In Spring 2019, I will be teaching a graduate class at the University of Connecticut about heat kernels in Dirichlet spaces and their applications. This will be the occasion to prepare a set of lecture notes on topics which have been close of my research interests in the last few years. I plan to cover the following topics:

1. Dirichlet forms and heat semigroups: Dirichlet forms, spectral theory of self-adjoint operators, Riesz-Thorin interpolation, theory of heat semigroups, heat kernels.
2. Sobolev inequalities: Ultracontractivity, Varopoulos’ approach to Sobolev inequalities.
3. Dirichlet spaces with Gaussian heat kernels: Regular Dirichlet forms, carre du champ operators and notions of gradients. The example of Riemannian manifolds with non-negative Ricci curvature will be explored in details.
4. Dirichlet spaces with sub-Gaussian heat kernels: Energy measures, notions of gradients. The example of fractals will be explored in details.

Besides the lecture notes for this course, the following references will be a good complement.

A. Grigor’yan: Heat kernels and function theory on metric measure spaces