When constructing a polynomial P whose Julia set has a desired "shape" S, a strategy is to make |P| roughly constant on S. When S is a disjoint union of smooth Jordan domains, this can be accomplished by equidistributing the roots of P in the boundary of S according to harmonic measure. Why does this work, and what are the actual values of these polynomials? I will discuss how answering this question involves relating the Poisson kernel (the density of harmonic measure) to contour integrals on various canonical conformal representations of S^c. Speaker(s): Kathryn Lindsey (UChicago)