We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t − \textrm{div}(A(x)\nabla_x))^s + q(x, t)$ for $s \in (0, 1)$ and show the unique recovery of q from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.

[1] A. Banerjee, S. Senapati. The Calderón problem for space-time fractional parabolic operators with variable coefficients, arXiv: 2205.12509, 2022.

Rich tomography refers generally to problems in which the image has more than just a scalar per voxel, and often the measurement is more than one scalar per source and detector pair. In this talk I will give a number of examples of real problems where the data collection systems exist and I will review their mathematical formulation, what is known and what is yet to be determined about the reconstruction problem (as well as sufficiency of data, range characterization and stability,

Small Angle X-ray Scattering (SAXS) tomography is an example where a diffraction pattern is measured for each ray, and the inverse problem is to determine the `reciprocal space map' a function of three variables at each point.

Several techniques involve the imaging of strain in a crystalline or polycrystalline material. I will show the formulation of the problem where the measurement uses neutrons and electrons. In polycrystalline materials the texture, or distribution of crystal orientations over a given scale, is often an `nuisance variable' but can be of interest in its own right. I will suggest some possible mathematical challenges.

Finally the imaging of magnetic fields is a vector tomography problem and I will contrast the method using polarimetric tomography with neutrons and also a method using electron tomography.

Medical imaging reconstruction is typically regularized with methods that lead to stability in an $L^2$ sense. However, we argue that the $L^2$ norm is not always a good metric with which to assess the quality of image reconstruction. For instance, two objects might be close in $L^2$, while one of them carries a localized, clearly visible artifact, not present in the other. While this issue has been raised for deep learning based algorithms, we show as an example, that the classical regularization method of compressed sensing for MRI is also not protected from such possible instabilities. This is joint work with Giovanni S. Alberti (University of Genoa) and Tandri Gauksson (ETH Zurich).

The inversion of the ray transform serves as an important mathematical tool for investigating object properties from external measurements with extensive applications spanning medical imaging to geophysics. However, the inversion of the ray transform on symmetric tensor fields is constrained by the presence of an infinite dimensional null space. One natural question is whether we can utilize supplementary data in the form of higher order moments of the ray transform for the explicit recovery of the entire tensor field. In this talk, we will focus our attention on normal operators associated to momentum ray transforms (the composition of the transform with its formal $L^2$ adjoint), and introduce an approach for the explicit reconstruction of entire symmetric $m$ tensor field from this data.