BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Dan Mangoubi (The Hebrew University of Jerusalem) DTSTART:20210113T121000Z DTEND:20210113T133000Z DTSTAMP:20260423T004911Z UID:GDS/23 DESCRIPTION:Title: A L ocal version of Courant's Nodal domain Theorem\nby Dan Mangoubi (The H ebrew University of Jerusalem) as part of Geometry and Dynamics seminar\n\ n\nAbstract\nLet u_k be an eigenfunction of a vibrating string (with fixed ends) \ncorresponding to the k-th eigenvalue. It is not difficult to show that \nthe number of zeros of u_k is exactly k+1. Equivalently\, the numb er of \nconnected components of the complement of $u_k=0$ is $k$.\n\nIn 19 23 Courant found that in higher dimensions (considering eigenfunctions \no f the Laplacian on a closed Riemannian manifold M) the number of connected \ncomponents of the open set $M\\setminus {u_k=0}$ is at most $k$.\n\nIn 1988 Donnelly and Fefferman gave a bound on the number of connected \ncomp onents of $B\\setminus {u_k=0}$\, where $B$ is a ball in $M$. However\, \n their estimate was not sharp (even for spherical harmonics).\n\nWe describ e the ideas which give the sharp bound on the number of connected \ncompon ents in a ball. The talk is based on a joint work with S. Chanillo\, \nA. Logunov and E. Malinnikova\, with a contribution due to F. Nazarov.\n LOCATION:https://researchseminars.org/talk/GDS/23/ END:VEVENT END:VCALENDAR