What is the hot spots conjecture in differential geometry?

A second Neumann eigenfunction u of the Laplacian approximates the temperature distribution on an insulated domain (in a Riemannian manifold) for large times. The 'hot spots' conjecture, due to Rauch, is the assertion that u does not assume its maximum value in the interior of the domain. In this talk, I will explain the physical intuition leading to the conjecture, and review old and new results.