Dilation theory, free semialgebraic geometry and matrix convex sets

线性系统工程和纯数学和应用数学分支中的许多问题都可以用涉及矩阵的不等式来建模。如果一个给定的矩阵不等式可以通过代数方法转换成一个新的矩阵不等式，它的解集有一个特别简单的形式，叫做“凸”，那么它在应用中是最有用的。将矩阵不等式转换为凸矩阵不等式的过程目前是在逐个案例的基础上完成的，并且在这个主题上有大量的工程文献，在特定的例子中取得了成功。在首席研究员和他的合作者的持续工作中，这个项目的目标是发展一种理论（所谓的自由实代数几何），以准确地识别那些可以减少以获得凸性的矩阵不等式，并在可能的情况下将这一过程自动化。该项目的进一步目标是开发数学和科学领域的工具和技术，这些工具和技术是自由实代数几何研究的基础。其中包括半定规划和线性矩阵不等式，这些学科在科学和工程的许多分支以及数学的一个分支膨胀理论中都有应用。这个项目的目标是开发一个自由的（自由非交换的）半代数几何的模拟，强调凸性。半代数几何是研究多项式不等式的学科。它的自由版本研究了由信号流程图控制的工程系统问题中出现的类型的矩阵不等式。里卡蒂不等式就是一个随处可见的例子。该项目的一个期望的长期成果是一个矩阵不等式的实际描述，直到多项式或有理的变量变化，凸解集。所采用的许多方法具有功能分析的性质，特别是涉及来自算子系统和空间的思想和技术。还使用了经典半代数几何、凸优化、半定规划以及几个复变量的技术和结果。该项目的进一步目标是从非自伴随算子代数的角度出发，开发用于研究函数代数的工具和技术。单位圆盘上有界解析函数的Hardy空间就是一个例子。特别是，结果将是对这些代数的表示和压缩矩阵值函数有更深的理解。所采用的技术和方法是函数理论、调和分析、黎曼曲面理论、算子理论和算子代数理论的混合。

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.