Minisymposia Abstracts

Speaker: Lauri Oksanen
University of Helsinki
Title: Inverse problem for the wave equation with one measurement and the scattering relation
Abstract: We consider the wave equation for Laplace–Beltrami operator on (Rn, g), where the metric g is known outside an open and bounded set M. We define a source f(t,x) = ∑j=1 aj δxj(x) δ(t), where the points xj, j ∈ Z+, form a dense set on the smooth boundary ∂M. We show that when the weights aj are chosen appropriately, u|R × ∂M determines the scattering relation on ∂M, that is, it determines for all geodesics which pass through M the travel times together with the entering and exit points and directions. The wave u(t,x) contains the singularities produced by all the point sources, but when aj = λj for some λ > 1, we can trace back the point source that produced a given singularity in the data. This gives us the distance in (Rn, g) between a source point xj and an arbitrary point y ∈ ∂M. In particular, if (M,g) is a simple Riemannian manifold and g is conformally Euclidean in M, these distances are known to determine the metric g in M. In the case when (M,g) is non-simple we present a more detailed analysis of the wave fronts yielding the scattering relation on ∂M.

Speaker: Leo Tzou
University of Arizona
Title: The Aharonov–Bohm effect and the Calderón problem for connection Laplacians
Abstract: The Aharonov–Bohm effect is a quantum mechanical phenomenon where electrons passing through a region of vanishing magnetic field gets scattered due to topological effects. It turns out that this phenomenon is closely related to the cohomology of forms with integer coefficients. We study this relationship from the point of view of the Calderón problem and see that it can be captured in how Cauchy data of the connection Laplacian determines uniquely the holonomy representation of the connection.

Speaker: Pilar Herreros
University of Munster
Title: Lens rigidity for surfaces with trapped geodesics
Abstract: In a Riemannian surface with boundary we consider the lens data, i.e. the exit time, point and direction of each geodesic going in. We say that a surface M is lens rigid if any other surface with the same boundary and lens data is isometric to M. We will prove lens rigidity for a flat cylinder and some other surfaces with trapped geodesics, and discus some of the differences with the higher dimensional case.

Speaker: Plamen Stefanov
Purdue University
Title: The identification problem in SPECT: uniqueness, non-uniqueness and stability
Abstract: We study the problem of recovering both the attenuation coefficient and the source term in the attenuated X-ray transform. The analysis of the linearization reveals an unexpected phenomenon: there is a natural Hamiltonian flow that determines the microlocal properties of the linearization. Recovery of perturbations supported in compacts that are non-trapping with respect to that flow is possible, at least up to a finite dimensional subspace, and stable. Non-trapping compact sets might be problematic; and in the radial case, they are. Based on that, we present local uniqueness and Holder stability results for the non-linear problem; and also non-uniqueness examples.

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