Series: Mathematics Colloquium

Date: Thursday, April 15, 2004

Time: 4:00 - 5:00 PM

Place: 102 McAllister Building

Hosts: Anna Mazzucato, Victor Nistor

Refreshments: 3:15 - 4:00 PM, in 212 McAllister

Speaker: Michael E. Taylor, University of North Carolina

Title: 

  Identifying a Region by How its Boundary Vibrates: Analytical and
  Geometrical Aspects

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
  occuring 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.