## Analysis and Geometric Analysis Seminar

The heat kernel is an object whose importance reaches across several fields of mathematics, in particular probability theory and the study of PDEs. In some spaces, we understand the behavior of the heat kernel very well; in other spaces, its behavior is more elusive. In many of the spaces where heat kernel estimates are known, they are of the form of two-sided Gaussian bounds. In this talk, we will explore a few examples where we have two-sided heat kernel estimates that are NOT of this form. In particular, we will look at the heat kernel on certain Riemannian manifolds with boundary with "nice" ends; we will take Dirichlet condition along some portion of the boundary and Neumann condition along the rest. This is joint work with Laurent Saloff-Coste.