The goal of the talk is to present a set of open problems about Fourier extension operators from a perspective motivated by features of the classical kinetic transport equation. This perspective naturally brings into play the Wigner transform, an ubiquitous operator in quantum mechanics which is closely connected to the classical Fourier transform. In joint work with Bennett, Gutierrez and Nakamura, we show how Sobolev estimates for the Wigner transform can be converted into certain tomographic bounds for the Fourier extension operator to the paraboloid (weaker variants of the classical Mizohata-Takeuchi conjecture for this manifold). We are also able to extend this analysis to a wide class of hypersurfaces, which requires finding and understanding a good "geometric" replacement for the classical Wigner transform. Our results depend on lower bounds for certain relative curvature quantities, rather than lower bounds for the Gaussian curvature of these manifolds, which contrasts the intuition behind the classical Fourier extension problem.