Heat equation

The heat equation is the canonical parabolic partial differential equation describing the diffusion of heat — or any conserved quantity — through a medium over time. Its geometric generalization, the Ricci flow, replaces the scalar temperature field with a Riemannian metric and the Laplacian with the Ricci tensor, producing a nonlinear evolution equation that has become the primary tool for classifying three-dimensional manifolds. The heat equation's linearity makes it analytically tractable; Ricci flow's nonlinearity makes it topologically powerful.