Lecture 1: Introduction, motivation, examples. Introduction to normal families, definition of Fatou and Julia sets.
Lecture 2: Basic properties of Fatou and Julia sets. Bloch's Principle and Zalcman's Lemma. Density of repelling periodic points.
Lecture 3: Classification of periodic Fatou components. Exampls of Baker and Wandering Domains.
Lecture 4: Local conjugacy results. Outlook.
Lecture 5: Quadratic polynomials and the Mandelbrot set: external rays, connectivity of the Mandelbrot set.
Lecture 1: Introduction to quasiconformal mapping and the Measurable Riemann mapping theorem.
Lecture 2: Introduction to Kleinian groups, and dictionary with dynamics of rational maps.
Lecture 3: Basic dynamics of Kleinian groups: density of hyperbolic points, minimality of the action on the limit set.
Lecture 4: The Ahlfors finiteness theorem, perhaps a discussion of the absence of invariant line fields.
Lecture 1: Branner-Hubbard motions and surgery on a hyperbolic component of a one parameter family.
Lecture 2: Straightening of polynomial-like maps and the Branner-Douady surgery
Lecture 3: The Douady Siegel surgery, consequences and variations.
Lecture 4: Trans-quasi-conformal surgery.
Lecture 1 (Rippon): The escaping set of a transcendental entire function: examples, connections with the Fatou and Julia sets, Eremenko's conjectures. Introduction to the fast escaping set.
Lecture 2 (Stallard): The fast escaping set of a transcendental entire function: equivalent definitions, connections with the Fatou and Julia sets, properties of components.
Lecture 3 (Stallard): Examples for which the escaping set and fast escaping set is connected. Components of the intersection of the Julia set and the fast escaping set.
Lecture 4 (Rippon): The escaping set of a transcendental meromorphic function: examples, connections with the Fatou and Julia sets, properties of the escaping set for a function with a direct tract.
I will give a one-hour lecture on local dynamics in several complex variables, stressing similarities and differences with one complex variable. In particular, I will explain how to construct Fatou-Bieberbach domains (which strictly speaking is not a local matter, but is a natural topic to touch upon when talking about attracting fixed points and their basin of attraction).