Real zero polynomials, linear matrix inequalities, free analogues and applications

半定规划提供了有效的算法来最小化受线性矩阵不等式约束的仿射线性函数。这就引出了几个数学问题：哪些集合可以用线性矩阵不等式来描述？哪些集合是其投影？这些问题位于代数几何，真实的代数几何和凸性之间的边界;它们导致一类多项式，所谓的真实的零多项式（称为双曲在齐次设置），这是一个多变量的推广多项式只有真实的零。哪些集合可以用线性矩阵不等式来描述的问题只在2维中得到了解决，即使在2维中也远未被完全理解。本研究的目的之一是在二维空间中进一步研究这个问题，并解决更高维的情况。它与最近证明的统计量子力学中的BMV猜想有关，并应用于算子代数理论中最有趣的公开问题之一- Connes嵌入问题。另一方面，可以考虑在自由非交换设置的线性矩阵不等式，替代元组的真实的对称矩阵的所有大小的非交换变量。这对于出现在系统和控制中的许多（如果不是大多数）优化问题是自然的，因为这些问题是维度无关的，即，自然变量是矩阵，并且问题涉及这些矩阵变量的有理表达式，因此这些矩阵变量具有与矩阵大小无关的相同形式。它也符合一个一般的范式，通过从交换设置的自由非交换设置，出现在过去二十年的过程中，在这样的不同领域的理论算子空间和自由概率。我们计划详细研究自由非对易情形。由于它比交换情形有更多的结构，可能导致更强的结果，我们也希望最终能够通过非交换提升来解决一些交换问题。

Semidefinite programming provides efficient algorithms for minimizing an affine linear function subject to a linear matrix inequality constraint. This leads to several mathematical questions: which sets can be described by a linear matrix inequality? which sets are projections thereof? These questions lie on the border between algebraic geometry, real algebraic geometry and convexity; they lead to a class of polynomials, so called real zero polynomials (called hyperbolic in the homogeneous setting), that are a multivariable generalizations of polynomials with only real zeroes. The problem of which sets can be described by a linear matrix inequality has been solved only in dimension 2, and even there it is far from being completely understood. It is one purpose of the proposed research to pursue this problem further in dimension 2 and to tackle the higher-dimensional situation. There are relations with the recently proved BMV conjecture from Statistical Quantum Mechanics, and applications to one of the most interesting open problems in the theory of operator algebras - Connes' embedding problem. On the other hand, one can consider linear matrix inequalities in the free noncommutative setting, substituting tuples of real symmetric matrices of all sizes for the noncommuting variables. This is natural for many if not most optimization problems appearing in systems and control since these problems are dimension-independent, i.e., the natural variables are matrices, and the problems involve rational expressions in these matrix variables which have therefore the same form independent of matrix sizes. It also conforms to a general paradigm of passing from the commutative setting to the free noncommutative setting that emerged over the course of the last two decades in such diverse areas as the theory of operator spaces and free probability. We plan to study the free noncommutative situation at length. As it has more structure than the commutative situation, leading possibly to stronger results, we also hope to be eventually able to tackle some commutative problems via a noncommutative lifting.