Semidefinite programming provides efficient algorithms for minimizing an affine linear function subject to a linear matrix inequality constraint. This leads to several mathematical questions: which sets can be described by a linear matrix inequality? which sets are projections thereof? These questions lie on the border between algebraic geometry, real algebraic geometry and convexity; they lead to a class of polynomials, so called real zero polynomials (called hyperbolic in the homogeneous setting), that are a multivariable generalizations of polynomials with only real zeroes. The problem of which sets can be described by a linear matrix inequality has been solved only in dimension 2, and even there it is far from being completely understood. It is one purpose of the proposed research to pursue this problem further in dimension 2 and to tackle the higher-dimensional situation. There are relations with the recently proved BMV conjecture from Statistical Quantum Mechanics, and applications to one of the most interesting open problems in the theory of operator algebras — Connes’ embedding problem. On the other hand, one can consider linear matrix inequalities in the free noncommutative setting, substituting tuples of real symmetric matrices of all sizes for the noncommuting variables. This is natural for many if not most optimization problems appearing in systems and control since these problems are dimension-independent, i.e., the natural variables are matrices, and the problems involve rational expressions in these matrix variables which have therefore the same form independent of matrix sizes. It also conforms to a general paradigm of passing from the commutative setting to the free noncommutative setting that emerged over the course of the last two decades in such diverse areas as the theory of operator spaces and free probability. We plan to study the free noncommutative situation at length. As it has more structure than the commutative situation, leading possibly to stronger results, we also hope to be eventually able to tackle some commutative problems via a noncommutative lifting.