This lecture will describe how one can paste together two rather skinny fractal sets, with no interior, to obtain a full 2-dimensional sphere. If f is a polynomial map from the complex numbers to themselves. then the "filled Julia set" K₆₀ is the set of complex numbers z such that the sequence z, f(z), f ((fz)), . .. is bounded. This is usually a complicated fractal set. Yet the operation of "mating", which pastes together two such filled Julia sets to yield a smooth Riemann sphere, is often defined. The lecture will study one particularly non-intuitive example of this construction.

The Mandelbrot set M can be thought of as the table of contents for a (very large) book which describes all possible kinds of dynamic behavior for quadratic polynomial maps. The various complicated geometric structures seen in M correspond to different types of behavior. This lecture will explain some of this structure by studying periodic orbits for quadratic maps.

Exploration of the larger world of rational maps from the Riemann sphere to itself.