Abstract: Modular forms are one of the central objects in modern number theory, while differential equations, of course, are fundamental in almost every part of mathematics. There are many interactions between the two subjects, but these are often not widely known. In these two talks I will present several of these connections. In particular,

(i) Every modular form satisfies a third order non-linear differential equation, the so-called Chazy equation being a famous example, and these appear in the theory of Painlevé equations and in connection with various enumerative problems of algebraic geometry.

(ii) After a suitable change of variable, every modular form also satisfies a linear differential equation with algebraic or polynomial coefficients. These play a key role in many questions of number theory and arithmetic algebraic geometry, notably in the proof of Apéry's theorem on the irrationality of $\zeta(3)$ and in the study of certain families of Calabi-Yau manifolds arising in mirror symmetry.

(iii) Modular forms also satisfy a third type of differential equation, now again linear but with coefficients that are themselves modular or quasimodular forms rather than polynomials. These have become important in conformal field theory and the theory of vertex operator algebras in recent years, and there is now a complete description (Joint work with K. Nagatomo and Y. Sakai).

The presentation will be as elementary as possible. I will not assume any prior knowledge of modular forms and will try to give many examples to illustrate the various kinds of differential equations that can arise and also some of their applications.