Dilation theory and convexity in free semi-algebraic geometry

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This project is centered on developing a theory of non-commutative semi-algebraic geometry, paralleling the classical commutative theory. Thus, while classical semi-algebraic geometry is the study of polynomial inequalities, non-commutative semi-algebraic geometry studies polynomial inequalities involving matrices, or non-commuting variables, as unknowns. While the subject has its own intrinsic mathematical interest, it is also motivated by the fact that non-commutative inequalities arise naturally from in a number of engineering systems problems that are modeled by a signal flow diagram. Because of the importance of convexity in optimization, of particular interest is the case where the solution, synonymously feasible set, is determined by a linear matrix inequality (LMI) or can be transformed via a non-commutative analytic map or fully matricial function, to a convex set or a set which is governed by an LMI.Many problems in mathematics, physics, and engineering are modeled using matrices. Unlike for numbers where the order of multiplication does not matter, matrix multiplication is not commutative - the order does matter. The project will develop a theory of non-commutative inequalities; i.e., a theory of inequalities involving matrix unknowns thus providing tools of use to researchers in mathematics, science, and engineering. Convex problems are especially important because they can be solved efficiently using numerical routines. A potential outcome of this project is the precise identification of those problems, particularly from systems engineering, which can be analyzed, perhaps after a change of variable, using convexity.