Parabolic Harnack inequalities on metric measure spaces

Parabolic PDE's describe time-dependent phenomena such as heat conduction or particle diffusion. Parabolic Harnack inequalities relate these phenomena to the geometry of the space. The relevance of the parabolic Harnack inequality in probability is due to its equivalence - in the case of the heat equation - with (sub-)Gaussian bounds for the transition density of Brownian motion. It has also been used to prove estimates for heat kernels with Neumann or Dirichlet boundary condition. In this talk I will describe recent work on Harnack inequalities for quasilinear parabolic equations on metric measure spaces.