REMINDER: IGAP SPECIAL LECTURES by DON ZAGIER - SECOND LECTURE TODAY Tuesday 19 December, 15:00-16:00 - VENUE: IGAP Building, room 205

[ICTP Math Section]

*_IGAP  S P E C I A L   L E C T U R E S_*


"Modular forms and differential equations"


Don Zagier (IGAP/MPIM)


Tuesday, 12 December 2023 (15:00 - 16:00)
Tuesday, 19 December 2023 (15:00 - 16:00)


Venue: IFPU/IGAP Building - room 205
Via Beirut 2, 34151 Trieste
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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 ζ(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.

https://www.igap-ts.it/2023/12/05/igap-special-lectures-by-prof-don-zagier-igap-mpi-modular-forms-and-differential-equations/

Everyone is welcome to attend. Kindly note that*today's lecture* is 
independent from the one given by Professor Zagier last week, so it *can 
be attended even in case one missed the first lecture.*