There continues to be great interest in the study of the heat equation on Riemannian manifolds. This may be due to the remarkable more recent work of Patodi . It may also be due in part to the asymptotic expansion of Minakshisundaram and Pleijel. The heat equation involves a parabolic partial differential equation that describes the distribution of heat in a given region over time. This equation has also appears in probability theory to describe random walks. The heat equation is also of importance in Riemannian geometry, topology and applied mathematics.
Bracken, Paul. "An Application of the Spectral Theorem To The Laplacian on a Riemannian Manifold." Proceedings of the Jangjeon Mathematical Society. Vol. 19. No. 3. 2016.
Proceedings of the Jangjeon Mathematical Society