turkmath.org

There is a large amount of work on the interplay between the spectrum of the Laplacian and geometric data of a Riemannian manifold. An especially powerful tool in this direction is the use of heat kernels. We will review some results related to heat kernels and their estimates. The same tools have some interesting applications for singular Schrödinger operators with potentials supported on points, curves and surfaces being embedded in Riemmanian manifolds. We will briefly touch upon their implications on some nonlinear problems inspired from many-body physics. Note: This is an introductory talk aiming at introducing the subject and the relevant problems for the nonexpert.