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.