Foreword * Preface * On the construction of isospectral manifolds, Werner Ballman * Statistical stability and time-reversal imaging in random media, James G. Berryman, Liliana Borcea, George C. Papanicolaou, and Chrysoul Tsogka * A review of selected works on crack identification, Kurt Bryan and Michael S. Vogelius * Rigidity theorems in Riemannian geometry, Christopher B. Croke * The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, R. Gulliver, I. Lasiecka, W. Littman, and R. Triggiani * Energy measurements and equivalence of boundary data for inverse problems on non-compact manifolds, A. Katchalov, Y. Kurylev, and M. Lassas * Ray transform and some rigidity problems for Riemannian metrics, Vladimir Sharafutdinov * Unique continuation problems for partial differential equations, Daniel Tataru * Remarks on Fourier integral operators , Michael Taylor * The Cauchy data and the scattering relation, Gunther Uhlmann * Inverse resonance problem for Z2-symmetric analytic obstacles in the plane, Steve Zelditch * List of workshop participants

This IMA Volume in Mathematics and its Applications GEOMETRIC METHODS IN INVERSE PROBLEMS AND PDE CONTROL contains a selection of articles presented at 2001 IMA Summer Program with the same title. We would like to thank Christopher B. Croke (University of Penn sylva­ nia), Irena Lasiecka (University of Virginia), Gunther Uhlmann (University of Washington), and Michael S. Vogelius (Rutgers University) for their ex­ cellent work as organizers of the two-week summer workshop and for editing the volume. We also take this opportunity to thank the National Science Founda­ tion for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume contains a selected number of articles based on lectures delivered at the IMA 2001 Summer Program on "Geometric Methods in Inverse Problems and PDE Control. " The focus of this program was some common techniques used in the study of inverse coefficient problems and control problems for partial differential equations, with particular emphasis on their strong relation to fundamental problems of geometry. Inverse coef­ ficient problems for partial differential equations arise in many application areas, for instance in medical imaging, nondestructive testing, and geophys­ ical prospecting. Control problems involving partial differential equations may arise from the need to optimize a given performance criterion, e. g. , to dampen out undesirable vibrations of a structure , or more generally, to obtain a prescribed behaviour of the dynamics.