Description:

The core analytical questions in inverse problems are those of uniqueness and stability, while in control problems the issues are of controllability and stabilizability. In fact, the issues of exact controllability and uniform stabilizability are intimately connected with observability/reconstruction estimates. Instead, the issue of approximate controllability and strong stabilizability are related to unique continuation properties. Thus, the type of estimates/inequalities one seeks in both fields are often closely related. These areas have recently seen the use of differential geometric methods to solve some outstanding questions in those fields. As an example, a combination of unique continuation results with the boundary control method has lead to the solution of the inverse problem of determining a metric of a Riemannian manifold (with boundary) from the dynamic Dirichlet-to-Neumann map associated with the wave equation. [The Dirichlet-to-Neumann map for the wave equation determines the boundary distance function (travel time along geodesics connecting points on the boundary of a Riemannian manifold). A natural question to ask if one can determine the metric from this data alone ; this question is at the center of the boundary rigidity problem studied in Riemannian geometry.] An example from control is the establishment of sufficient conditions for controllability and stabilization, expressed in terms of geometry of the domain. There are also interesting inverse problems for semilinear elliptic partial differential equations where the uniqueness issue is intimately tied with the geometry of the domain. The boundary control method used in inverse problems as well as some deep results in exact controllability are based on the study of propagation of singularities for partial differential equations. The propagation in turn depends on geometric properties of the domain. While there has been some remarkable success, many open problems still remain.

While the importance of differential geometry in the control of PDE and inverse problems has been established, there has never been a concerted effort to bring geometers to interact with experts in these other areas. The workshop goals are to bring together geometers with researchers in inverse problems and control of PDE to facilitate exchange of ideas and encourage collaboration; to make tools of differential geometry known to those working in inverse problems and control, and to open new areas of research in geometry. The workshop will for instance explore the use of inverse problem- and control methods to rigidity problems in geometry.

We are planning a 2-week workshop in which Week 1 will emphasize the Geometric and PDE methods and Week 2 will focus on applications of these techniques to inverse and control problems. Both weeks will begin with overview talks designed to set the stage for the workshop. The presence of several talks of a survey nature will make this workshop an ideal venue for young researcher wanting to get involved in these fields. The format of the workshop will consist in invited talks (both of a survey and of a more specialized nature), two sessions of contributed talks and a panel discussion. The Proceedings of the summer program will be published as an IMA Volume.

Topics that will be covered in this program include:

1. Boundary Rigidity
2. Conjugacy of Geodesic Flows
3. Integral Geometry
4. Wave propagational inverse problems (Maxwell equations, elasticity, anisotropy)
5. Propagation of sigularities (Low regularity of coefficient and domain)
6. Boundary Control methods
7. Observability/controllability inequalities by differential geometric methods.
8. Unique continuation
9. Elliptic inverse problems with rough coefficients

LIST OF CONFIRMED PARTICIPANTS

as of 7/17/2001

schedule.pdf    schedule.ps   schedule.tex

Material from Talks   Abstracts

2001-2002 IMA Thematic Year on Mathematics in the Geosciences