In 1996 A. Connes proposed a beautiful formula for the Hausdorff measure of certain Julia sets in terms of singular traces.  A proof of the formula has only recently been published in joint work of the two speakers with F. Sukochev and A. Connes. This two-part talk briefly describes the proof. The first half will cover the necessary operator-theoretic preliminaries and the second will discuss the theory of Julia sets of polynomials, with a view towards the proof of Connes' formula.