In 1822, the French mathematician and physicist Joseph Fourier published his seminal treatise Théorie analytique de la chaleur in which he introduced expansions of periodic functions and solved the heat equation on R. Since then,
Fourier series and Fourier transforms have been used successfully and extensively in many areas of fundamental and applied sciences. Harmonic analysis aims to generalize Fourier analysis on topological spaces on which a group acts. Group actions lay at the confluences of Mathematics and Physics, their study is the purpose of representation theory of Lie groups. I will explain the ins and the outs of Lie theory, and sketch a few of its many tremendous applications. In particular, I will describe how representation theory relates objects of different nature such as orbits, characters and Dirac operators. These recent results are based on joint work with Pavle Pandži´c, David Vogan and Roger Zierau.