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SUMMARY:The hot-cold distance  - Lawford Hatcher (Indiana University)
DTSTART:20260204T114500Z
DTEND:20260204T121500Z
UID:TALK239809@talks.cam.ac.uk
DESCRIPTION:Given a solution to the heat equation on a Euclidean domain\, 
 Riemannian manifold\, or discrete graph\, it is natural to ask where the h
 ottest and coldest points are located&nbsp\;over large time scales. Rauch'
 s hot spots conjecture states that\, in the Euclidean setting with Neumann
  boundary conditions\, these points should all tend toward the boundary of
  the domain as time tends to infinity. Motivated by this conjecture\, we s
 tudy the long-time distance between the sets of hottest and coldest points
 \, i.e. the&nbsp\;hot-cold distance\, in several geometric settings. In se
 veral cases\, these distances agree with the geodesic diameter of the obje
 ct in question\, leading to an affirmative resolution of Rauch's conjectur
 e. Counterintuitively\, however\, we also construct some situations in whi
 ch the hot-cold distance is strikingly small with respect to the diameter.
 &nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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