\noindent In my colloquium talk, I will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing. I will begin by describing our solution to the problem of finding a canonical orthonormal basis of eigenfunctions of the discrete Fourier transform (DFT). Then I will explain how to generalize the construction to obtain a larger collection of functions that we call "The oscillator dictionary". Functions in the oscillator dictionary admit many interesting properties, in particular, I will explain two of these properties which arise in the context of problems of current interest in communication theory. This is joint work with Shamgar Gurevich (Berkeley) and Nir Sochen (Tel Aviv). \\ \noindent There is a sequel to my colloquium talk, which will be slightly more specialized and will take place during the algebraic geometry seminar. Here, my main objective is to introduce the geometric Weil representation which is an algebra-geometric ($ \ell $-adic Weil sheaf) counterpart of the Weil representation. Then, I will explain how the geometric Weil representation is used to prove to main result stated in my colloquium talk. In the course, I will explain Grothendieck's geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects.