Sugata Mondal, Indiana University

The hot spots conjecture was made by J. Rauch at a conference in 1974. One of the (many) versions of the conjecture says the following. Let D be a domain in a Euclidean space with piece-wise smooth boundary. Then a second Neumann eigenfunction u for D can not attain its global maximum at an interior point of D. The conjecture is known to be false for domains with holes. Positive results are known in many situations due works of K. Burdzy and his collaborators, D. Jerison-N. Nadirashvilli and many others. This talk will be focused on the hot spot conjecture for Euclidean triangles. Obtuse triangles known to satisfy the conjecture, due to works of Burdzy-Banuelos. A class of acute triangles also known to satisfy the conjecture, due to works of Miyamoto and Siudeja. In this talk I will try to explain a proof of the conjecture for all Euclidean triangles. This a joint work with Chris Judge.