The Calderón problem for the fractional Schrödinger equation, introduced by T. Ghosh, M. Salo and G. Uhlmann, satisfies global uniqueness with only one single measurement. This result exploits the antilocality property of the fractional Laplacian, that is, if a function and its fractional Laplacian vanish in a subset, then the function is zero everywhere.

Nonlocal operators which only see the functions in some directions and not on the whole space cannot satisfy an analogous antilocality property. In theses cases, only directional antilocality conditions may be expected.

In this talk, we will consider antilocality in cones, introduced by Y. Ishikawa in the 80s, and its possible implications in the corresponding Calderón problem. In particular, we will see that uniqueness for the associated Calderón problem holds even with a singe measurement, but new geometric conditions are required.

This is a joint work with G. Covi and A. Rüland.

The study of nonlinear equations arises in many physical phenomena and is an active direction in the field of inverse problems. In this talk, we will discuss inverse problems for the fractional magnetic Schrodinger equation with nonlinear potential and address the crucial techniques that are applied to reconstruct the unknown coefficient from measurements.

In this talk I will consider two kinds of anisotropic viscoelastic systems. One is an anisotropic viscoelastic systems which is described as an integro-differential system (ID system) and the other is the so-called extended Maxwell model (EM system) which is schematically described using springs and dashpots. There is a relation between them but they are different systems. I will discuss about their relation and the large time behavior of their solutions. Further, for the EM system, I will discuss about a generation of semigroup and the limiting amplitude principle. The method of proof for the ID system is an energy estimate, and that for the EM system is mainly based on several resolvent estimates. The limiting amplitude principle could be useful for setting up the inverse problem when the measurements may use a sequence of several different time harmonic inputs such as the magnetic resonance elastography.

This is a joint work with Maarten de Hoop, Ching-Lung Lin for the EM system and also including Masato Kimura for the EM system.

We shall discuss some recent progress on the fractional anisotropic Calderon problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we show that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderon problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher. This is joint work with Ali Feizmohammadi, Tuhin Ghosh, and Gunther Uhlmann.