Introduction to microlocal analysis using groupoids

Taking the Fourier transform of the derivative is the same as multiplying the Fourier transform by the frequency variable. This easily demonstrable fact (via integration by parts) is of great help in solving partial differential equations (PDEs) with constant coefficients. Following the idea of Van Erp and Yuncken, I will use a technique to freeze the coefficients (this is the microlocal part), and I will then show how to extend this method to linear PDEs with non-constant coefficients. This approach will lead us to discover the world of groupoids, which, briefly, are groups with several identity elements. In particular, the peculiar structure of Connes’ tangent groupoid will allow us to solve a certain class of PDEs (namely elliptic ones) using pseudo-differential operators, which we will define.