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 (R^{n}, g), where the metric g is known outside an open and bounded set M. We define a source f(t,x) = ∑^{∞}_{j=1} a_{j} δ_{xj}(x) δ(t), where the points x_{j}, j ∈ Z_{+}, form a dense set on the smooth boundary ∂M. We show that when the weights a_{j} 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 a_{j} = λ^{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 (R^{n}, g) between a source point x_{j} 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 nonsimple 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, nonuniqueness and stability 
Abstract:  We study the problem of recovering both the attenuation coefficient
and the source term in the attenuated Xray 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 nontrapping with respect to that flow is possible, at least
up to a finite dimensional subspace, and stable. Nontrapping 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 nonlinear problem; and also
nonuniqueness examples.

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