Identifying a region by how its boundary vibrates: analytical and geometrical aspects


Michael Taylor

Univ. of North Carolina at Chapel Hill

Northeastern University

Thursday, February 24, 2005


Talk at 4:30 p.m. in 509 Lake Hall

Tea at 4:00 p.m. in 544 Nightingale Hall


Abstract:   A problem formulated by I.M. Gelfand in the 1950s is to reconstruct the metric tensor of a compact Riemannian manifold with boundary, from data on the spectrum of its Laplace operator, with the Neumann boundary condition, and the behavior at the boundary of the normalized eigenfunctions.

The first ingredient that goes into the resolution of such an ``inverse problem'' is a uniqueness theorem, but further work beyond establishing uniqueness is required. This arises because of the ``ill posedness'' associated with inverse problems. That is, various "large'' perturbations of the unknown region can yield small perturbations of the observed data. The key to stabilizing an ill-posed inverse problem is to have appropriate a priori knowledge of the unknown domain so that a search for the solution can be confined to a "compact'' family of possible domains. In this context, the suitable notion is that of Gromov compactness, and one key to stabilizing Gelfand's inverse problem involves establishing such compactness. This is done under fairly weak hypotheses on the geometry of the unknown domain, including bounds on its curvature (to be precise, its Ricci tensor) and on the curvature of its boundary. Estimates for solutions to a naturally occurring elliptic boundary value problem for the metric tensor play a central role.

The speaker will discuss some of these matters, which have been treated in joint work with M. Anderson, A. Katsuda, Y. Kurylev, and M. Lassas.

Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall.

There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from Maxim Braverman.