The hot spots conjecture concerns where an eigenfunction for the first positive Neumann eigenvalue of the Laplacian on a bounded domain attains its global maximum and minimum. The boundary condition is zero outward normal derivative. The relevant assertion for a smooth bounded simply connected planar domain places both extrema on the boundary.
Burdzy (2026) gave a conditional extension to bounded, simply connected planar domains without boundary smoothness, assuming compact Neumann resolvent and a simple first positive eigenvalue. The conclusion places the extrema on the boundary in the non-strict sense. The smooth-domain theorem excluding interior critical points is an external input, so the extension is not an independent proof of the conjecture.
Burdzy (2026) states that ChatGPT generated the argument and that he has not checked every detail. He also reports an independent, nearly identical ChatGPT-generated argument obtained by Rodrigo Bañuelos and asks that both researchers receive credit when the result is cited.
