Complex dynamics: the Mandelbrot set and Newton's Method. On useless and useful Mathematics

Complex dynamics (in the sense of iteration of complex-differentiable functions) is a young and active field with many aspects and relations to other fields. We discuss two of them:
1. First we introduce the iteration theory of quadratic polynomials and the classical “Mandelbrot set” as an area of mathematics that is interesting, deep, and beautiful, but that seems apparently useless.
2. Then we discuss Newton's root-finding methods, especially in order to find roots of complex polynomials: this is a classical theory that is generally known as useful. In particular, we show some very recent results on the theory of Newton's method that remained open for many years.
3. Finally, we show that in order to understand many deep questions on Newton's method, one has to understand the Mandelbrot set and many of the “useless” aspects of research.
The moral of the story is, of course, that it is quite useless to distinguish useless and useful aspects of mathematics, and beautiful or interesting questions of mathematics are likely to find their use sooner or later: and the more unexpectedly this happens, the more useful this may be in the end.
Strong graduate students are always welcome in the field, also in joint projects with Vladlen Timorin from the Moscow University of Higher Economics.