Central in harmonic analysis is the study of extension operators: Fourier transforms of functions defined on curved surfaces. Such objects are sums of waves, evolving inside tubes in space-time. The restriction conjecture (Stein, 1960s) claims that these waves in principle interfere with each other destructively, forcing the extension operators to have small $p$-norms (for low $p$). The restriction conjecture is intimately connected with geometric measure theory, PDE, number theory and incidence geometry. In 1973, Hörmander asked whether more general oscillatory integral operators have similarly small $p$-norms. Hörmander's operators can be viewed again as Fourier transforms of functions defined on curved surfaces, with the difference that the surfaces vary as we change the point where we calculate the Fourier transform. In this paper and in subsequent work, we settled Hörmander's question according to the curvature signature of the involved varied surfaces, exploiting the algebraic nature that underlies the problem. In the talk we will discuss Hörmander's question, applications and aspects of the solution.