In these five lectures on the spectral, and more particularly the inverse spectral theory of the Dirichlet problem in planar domains I wish to show how the investigation of a concrete mathematical question can draw on quite extensive areas of mathematics. Thus I hope that these talks will help to bind together various parts of the material you will have seen, or be seeing, elsewhere in this Workshop.
Each of the lectures will centre on one or two explicit 'results' which I will try to put into context. Then, as time permits, I will outline some of the proofs and discuss extensions and refinements of the various questions. For the most part I will refer elsewhere for details.