A "hot spot" on a sufficiently regular domain is an interior extremum of the first nonconstant Neumann eigenfunction of the Laplace operator. The Hot Spots conjecture states that hot spots do not exist on convex planar domains.

Chris Judge and Sugata Mondal have settled the Hot Spots conjecture in the affirmative for all Euclidean triangles: Euclidean triangles have no hot spots, Annals of Mathematics 191-1 (2020) 167-211. (preprint)

This conjecture was the subject of Polymath 7.