Operator Theory and Matrix Inequalities

许多实际问题和数学问题都可以用多项式不等式来描述。这些是代数不等式，其变量（未知数）表示数字。 由于其重要性，多项式不等式已经被深入研究了许多世纪。 矩阵不等式，即以矩阵为变量的代数方程，也出现在许多应用领域，如线性系统工程和数学领域。多项式和矩阵不等式之间的关键区别在于，对于矩阵X和Y，与真实的数不同，XY和YX可能不相同，并且如果X和Y是“正”的，则XY不需要是“正”的。 这个项目的一个主要主题是发展一个数学理论的矩阵不等式，强调凸性。 一个集合是凸的，如果连接集合中任何两点的线段完全位于集合中。 虽然很容易画出两个变量的凸集，但这个概念在有大量变量的情况下是有意义的，并且非常重要，例如最小化成本或在设计问题中，它对确定最坏情况的性能至关重要。在许多情况下，当在凸集上搜索时，存在唯一的最小值（或最坏情况），并且存在非常可靠的计算机算法来找到最小值。 另一方面，当在一个非凸的集合上寻找最小值时，很难确定找到了最小值，而不是一个简单地小于附近值的值。线性矩阵不等式的解集是凸的，但一般的矩阵不等式的解集不是凸的。主要研究者和他的合作者的目标是确定矩阵不等式的解集何时是凸的，可以以易于处理的方式转换为或合理地近似为以某种方式由线性矩阵不等式描述的凸集。这个项目将有助于凸矩阵不等式的数学基础，发展的主题都平行半代数几何和作为一个非线性版本的理论的运营商系统和空间和完全积极/压缩地图。动力来自于与工程系统理论等应用的深层科学互动。它将推进对线性矩阵不等式（现在是科学和工程中的标准工具）和系统工程问题中出现的矩阵不等式的理解，特别是对于许多从信号流图中出现的矩阵不等式。一个重要的目的是确定和自动识别的矩阵不等式与凸解集，映射一个非凸解集凸（目前做的一些特殊情况下，在系统工程文献的特设基础上），否则近似非凸解集凸的。所采用的方法涉及的思想，从功能分析，算子理论，复杂的分析和几个复杂的变量，运营商系统和空间，完全正映射，半定规划和半代数几何。相反，一个重要的方面，这一建议是发展的技术和成果，有助于理论的运营商系统，空间和代数和完全积极的地图和（自由）noncommutative类似物的理论解析函数在一个和几个变量。在另一个方向，积极和完全积极的映射之间的区别自然算子代数的功能产生的一个和几个复杂的variable.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Many practical and mathematical problems can be described by polynomial inequalities. These are algebraic inequalities whose variables (unknowns) represent numbers. Because their importance, polynomial inequalities have been studied intensely for many centuries. Matrix inequalities, algebraic equations whose variables are matrices, also appear in many applied areas, such as linear systems engineering, and mathematical fields. A key difference between polynomial and matrix inequalities is that for matrices X and Y, unlike real numbers, it can happen that XY and YX are not the same and if X and Y are "positive", XY need not be "positive". A major theme of this project is the development of a mathematical theory of matrix inequalities with an emphasis on convexity. A set is convex if the line segment joining any two points in the set lies entirely in the set. While it is easy to draw pictures of convex sets in two variables, the notion makes sense and is immensely important in the cases where there are a large number of variables, for instance minimizing cost or in design problems where it is crucial to identifying worst case performance. In many settings, when searching over a convex set, there is a unique minimum (or worst case) and there are very reliable computer algorithms to find the minimum. On the other hand, when searching for a minimum over a set that is not convex it is difficult to be certain that the minimum, and not a value that is simply smaller than those nearby, has been found. The solution set of a linear matrix inequality is convex, but in general solution sets to matrix inequalities are not. A goal of the principal investigator and his collaborators is to determine when the solution set of a matrix inequality is convex, can be converted in a tractable way to, or approximated reasonably by, a convex set described in some fashion by a linear matrix inequality.This project will contribute to the mathematical foundations of convex matrix inequalities, developing the subject both in parallel to semialgebraic geometry and as a non-linear version of the theory of operator systems and spaces and completely positive/contractive maps. Motivation flows from the deep scientific interactions with applications such as engineering systems theory. It will advances the understanding of both linear matrix inequalities, now standard tools in science and engineering, and matrix inequalities that arise in systems engineering problems, particular for many of those that arise from a signal flow diagram. An important aim is to determine and automate the identification of matrix inequalities with convex solution sets, mapping a non-convex solution set to a convex one (currently done on an ad-hoc basis for some special cases in the systems engineering literature), and otherwise approximating non-convex solution sets by convex ones. The methods employed involve ideas from functional analysis, operator theory, complex analysis and several complex variables, operator systems and spaces, completely positive maps, semidefinite programming and semialgebraic geometry. Conversely, a significant aspect of this proposal is the development of techniques and results that contribute to the theories of operator systems, spaces and algebras and completely positive maps and of (freely) noncommutative analogs of the theory of analytic functions in one and several variables. In another direction, the distinction between positive and completely positive maps on natural operator algebras of functions arising in one and several complex variables will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.