Heat kernels appear as a fundamental object in many fields. Among their many faces, they are the fundamental solutions of the heat equations. The Gaussian functions formulate the standard heat kernels on $\mathbb{R}^n$, and are closely related to sharp two-sided estimates for heat kernels on a large class of manifolds. Other ingredients in such bounds are geometrical. In this topics in analysis course we will learn from the book Aspects of Sobolev-Type Inequalities how to study heat kernels from a functional analytic point of view (Harnack inequalities, Nash-Moser iteration, functional inequalities under the names Sobolev, Poincare, Nash, and so on). One highlight is the characterization of all complete Riemannian manifolds that satisfy the following sharp heat kernel bounds
$$\frac{c_1}{V(x,\sqrt{t})}\exp{\left(-C_1\frac{d(x,y)^2}{t}\right)}\leq p(t,x,y)\leq \frac{c_2}{V(x,\sqrt{t})}\exp{\left(-C_2\frac{d(x,y)^2}{t}\right)}$$
By Dirichlet form comparison techniques, we could also treat uniformly elliptic second order differential operators.