posedness and scattering for cubic Dirac equations and the wave maps equation in scaling-critical spaces. This is based on joint work with Timothy Candy.