The arithmetic and topology of differential equations

Abstract

This survey paper attempts to present in as elementary a way as possible a wide panorama of results concerning the relations between differential equations on the one hand and algebraic geometry, number theory and topology on the other. We use the famous Apéry numbers as a running example to illustrate the connections with, among other things, the theory of periods (Picard–Fuchs differential equations), the theory of modular forms (and the special values of their L-series), the theory of motives (starting with counting points on varieties over finite fields), and mirror symmetry (in particular, the Gamma Conjecture relating the asymptotics of solutions of differential equations to the multiplicative “Gamma class” of a variety). A number of relations to works by Friedrich Hirzebruch, in whose honor the lecture was given, are also described.