Special Lecture: Modular forms and differential equations - Don B. Zagier

[ICTP Math Section]

*_Special Lecture: Modular forms and differential equations_*

*Don B. Zagier (MPI, Bonn - ICTP-IGAP-SISSA)*

Tuesday, 17 March at 16:30 -  Leonardo Building, Budinich Lecture Hall 
(in-person) - https://indico.ictp.it/event/11309/

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*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/*

All are welcome to join.