Department of Mathematics

A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conformal conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to $\mathbb C$. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we will begin with an overview of complex dynamics, focusing on the moduli space of quadratic rational maps, which is isomorphic to $\mathbb C^2$. We will explore this space, finding many interesting objects along the way. We will then focus on special algebraic curves, called “Milnor curves” in this space. In general, it is unknown if Milnor curves are irreducible over $\mathbb C$. Because these curves are smooth, this is equivalent to asking whether they are connected. We will exhibit an infinite collection of Milnor curves that are connected. This is joint work with X. Buff and A. Epstein. Tea at 3:30pm in SC 1425. (Contact Person: Spencer Dowdall)