Dilation theory, free semialgebraic geometry and matrix convex sets

University Of Florida, Gainesville FL

Investigators

Abstract

Many problems in linear systems engineering and branches of mathematics both pure and applied can be modeled by inequalities involving matrices. A given matrix inequality is most useful in applications if it can be converted, through algebraic means, to a new matrix inequality whose solution set has a particularly simple form called "convex." The process of converting a matrix inequality to a convex matrix inequality is currently done on a case-by-case basis, and there is an extensive engineering literature on this subject with successes in particular examples. Continuing work of the principal investigator and his collaborators, a goal of this project is the development of a theory (so-called free real algebraic geometry) to identify exactly those matrix inequalities that can be reduced to obtain convexity and to automate this process when it is possible. A further goal of this project is the development of tools and techniques in the mathematical and scientific fields that underlie the study of free real algebraic geometry. These include semidefinite programming and linear matrix inequalities, subjects that find use in many branches of science and engineering, as well as a branch of mathematics called dilation theory. A goal of this project is the development of a free (freely noncommutative) analog of semialgebraic geometry with an emphasis on convexity. Semialgebraic geometry is the study of polynomial inequalities. Its free version studies matrix inequalities of the type that arise in engineering systems problems governed by a signal flow diagram. The Riccati inequality is a ubiquitous example. A desired long-term outcome of the project is a practical description of matrix inequalities with, up to polynomial or rational change of variable, convex solution sets. Many of the methods employed are of a functional analytic nature, involving ideas and techniques from operator systems and spaces, specifically. Techniques and results from classical semialgebraic geometry, convex optimization, semidefinite programming, as well as several complex variables, are also used. A further objective of this project is the development, from the perspective of non-self-adjoint operator algebras, of tools and techniques useful in the study of function algebras. The Hardy space of bounded analytic functions on the unit disc is an example. In particular, an outcome will be a deeper understanding of the representations and contractive matrix-valued functions of such algebras. The techniques and methods employed are a blend of function theory, harmonic analysis, Riemann surface theory, operator theory, and the theory of operator algebras.