LMI Representations of Convex Semialgebraic Sets and Determinantal   Representations of Algebraic Hypersurfaces: Past, Present, and Future
  10 years ago or so Bill Helton introduced me to some mathematical problems arising from semidefinite programming. This paper is a partial account of what was and what is happening with one of these problems, including many open questions and some new results. 
Victor Vinnikov
math.OC