Dimitri Cobb

The topic of this talk will be the properties of the eigenvalues of the Laplace operator on a domain. More precisely, we will explore how the geometry of the said domain affetcs the eigenvalues. As we will see, this question is linked to heat transfer problems of various types: for instance, how long will a piece of hot metal take to cool down? how does this cooling time depend on the geometry of the object? What if we add an insulator around the object?

We will mainly approach these questions with an intuitive and geometric point of view, providing as many examples as possible in order to illustrate some of the important concepts of the theory (Rayleigh quotients, test functions, elementary surgery, etc.). This will allow us to discuss issues such as sensitivity of the spectrum to perturbations of the domain or the importance of boundary conditions.