The Bochner-Riesz conjecture and the Fourier restriction conjecture concern fundamental properties of the Fourier transform and have both been widely open for decades. They can both be efficiently studied by the theory of oscillatory integrals and are known to share some key features. Building on prior research on a question posed by Hörmander, in a recent joint work with Shaoming Guo and Hong Wang we formulated a new conjecture that unifies the two conjectures and found some evidence. I will talk about the history and mention some interesting components in the proof which have nice connections to algebra and geometry.