Let D₀ and be two concentric plane, open disks. It is well known that the restriction to D₀ of a function u supported in D cannot be determined from the restriction of its Radon transform Ru(L) to the set of lines L that intersect D₀. It has been conjectured that the same is true if D₀ and are arbitrary open, bounded, convex subsets of the plane. A theorem related to this conjecture is presented.

Generalizing Lemma 28 from Newton's "Principia" [25], Arnold [10] asked for a complete characterization of algebraically integrable domains. In this chapter we describe the current state of Arnold's problems. We also consider closely related problems involving the Radon transform of indicator functions.

In two recent papers, Boman (J Geom Anal 31:2726–2741, 2020, https://doi.org/10.1007/s12220-020-00372-8, J Ill-posed Inverse Probl 2021, https://doi.org/10.1515/jiip-2020-0139), we proved that the Radon transform of a compactly supported distribution can be supported in the set of supporting planes to a bounded, convex domain D⊂Rⁿ only if the boundary of D is an ellipsoid. Using closely related methods we study here the relationship between the analytic wave front set for the characteristic function, _χD, of a domain D⊂Rⁿ and singularities of the boundary ∂D of the domain. For instance we prove that the boundary surface must be real analytic in a neighborhood of a point z∈∂D∈C¹, if the analytic wave front set of _χD at z contains no other elements than the conormals to ∂D at z.

This article gives a brief overview of the research in microlocal analysis, tomography, and integral geometry of Professor Eric Todd Quinto, Robinson Professor of Mathematics at Tufts University, along with the collaborators and colleagues who influenced his work. 

If the Radon transform of a compactly supported distribution f not equal 0 in Rn is supported on the set of tangent planes to the boundary partial derivative D of a bounded convex domain D, then partial derivative D must be an ellipsoid. As a corollary of this result we get a new proof of a recent theorem of Koldobsky, Merkurjev, and Yaskin, which settled a special case of a conjecture of Arnold that was motivated by a famous lemma of Newton.

If the Radon transform of a compactly supported distribution f not equal 0 in R-n is supported on the set of tangent planes to the boundary partial derivative D of a bounded convex domain D, then partial derivative D must be an ellipsoid. The special case of this result when the domain D is symmetric was treated in [J. Boman, A hypersurface containing the support of a Radon transform must be an ellipsoid. I: The symmetric case, J. Geom. Anal. (2020), DOI 10.1007/s12220-020-00372-8]. Here we treat the general case.

In this paper we study Schrodinger operators with absolutely integrable potentials on metric graphs. Uniform bounds-i.e. depending only on the graph and the potential-on the difference between the eigenvalues of the Laplace and Schrodinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schrodinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schrodinger operator to the Euler characteristic of the underlying metric graph.

We consider the inverse problem for the 2-dimensional local Radon transform R[f], where f is supported in y >= x(2) and R[f](xi, eta) = integral f (x, xi x + eta) dx is defined near (xi, eta) = (0, 0). We give logarithmic estimates of f in terms of R[f] for functions f that satisfy an a priori bound. For a certain class of smooth, positive weight functions m we give similar estimates for the weighted Radon transform R-m vertical bar f](xi, eta) = integral f (x, xi x + eta) m(xi, eta, x) dx.

We study the X-ray transform I of symmetric tensor fields on a smooth convex bounded domain Omega C R-n. The main result is the stability estimate parallel to(s)f parallel to(L2) <= C parallel to If parallel to(H1/2), where (s)f is the solenoidal part of the tensor field f. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.

We give a simple proof of an important special case of the famous theorem of Jósef Siciak on separate analyticity