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Título: A Mini-Course on Modular Vector Fields attached to Calabi-Yau Manifolds


Expositor: Younes Nikdelan
Universidade do Estado do Rio de Janeiro (UERJ), 
Instituto de Matemática e Estatística (IME)
Departamento de Análise Matemática, Rio de Janeiro, Brazil.

Horarios: 
Lunes 10, miércoles 12 y viernes 14 de octubre, de 16:00 a 18:00
Cuarta sesión a determinar con el público asistente

Modular vector field is a part of bigger project calling Gauss-Manin connection in disguise, GMCD for short, introduced by Hossein Movasati, see [Mov16], which aims to unify mod- ular and automorphic forms with toplogical string partitions of string theory. Initially, in [Nik15], instead of modular vector field we used the name Darboux-Halphe-Ramanujan vector field, because of historical relationships. But then, since in low dimensions 1 and 2 the solutions of the vector fields are in terms of (quasi-)modular forms, we called them modular vector fields.
This mini-course is based on four consecutive lectures in four days at IMCA, Peru. I will start with an introductory example and then proceed with some basic definitions and facts about Hodge theory that are necessary to understand GMCD. I will finish my lectures with the works about modular vector fields 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.
Lecture 2. Intersection form, Picard-Fuchs equation, Calabi-Yau manifolds, Dwork family.
Lecture 3. Modular vector field, Enumerative properties of modular vector fields. 
Lecture 4. Sketch of proofs.

For more details one is referred to [Mov16, MN16, Nik15] and references given there.

References
[Mov16] Hossein Movasati. Gauss-Manin connection in disguise: Calabi-Yau modular forms. To appear in Surveys of Modern Mathematics, IP, Boston. Available online at http://w3.impa.br/hossein/myarticles/GMCD-MQCY3.pdf, 2016.
[MN16] H. Movasati and Y. Nikdelan. Gauss-Manin connection in disguise: Dwork Family. arXiv:1603.09411, 2016.
[Nik15] Younes Nikdelan. Darboux–Halphen–Ramanujan vector field on a moduli of Calabi–Yau manifolds. Qual. Theory Dyn. Syst., 14(1):71–100, 2015.