A new approach to the Fourier extension problem for the paraboloid

The goal of the talk is to motivate and introduce the main questions regarding the Fourier restriction operator for the paraboloid, as well as to present some recent developments that we obtained in joint work with Camil Muscalu. We will describe some connections between the study of this operator and questions in dispersive PDEs and geometric measure theory. Once we set up the background and comment on the current tools employed to treat the main conjectures in this area, we will change our focus to a different framework that unifies the treatment of the linear and multilinear theory previously presented. In a nutshell, this framework allows to show that all (linear and multilinear) Fourier extension conjectures for the paraboloid are true up to the endpoint as long as one of the functions involved is of tensor-type.