Heat Kernel Asymptotics, Weyl's Law and Isospectrality

Following Chapter 6 of Levitin–Mangoubi–Polterovich, we will discuss the heat kernel and the asymptotics of its trace. We will leverage this theory to prove Weyl's law on a Riemannian manifold, albeit with a less-than-optimal remainder estimate. Time permitting, we will discuss isospectral manifolds.