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.
Many problems in mathematics and its applications including systems engineering, the field of engineering that designs automatic controllers such as autopilots, are modeled by matrix inequalities. Matrix inequalities are more powerful than the more commonly known polynomial inequalities. They are also more complicated, in part because, unlike for real numbers, matrix multiplication does not commute. For matrices X and Y, it can happen that XY is not the same YX. For many applications it is important that the matrix inequality have a convex solution set. In the case that this solution set is not convex, alternatives include approximating it by a convex set or finding a tractable change of variable which converts it to a convex set. One of the main outcomes of this project was the demonstration that, in many contexts, if the solution set of a matrix inequality is convex, then the matrix inequality can be taken to have a particularly simple form called a linear matrix inequality. Practically, this simplifies determining whether a given matrix inequality has a convex solution set and, when it is not, the process of modifying the inequality to produce a convex solution set. This project also produced initial results on finding suitable convex replacements for non-convex solution sets of matrix inequalities. A related outcome of the project was a generalization of what is know as the matrix cube problem from mu synthesis in systems engineering. The theories from the mathematical subjects of operator theory and operator algebras, systems and spaces as well as dilation theory are intimately connected with both quantum physics and systems engineering. They were pivotal in producing the outcomes described in this paragraph. Conversely, results of this project also contributed to the further development of these theories. Further outcomes of this project accrued to Pick interpolation and dilation theory, inextricably intertwined subjects which lie at the intersection of complex analysis and operator theory. Complex analysis - the theory of analytic functions of a complex variable - is ubiquitous in science and engineering and is one of the most classical and useful branches of mathematics. It finds applications in areas from number theory to hydrodynamics. Transfer functions, which figure prominently in electrical engineering, are special types of analytic functions. Pick interpolation provides a mathematical model of the problem of constructing a transfer function with minimum size amongst functions meeting specified design criteria. A distinguished variety is a generalization of the complex numbers. An outcome of this project was the solution to the Pick interpolation problem on distinguished varieties in the bidisk. A further result of the project provided a measure of the difficulty of the problem rational dilation on distinguished varieties. While not funded directly by this grant, the investigations did provide research and training opportunities. During the period of the grant the PI had three PhD students. Two finished their degrees this Spring (2014) working on aspects of the project. One student studied Hilbert spaces of Dirichlet series in one and several variables. Dirichlet series are basic objects in number theory, a subject known for its connections with cryptography. His results contributed to our understanding of multipliers on reproducing kernel Hilbert spaces, an important topic in operator theory with connections to Pick interpolation. The other student studied Toeplitz operators. Toeplitz operators arise in a number of contexts, including stationary stochastic process in engineering and in index theory, a topic which ties together many important areas of mathematics including differential geometry, complex analysis and operator algebras. His thesis provided a totally new approach to studying the eigenvalues of Toeplitz operators with real symbols on multiply connected domains in the complex plane, giving a systematic approach to many of the results in the literature pertaining to this subject and producing many new results. Sturm–Liouville differential operators (equations) naturally arise in many physical contexts including the modeling of vibrating strings. A basic problem in this theory is finding a non-vanishing solution to Sturm– boundary value problems. The third student is exploring operator theory approaches to producing such non-vanishing solutions and generalizing to the multi-variable setting.
