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Event Type:

Applied Mathematics and Computation Seminar

Date/Time:

Thursday, January 26, 2017 - 15:00 to 16:00

Location:

KEAR 212

Guest Speaker:

Institution:

Simon Fraser University

Abstract:

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.

Abstract:

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).

New link

Link: https://global.gotomeeting.com/join/778150365