BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nilima Nigam (Simon Fraser University) DTSTART:20200728T140000Z DTEND:20200728T150000Z DTSTAMP:20260423T024545Z UID:CRM-CAMP/5 DESCRIPTION:Title: A modification of Schiffer's conjecture\, and a proof via finite elements \nby Nilima Nigam (Simon Fraser University) as part of CRM CAMP (Compu ter-Assisted Mathematical Proofs) in Nonlinear Analysis\n\n\nAbstract\nApp roximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirich let Laplacian in certain domains. If these are to be used as part of a pro of\, care must be taken to ensure each step of the computation is validate d and verifiable. In this talk we present a long-standing conjecture in sp ectral geometry\, and its resolution using validated finite element comput ations. Schiffer’s conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann eigenfunction which is a constant on the bound ary\, then Ω must be a ball. This conjecture is open. A modification of S chiffer’s conjecture is: for regular polygons of at least 5 sides\, we c an demonstrate the existence of a Neumann eigenfunction which does not cha nge sign on the boundary. In this talk\, we provide a recent proof using f inite element calculations for the regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains o f the triangle with mixed Dirichlet and Neumann boundary conditions. We us e a learning algorithm to find and optimize this sequence of subdomains\, and use non-conforming linear FEM to compute validated lower bounds for th e lowest eigenvalue in each of these domains. The linear algebra is perfor med within interval arithmetic. This is joint work with Bartlomiej Siudeja and Ben Green at University of Oregon.\n LOCATION:https://researchseminars.org/talk/CRM-CAMP/5/ END:VEVENT END:VCALENDAR