Given a polynomial f(z) with complex coefficients, we can ask for which complex numbers p is the set {f(p), f(f(p)), f(f(f(p))),...} bounded, that is to ask which complex numbers have bounded forward orbit under f(z)? Alternatively we can turn the question around and ask for a fixed complex number p, for which (complex) polynomials is the forward orbit of p bounded? Finite? Periodic? These questions give

Amazingly many of the tools needed to approach these questions, arose well before computers allowed us to generate images like the one above. In this series of talks we will explore some of the basic tools and results of complex dynamics - the field of math motivated by questions similar to the ones above. In particular, the lectures should roughly proceed as:

    + Lecture 1: Fatou and Julia Sets and Cyles

    + Lecture 2: Bottcher’s Theorem, Filled Julia Set, Connectedness

    + Lecture 3: The Mandelbrot set

As a results of time constraints often times results presented in the lectures will be stated without proof, however, I will try to make sure to provide citations for any such results in the notes. Additionally. through out the lectures the interplay between complex dynamics and other fields like complex analysis, algebra, and number theory will be discussed.

Prerequisites: Basic familiarity with complex analysis and topology.