Recent progress in the anisotropic electrical impedance problem.
Inverse boundary value problems for partial differential equations.
Recent progress on the boundary rigidity problem.
Generalized Backscattering and the Lax-Phillips Transform
2000 Mathematics Subject Classification: 35P25, 35R30, 58J50. Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle w = Sq in terms of the incoming angle with S orthogonal and Id-S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show...
Inverse Problems: Visibility and Invisibility
This survey article expands on the lectures given at Biarritz in June, 2012, on “Inverse Problems: Visibility and Invisibility". The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography (EIT) and also Calderón’s problem since the famous analyst proposed it in the mathematical literature []. The second is on travel time tomography. The question...
30 Years of Calderón’s Problem
In this article we survey some of the most important developments since the 1980 paper of A.P. Calderón in which he proposed the problem of determining the conductivity of a medium by making voltage and current measurements at the boundary.
Global uniqueness for an inverse boundary problem arising in elasticity.
Electrical impedance tomography in nonlinear media
Composition of some singular Fourier integral operators and estimates for restricted -ray transforms
We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations . These canonical relations, which arise naturally in integral geometry, are such that : is a Whitney fold and : is a blow-down mapping. If , , then a class of pseudodifferential operators with singular symbols. From this follows boundedness of with a loss of 1/4 derivative.
On determining a riemannian manifold from the Dirichlet-to-Neumann map
Complex geometrical optics solutions for Lipschitz conductivities.
Mathematics of Invisibility
We will describe recent some of the recent theoretical progress on making objects invisible to electromagnetic waves based on singular transformations.
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