Title: Separation of roots of random polynomials
ABSTRACT: Let f_n be a degree n polynomial with independent and identically distributed coefficients. What do its roots tend to look like? Classical results of Erdos-Turan and others tell us that most roots are near the unit circle and that they are approximately rotationally equidistributed. Qualitatively, the roots tend to "repel" each other. In order to quantify this repulsion, we will discuss the smallest distance m_n between pairs of roots and prove that m_n is of order m^{-5/4} and satisfies a non-degenerate limit theorem when rescaled. We will discuss general strategies for proving facts about macroscopic behavior (such as approximate equidistribution) as well as discussing where additive combinatorics comes into play.