A Short Introduction to the Analysis on Manifolds with Conical Singularities

These notes recall central elements of the cone calculus. The focus lies on conically degenerate differential operators and the Laplace-Beltrami operator with respect to a conically degenerate metric as a prototypical example. The topics include manifolds with conical singularities, the Mellin transform, cone Sobolev spaces, and the notion of ellipticity in terms of the invertibility of the principal pseudodifferential symbol and the principal Mellin symbol. The notes end with a sketch the full cone calculus. Lecture notes for a 3 hour course during the summer school 'Modern Problems in PDEs and Applications' at Ghent University, Belgium, August 23 - September 2, 2023.