BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Neumann heat kernels and spectral zeta: mirror coupling on spheric
 al caps and hyperbolic disks - Richard Laugesen (University of Illinois at
  Urbana-Champaign)
DTSTART:20260323T153000Z
DTEND:20260323T163000Z
UID:TALK241123@talks.cam.ac.uk
DESCRIPTION:The shape of the Neumann heat kernel p(t\,x\,y) on the unit ba
 ll is more easily visualized than analyzed. (Dirichlet conditions are bett
 er understood.) And so we examine a conjecture restricted to the spatial d
 iagonal: for fixed t\, is p(t\,x\,x) increasing as a function of r=|x|? Th
 at is\, for reflected Brownian motion in the ball\, does the probability o
 f returning to one&rsquo\;s initial point get higher when that initial poi
 nt is located closer to the boundary?\nLaugesen and Morpurgo raised the co
 njecture 30 years ago and then Pascu and Gageonea proved it 15 years ago f
 or Euclidean space\, by an ingenious application of probabilistic mirror c
 oupling (a technique used by Atar and Burdzy for hot spots). We extend the
  theorem to geodesic balls in hyperbolic space and the hemisphere.\nAs a c
 orollary\, we deduce monotonicity wrt curvature of the Neumann spectral ze
 ta function on a geodesic disk of fixed area &mdash\; a result known previ
 ously only in the large-parameter limit\, by Bandle&rsquo\;s theorem on th
 e first nonzero eigenvalue.\n[Joint with Jing Wang\, Purdue University.]
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR