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

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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.