We are happy to announce the second talk in the SIAM PNW Section 2016-17 Seminar Series. See https://sites.google.com/site/siampnwsection/home for more informaiton about the seciton and how to contact the officers.
Who: Prof. Nilima Nigam, Mathematics, Simon Fraser University
When: Thursday, January 26, 3pm (PST)
Where: KEAR 212 (for local attendees)
Title: A MODIFICATION OF SCHIFFER’S CONJECTURE, AND A PROOF VIA FINITE ELEMENTS.
Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues
for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the
computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution
using validated finite element computations. Schiffer’s conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann
eigenfunction which is a constant on the boundary, then Ω must be a ball. This conjecture is open. A modification of Schiffer’s conjecture
is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which is
does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations for the regular
pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet
and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-
conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed
within interval arithmetic. This is joint work with Bartlomiej Siudeja and Ben Green, at U. Oregon.
Notes: The seminar can be viewed via the online link provided below. However, we strongly encourage local attendees to watch it from
Kear 212. (The first talk in the series in October was quite heavily subscribed and some groups from various universities were unable to watch
due to the limitations on the number of available connections).