Dilation theory and convexity in free semi-algebraic geometry

这个项目的中心是发展一种非对易半代数几何理论，与经典的对易理论平行。因此，经典的半代数几何研究的是多项式不等式，而非交换的半代数几何研究的是涉及矩阵或非交换变量作为未知数的多项式不等式。虽然这门学科有它自己内在的数学兴趣，但它也受到这样一个事实的驱使，即在许多由信号流图建模的工程系统问题中自然地出现了非对易不等式。由于凸性在最优化中的重要性，尤其令人感兴趣的情况是解，即同义可行集，由线性矩阵不等式(LMI)确定，或者可以通过非交换解析映射或全矩阵函数变换为凸集或由LMI控制的集。数学、物理和工程中的许多问题都是使用矩阵建模的。与乘法顺序无关紧要的数字不同，矩阵乘法不是可交换的--顺序很重要。该项目将发展一种非交换不平等理论，即涉及矩阵未知数的不等理论，从而为数学、科学和工程研究人员提供使用工具。凸性问题特别重要，因为它们可以用数值例程有效地解决。这个项目的一个潜在结果是准确地识别这些问题，特别是从系统工程中，可以使用凸性来分析这些问题，也许在变量改变之后。

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.