Talk by Stefan Steinerberger

The Hot Spots conjecture says that if you have a nice domain in Euclidean space and run the heat equation for a long time, then the hottest and the coldest spots are on the boundary. One can also phrase it as a question for the first nontrivial Laplacian eigenfunction. It is widely conjectured to be true for all convex domains and a really tricky problem. One can ask a similar question on Graphs and that's where things get really interesting: PDE difficulties are replaced by Linear Algebra and it is possible to do things. One would also hope that by understanding the Graph problem well enough, one can make some progress on the continuous problem. I will discuss many open problems.