The Mandelbrot set and its cubic analog

The Mandelbrot set is perhaps the most well known mathematical fractal outside of the mathematical community. This set describes how the dinamics of a quadratic polynomial $z^2+c$ varies with the complex parameter $c$. Just by looking at the location of $c$ relative to the Mandelbrot set, we can say a lot about the dynamical properties of $z^2+c$ (whereas having an explicit expression for $c$, say, $c=-1.5$, is by far less convenient). We will discuss the structure of the Mandelbrot set and, in particular, its (conjectural) topological model. If time permits, we may very briefly overview the new and active area of research dealing with the structure of the cubic Mandelbrot set.