**Internal Addresses of the Mandelbrot Set
and Galois Groups of Polynoials**

I will start this talk with a brief synopsis of my research in
``past, present, and future'', as in my IUB job talk of last year.
My main focus will be on a link between symbolic dynamics, algebra,
and the Mandelbrot set: we determine the Galois groups of certain
dynamically defined polynomials (or equivalently, show connectivity
of a Riemann surface defined by periodic points of quadratic polynomials).
This will be done using symbolic dynamics of quadratic polynomials for
which the Julia set is a Cantor set, and the problem is reduced to an
existence problem in symbolic dynamics (``kneading sequences''). It
turns out that the solution of this problem, in turn, requires a deep
understanding of the structure of the Mandelbrot set.

*Dierk Schleicher*