Wolff Lecture - General Lecture

Dynamics of quadratic polynomials, real and complex, has been an area of active research since the late 1970s. Over the complex, it is described by beautiful fractal objects, like Julia sets and the Mandelbrot set M, exhibiting surprising universal features. Over the reals, it exhibits an intricate intertwining between regular and chaotic dynamical regimes. Many of these features are related to the Renormalization phenomenon that controls small scale structure of the dynamics. By now, there is a thorough understanding of this area, though some deep problems are still awaiting for resolution. In the talk, we will give an overview of this field over the 40 years period, starting with the definition of a quadratic polynomial (z^2+c) and a description of the Douady-Hubbard-Thurston topological model for M.