  One of the main applications of semidefinite programming lies in linear
systems and control theory. Many problems in this subject, certainly the
textbook classics, have matrices as variables, and the formulas naturally
contain non-commutative polynomials in matrices. These polynomials depend only
on the system layout and do not change with the size of the matrices involved,
hence such problems are called "dimension-free". Analyzing dimension-free
problems has led to the development recently of a non-commutative (nc) real
algebraic geometry (RAG) which, when combined with convexity, produces
dimension-free Semidefinite Programming. This article surveys what is known
about convexity in the non-commutative setting and nc SDP and includes a brief
survey of nc RAG. Typically, the qualitative properties of the non-commutative
case are much cleaner than those of their scalar counterparts - variables in
R^g. Indeed we describe how relaxation of scalar variables by matrix variables
in several natural situations results in a beautiful structure.
