Dilation theory and convexity in free semi-algebraic geometry

本计画以发展非交换半代数几何理论为中心，平行于经典交换理论。因此，经典半代数几何研究的是多项式不等式，而非交换半代数几何研究的是涉及矩阵或非交换变量的多项式不等式。虽然这门学科有自己内在的数学兴趣，但它也受到这样一个事实的推动，即非交换不等式自然地出现在许多由信号流图建模的工程系统问题中。由于凸性在优化中的重要性，特别令人感兴趣的是解，即可行集，由线性矩阵不等式（LMI）决定或可以通过非交换解析映射或全矩阵函数转换为凸集或由LMI控制的集合的情况。数学、物理和工程中的许多问题都是用矩阵建模的。不像数字的乘法顺序不重要，矩阵乘法是不可交换的——顺序是重要的。该项目将发展一个非交换不等式理论；例如，一种涉及矩阵未知数的不等式理论，从而为数学、科学和工程研究人员提供了使用工具。凸问题特别重要，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的结果是精确地识别那些问题，特别是来自系统工程的问题，这些问题可以被分析，也许在变量改变之后，使用凸性。

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.