the talk pertains to linear time invariant systems of ode. the standard engineering approach for the last 20 years has been to convert these problems to matrix inequalities. these consist of polynomials in matrices and one must find (by numerical computation) choices of matrices which make the polynomial positive definite. the trouble is polynomials which are easily produced from your system problem, by turn the crank methods, are horrible. the holy grail is to somehow convert your system of polynomial matrix inequalities to linear matrix inequalities, lmi. there are many lmi solvers and lmis are certainly convex. the talk concerns the converse: which convex problems give lmis? the focus is on this rather than the systems engineering context. next comes, the issue of converting a problem to a convex one. this is daunting and requires some new kind of noncommutative (free) real algebraic geometry, and will be a long time in coming, but there has been serious progress in the last 10 years.