Dilation theory, free semialgebraic geometry and matrix convex sets

线性系统工程和数学分支中的许多问题，无论是纯数学还是应用数学，都可以用涉及矩阵的不等式来建模。如果一个矩阵不等式可以通过代数方法转化为一个新的矩阵不等式，它的解集具有一种称为"凸"的特别简单的形式，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个案基础上完成的，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。继续工作的主要研究者和他的合作者，这个项目的目标是发展一个理论（所谓的自由真实的代数几何），以确定正是这些矩阵不等式，可以减少获得凸性和自动化这一过程时，它是可能的。 该项目的另一个目标是在数学和科学领域开发工具和技术，作为自由真实的代数几何研究的基础。这些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中都有应用，还有一个称为膨胀理论的数学分支。 这个项目的一个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。 它的自由版本研究在由信号流图控制的工程系统问题中出现的类型的矩阵不等式。Riccati不等式是一个普遍存在的例子。该项目的一个预期的长期成果是矩阵不等式的实用描述，直到多项式或变量的合理变化，凸解集。所采用的许多方法是一个功能分析的性质，涉及的思想和技术，从运营商系统和空间，具体。技术和结果从经典的半代数几何，凸优化，半定规划，以及几个复杂的变量，也被使用。 这个项目的另一个目标是从非自伴算子代数的角度，开发在函数代数研究中有用的工具和技术。单位圆盘上有界解析函数的哈代空间就是一个例子。特别是，一个结果将是更深入地了解这种代数的表示和压缩矩阵值函数。所采用的技术和方法是融合了函数论、调和分析、黎曼曲面理论、算子理论和算子代数理论。

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.