I have been working with transmission eigenvalues for the past 2 years, before turning to sparse tomography. If you do not know what I am talking about, read this brief introduction first.
A breakthrough in such field was reached when the question whether transmission eigenvalues and non-scattering energies coincide or not (*) was solved. The answer is no, even though new questions arise. Since in the radially symmetric case (that is when the medium is a sphere and the index of refraction 
is a radial map) transmission eigenvalues and non-scattering energies coincide, the question under which regularity conditions such discrepancy happens, arise naturally.
In the recent years, several cases were investigated: a medium with cavities (**), inhomogeneous mediums (**), absorbing mediums (**), and surely some I did not read about. In such cases, both existence, infiniteness and discreteness of transmission eigenvalues has been proven (under certain assumptions, not always "convenient" (***)).
Up to date (and to my knowledge), it is possible to compute only real transmission eigenvalues, through the far field equation. Also, so far researchers have been able to exploit only the first transmission eigenvalue (or them all!) to estimate some medium's properties. A recent work shows that the first transmission eigenvalue is related to the size of the void the cavities case. In another recent paper, Cakoni, Colton, and Gintides prove that in the radially symmetric case, the knowledge of all transmission eigenvalues (possibly also the complex ones) helps recover the radial index of refraction. If such index of refraction is constant, the smallest (in norm) transmission eigenvalue is sufficient to recover it. Another study of the relationship between transmission eigenvalues and the material properties was published by Giorgi and Haddar, where knowledge of a (real) transmission eigenvalue can estimate a constant index of refraction.
Many questions are still open. In particular, I would be extremely interested in seeing some real-life applications of such results (for instance, in non-destructive testing) and to hear some hypothesis by physicists regarding what such eigenvalues may be. Can research in the medium with cavities case go on and exploit more transmission eigenvalues? Can researchers find a way to compute complex ones, and will they be useful?
(*) See this paper.
(**) Prof. Fioralba Cakoni's publications list is an excellent resource to find papers about such studies. For the inhomogeneous case, see for instance this paper. For the absorbing medium case, see this paper.
(***) In the absorbing case, in all major results one has to make some assumptions on the index of refraction. On the other hand, if such a priori knowledge is granted, a result helps locating the complex transmission eigenvalues.
The featured image of this post is taken from www.math.louisville.edu.
