Abstract

The concept of Gauss-Manin connection in disguise, GMCD for short, introduced by Hossein Movasati, which aims to unify modular and automorphic forms with toplogical string partitions of string theory. This minicourse is based on three consecutive lectures in three days at IPM. I will start with an introductory example and then present some basic definitions and facts about Hodge theory that are necessary to understand GMCD. I will finish my lectures with the works that I have contributed so far in the project of GMCD on a certain family of Calabi-Yau manifolds, in particular Dwork family. Following I state subjects that I am going to present in any lecture.

Lecture 1: An introductory example: Ramanujan's vector field, de Rham cohomology and Hodge decomposition theorem, Gauss-Manin connection, Intersection form.
Lecture 2: Picard-Fuchs equation, Calabi-Yau manifolds, Dwork family.
Lecture 3: Modular vector field (that initially we called it Darboux-Halphen-Ramanujan vector field), Enumerative properties of modular vector fields.



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