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In the heart of harmonic analysis lies the restriction problem: the study of Fourier transforms of functions that are defined on curved surfaces. The problem came to life in the late 60s, when Stein observed that such Fourier transforms have better behaviour than if the surfaces were flat. Soon after, Hörmander conjectured that oscillatory integral operators with more general phase functions should also demonstrate similar agreeable behaviour. Surprisingly, 20 years later Bourgain disproved Hörmander's conjecture. However, under additional assumptions on the phase function one can expect better estimates than the sharp ones by Bourgain. In this talk, we present such better estimates in the sharp range, under the assumption that the phase function is positive definite. This is joint work with Larry Guth and Jonathan Hickman, and builds on recent work of Guth that improved on the restriction problem via the polynomial method