Mini course: The Uncertainty Principle in Fourier Analysis, by Torunn Stavland Jensen

Abstract: The principle roughly says that it is not possible for both a function f and its Fourier transform \hat{f} to be localized on small sets. We will mainly follow chapter 10 in the book "Classical and Multilinear Harmonic Analysis" by Muscalu and Schlag. We will start with looking at the Heisenberg Uncertainty principle from physics and prove the mathematical version of this. We will also look at other versions of the uncertainty principle, such as the  Payley Wiener theorem, Amrein Berthier Theorem, Logvinenko-Serada theorem, and how we can apply this principle to proving existence of a solution to a PDE problem.