[1] Douady was a student of Henri Cartan at the École normale supérieure, and initially worked in homological algebra.

Subsequently, he became more interested in the work of Pierre Fatou and Gaston Julia and made significant contributions to the fields of analytic geometry and dynamical systems.

Together with his former student John H. Hubbard, he launched a new subject, and a new school, studying properties of iterated quadratic complex mappings.

They made important mathematical contributions in this field of complex dynamics, including a study of the Mandelbrot set.

One of their most fundamental results is that the Mandelbrot set is connected; perhaps most important is their theory of renormalization of (polynomial-like) maps.