Titles and abstracts for the conference on ``Scattering theory and singular spaces''

• Hiroshi Isozaki: ``Hyperbolic geometry and inverse boundary value problems''

  
  Abstract: One can use the hyperbolic space structure as a tool for solving the inverse boundary value problem for Schroedinger operators in the Euclidean space. After briefly explaining this idea, I am going to talk about the following two subjects, both of which have a deep connection to the practical problem of electrical impedance tomography in, e.g., medical science. (1) The inverse boundary value problem in the horosphere. (This is a justification of the well-known Barber-Brown algorithm in EIT). (2) The inclusion detection problem. (This is an application of hyperbolic geometry to numerical computation).
  
  

• Lizhen Ji: ``Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups''





• Rafe Mazzeo: ``Scattering and locally symmetric spaces''

  
  Abstract: I will describe continuing joint work with Andras Vasy concerning the resolvent of the Laplacian on globally and locally symmetric spaces.
  
  

• Richard Melrose: ``Scattering, conformal metrics and corners''

  
  Abstract: Starting from the traditional context of Euclidean scattering I will describe successive generalizations of the setting, especially the geometry, to
  • Scattering metrics on compact manifolds with boundary
  • Big potentials
  • Conformal scattering metrics
  • Totally scattering metrics on manifolds with corners
  and, to the extent that it is known, give descriptions of the spectrum of the Laplacian and the scattering matrices.
  
  

• Werner Muller: ``Harmonic analysis on locally symmetric spaces and scattering theory''

  
  Abstract: I will mainly focus on the case of locally symmetric spaces of noncompact type. There are canonical compactifications of these spaces, which are singular, and the analysis of invariant differential operators on these spaces is closely related to the structure of the compactification.
  
  

• Leslie Saper: ``L^2-hamonic forms on locally symmetric spaces''





• Michael Taylor: ``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.
  
  

• Gunther Uhlmann: `` Boundary Rigidity and the Dirichlet-to-Neumann Map''

  
  Abstract: In this lecture we will discuss some recent results on the boundary rigidity problem. The problem consists in determining a Riemannian metric on a compact manifold with boundary by knowing the distance function between boundary points. This problem arises also in geophysics in an attempt to determine the structure of the interior of the Earth by measuring the travel times of seismic waves going through the Earth. We will also discuss a connection between boundary rigidity and the inverse boundary problem of determining a Riemannian metric from the Dirichlet-to-Neumann map associated to the Laplace-Beltrami operator.
  
  

• Zhihong Jeff Xia: ``Interesting solutions in the classical n-body problems''