In 1974 Jeff Rauch conjectured that for the generic initial
temperature distribution on an insulated object, the maximum of the
temperature distribution tends towards the insulated boundary.
This conjecture is essentially equivalent to the assertion
that the second eigenfunction of the Neumann Laplacian has no interior
global extrema. This mathematical conjecture is still wide open. Sugata
Mondal and I have shown that the conjecture holds for triangular objects
thus resolving Polymath 7. In this talk, I will motivate and discuss the
conjecture and describe some of the methods that have been used to attack it.