$\Delta$: Equilateral triangle? Alt-J? Symmetric difference? Your preferred airline?

The heat equation and its component differential operator, the Laplacian, are classical (established) and classic (exemplary). We'll review why. Mostly, we'll examine the situation on graphs.

Eventually, we'll discuss the elliptic and parabolic Harnack inequalities, which give bounds on harmonic functions and solutions to the heat equation, respectively. In particular, we'll prove(ish) an elegant connection between these stochastic bounds and the geometry of the graph.