Program
Modular and automorphic forms & beyond
Student No.:40
Time:Mon & Wed 15:20-16:55, Jun.4-Jul.18
Instructor:Hossein Movasati  
Place:Lecture Hall, 3rd floor of Jin Chun Yuan West Bldg.
Starting Date:2018-6-4
Ending Date:2018-7-18

The guiding principal in this lecture series is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, including Calabi-Yau modular forms with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. We will first use the available tools in Algebraic Geometry, such as Geometric Invariant Theory, and construct the moduli space T of projective varieties enhanced with elements in their algebraic de Rham cohomology ring.


The new theory of modular forms lives on the moduli space T. It turns out that such moduli spaces are of high dimension and enjoy certain foliations, called modular foliations, which are of high codimension, and are constructed from the underlying Gauss-Manin connection. The mincourse will be mainly focused on three independent topics:
1. Hilbert schemes and actions of reductive groups and the construction of the moduli space T.
2. The theory of foliations of arbitrary codimensions on schemes and its relation with Noether-Lefschetz and Hodge loci in the case of modular foliations.
3. To rewrite available theories of automorphic forms, such as Siegel modular forms, Hilbert modular forms, modular forms for congruence groups, and in general automorphic forms on Hermitian symmetric domains, using the moduli space T. This will produce a geometric theory of differential equations of automorphic forms.
The seminar is based on a book that I am writing and its preliminary draft will be distributed between participants. It involves many reading activities on related topics, and contributions are most welcome.


Prerequisite: Basic Algebraic Geometry, Complex analysis.


Reference:

[1] H. Movasati, Modular and automorphic forms & beyond, manuscript under preparation.
[2] H. Movasati, Gauss-Manin connection in disguise: Calabi-Yau modular forms (Book), with appendices by Khosro Shokri and Carlos Matheus, Surveys in Modern Mathematics, Vol 13, International Press, Boston.
[3] B. Haghighat H. Movasati, S.-T. Yau. Calabi-Yau modular forms in limit: Elliptic fibrations, Communications in Number Theory and Physics, Vol. 11, Number 4, 879-912, 2017.
[4] M. Alim, H. Movasati, E. Scheidegger, S.-T. Yau. Gauss-Manin connection in disguise: Calabi-Yau threefolds, Comm. Math. Phys. 344, (2016), no. 3, 889-914.
[5] H. Movasati. Quasi-modular forms attached to elliptic curves, I, Annales Mathematique Blaise Pascal, v. 19, p. 307-377, 2012.