Abstract: Modular forms are among the most beautiful and most fertile objects in all of number theory, with innumerable surprising properties. One of these, going back to the work of Eichler and Shimura in the 1960s and of my recently deceased colleague Manin in the 1970s, is that one can associate to every modular form a certain polynomial, its so-called period polynomial, without losing any information, thus in some sense reducing an a priori completely transcendental theory to the study of finite and algebraic objects.
The lecture will be divided into two parts, with a break in between to allow people to escape. In the first part I will explain briefly the definition and basic properties of modular forms and their period polynomials and will describe one or two particularly nice applications. The second part will address the question of reconstructing a cusp form from its period polynomial. I will indicate a complete solution of this problem for the case of the full modular group relying on an elementary but not obvious lemma about the continued fraction expansion of real numbers, and will also explain how the attempt to generalize this statement to other modular groups led in a natural way to a conjecture about the dynamics of general Fuchsian groups that has been checked in many cases but is still open after more than 25 years.