Abstract:

In the theory of special functions, the central question is usually whether they satisfy any kind of differential equation, like Bessel or Legendre functions. Strangely enough, modular forms, which are ubiquitous both in pure mathematics and in mathematical physics, do satisfy differential equations, and even of several different kinds, but these facts are not widely known and used.

I will give a very brief introduction to modular forms, followed by a survey of the three principal kinds of dfferential equations, each of which is important in several parts of mathematics and mathematical physics:

(i) Non-linear differential equations of Painleve type, the proto-example being the so-called Chazy equation.  These are relevant in the theory of integrable systems, which also has many applications in the study of moduli spaces and in string theory.

ii) Linear equations of Picard-Fuchs type. These have applications both in number theory (e.g. in Apery's proof of the irrationality of zeta(3)) and in the study of Calabi-Yau manifolds, again important in string theory.

iii) Finally, the so-called modular linear differential equations that are now playing an increasing role in conformal field theory and the study of VOAs.


This lecture is part of the KMPB-Ukraine Workshop programme (in Berlin): https://indico.desy.de/event/52099/