Working on the Seminar on Fourier Analysis at Universidad de los Andes. We started showing the existence of the Haar measure for abelian locally compact groups and we proved Plancherel’s Theorem. By now, we want to study some Fourier analysis on non abelian locally compact topological groups so we started working on representations. I started talking on Matrix Groups using Deitmar’s text on harmonic analysis, but now I am asked to do a bit more hard work, so I was asked to reach the Peter-Weyl Theorem. Really, I do not feel myself so confident on algebraic topology but one of the most valuable issues I am learning from my advisor is dealing with a lot of topics not neccesarily the ones we are quite familiar. With that scope we are going to talk on Segal’s Lectures on Lie Groups and Lie Algebras in order to find Fourier series based on representations.