Studies in the analytic theory of polynomials and matrix analysis with applications to the majorana representation in quantum physics

多项式和矩阵是两个最基本的数学对象，有着悠久而相互关联的历史。本课程旨在利用多项式解析理论的思想来研究矩阵的谱性质，如两个正规矩阵的特征值及其乘积之间的关系。其中一些相同的技术可以用于研究量子物理学中的马约拉纳表示。1932年，物理学家埃托雷·马约拉纳（Ettore Majorana）发现了一种用球体上的s个点来表示自旋s/2状态的自然方法。很容易看出，当且仅当对称自旋s态的马约拉纳表示的所有点都在球上的一个点上时，对称自旋s态是非纠缠态或相干态。最近已经注意到，高度纠缠的对称自旋态有它们的马约拉纳点散布在球体上。我们将研究马约拉纳表示的几何形状与自旋态的物理性质之间的关系。这个项目旨在发现数学三个领域之间的一些有趣的联系：多项式的解析理论，球面上点的构型和量子自旋态理论。其中一些结果将应用于矩阵理论、量子计算和数值分析。

Polynomials and matrices are two of the most basic mathematical objects with a long and interconnected history. This program aims to use ideas from the analytic theory of polynomials to study the spectral properties of matrices such as the relationship between the eigenvalues of two normal matrices and their product. Some of these same techniques can be used to study the Majorana representation in quantum physics. In 1932, the physicist Ettore Majorana discovered a natural way of representing a spin s/2 state as s points on a sphere. It can be easily seen that a symmetric spin s state is unentangled or coherent if and only if its Majorana representation has all of its points in one spot on the sphere. It has recently been noted that highly entangled symmetric spin states have their Majorana points spread out on the sphere. We will study the relationship between the geometry of the Majorana representation and the physical properties of the spin state. This program aims to discover some interesting interconnections between three areas of mathematics: the analytic theory of polynomials, configurations of points on a sphere and the theory of quantum spin states. Some of the results will have applications to Matrix Theory, Quantum Computing and Numerical Analysis.