TITLE: Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?
QUESTION [5 upvotes]: I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational difficulty of reconstructing matrices from prescribed spectral data (e.g., prescribed spectrum).
Are these problems sort of dead right now, like hopeless?  Or perhaps IEPs is actively being researched today?  
Most of the literature that I have found on IEPs with the most partial progress seems to be very old - late 70s to mid 80s.  (Newer papers seem to just reiterate old results.) 
For the nonnegative inverse eigenvalue problem, the problem is open for $n\ge 5$.
Thanks in advance for any advice or suggestions.

REPLY [3 votes]: Although I work exclusively on the nonnegative inverse eigenvalue problem (NIEP), I can assure you that IEPs are far from dead, both on the theoretical side and the applied side (there has been a flurry of activity in recent years, along with a special session at the 2017 Meeting of the International Linear Algebra Society).
I would like to point out that in the paper by Charlie Johnson and I [MR3452738; *Linear Algebra Appl.* 493 (2016), 281–300], connections between the real NIEP to Bose-Mesner algebras and association schemes were established.
For more details on current developments on the NIEP, see the recent survey paper (which will appear later this year in Operator Theory: Advances and Applications by Johnson et al (it is worth noting that this survey already dated needs updating).