Spectral methods in the geometry of polynomials

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In the 1960's the mathematican B. Sendov conjectured that given any polynomial of degree two or greater all of whose zeros lie in the unit disk on the complex plane, each root of the polynomial is a distance of at most one from a zero of the derivative of that polynomial. Sendov's conjecture remains unsolved (although it has been proven for polynomials of degree less than or equal to eight) and is considered an important problem in complex function theory. In the past two years, researchers have devised ways of using matrices to study the relationship between the zeros of a polynomial and that of its derivative. I propose to develop techniques in matrix analysis, especially in spectral perturbation theory, that may be useful in solving Sendov's conjecture. While this is primarily curiosity-driven research, spectral perturbation theory has important applications in the physical sciences and computation and the techniques developed may find application in these areas.