This is a research plan for a collaborative work involving colleagues from Brazil, France and Portugal. Namely, the members of this project are the following: Brazil: H. Movasati (IMPA) and Y. Nikdelan (UERJ); France: J. Rebelo (IMT) and D. de la Rosa (IMT - graduate student); Portugal: H. Reis (UP). Among previous works directly related to the questions raised in this project, we may quote: [G], [AMSY], [RR], [Mov3], and [MN]. Ultimately the purpose of our project is to conduct a dynamical study of certain differential equations appearing in Mathematical-Physics (especially in Mirror Symmetry) as well as in some other contexts, including representations of $\mathfrak{sl}_2(\C)$ and Number Theory. We believe that a good dynamical understanding of the mentioned equations will have a few interesting applications, not least to the study of certain families of Calabi-Yau manifolds along with corresponding Mirror maps.

References:

[AMSY] M. Alim, H. Movasati, E. Scheidegger and S.-T. Yau, "Gauss-Manin connection in disguise: Calabi-Yau threefolds", Commun. Math. Phys., 344, 3 (2016), 889-914.

[COGP] P. Candelas, X.C. de la Ossa, P.S. Green and L. Parkes, "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory", Nuclear Phys. B, 359, 1 (1991), 21-74.

[D] G. Darboux, "Sur la théorie des coordonnées curvilignes et les systémes orthogonaux", Ann Ecole Normale Supérieure, 7 (1878), 101-150.

[Dwo1] B. Dwork, "A deformation theory for the zeta function of a hypersurface", Proc. Internat. Congr. Mathematicians (Stockholm), (1962), 247-259

[Dwo2] B. Dwork, "On the zeta function of a hypersurface III", Ann. of Math., 83, 2 (1966), 457-519.

[G] A. Guillot, "Sur les équations d'Halphen et les actions de SL2(**C**)", Publ. Math. Inst. Hautes Études Sci., 105 (2007), 221-294.

[GR] A. Guillot and J.C. Rebelo, "Semicomplete meromorphic vector fields on complex surfaces", J. reine angew. Math., 667 (2012), 27-65.

[GMP] B.R. Greene, D.R. Morrison and M.R. Plesser",Mirror manifolds in higher dimension",Comm. Math. Phys., 173 (1995), 559-598.

[Hal] G.H. Halphen, "Sur un systéme d'équations différetielles", C. R. Acad. Sci Paris, 92 (1881), 1101-1103.

[Mov1] H. Movasati, "Multiple Integrals and Modular Differential Equations",28th Brazilian Mathematics Colloquium, Instituto de Matemática Pura e Aplicada, IMPA, 2011.

[Mov2] H. Movasati, "Modular-type functions attached to mirror quintic Calabi-Yau varieties", Math. Zeit., 281, 3 (2015), 907-929.

[Mov3] H. Movasati, "Gauss-Manin connection in disguise: Calabi-Yau modular forms", to appear in Surveys of Modern Mathematics, IP, Boston

[MN] H. Movasati and Y. Nikdelan, "Gauss-Manin Connection in Disguise: Dwork-Family", arXiv:1603.09411 [math.AG]}, 2016.

[Nik] Y. Nikdelan, "Darboux-Halphen-Ramanujan vector field on a moduli of Calabi-Yau manifolds",Qual. Theory Dyn. Syst., 14, 1 (2015),71-100.

[Ram] S. Ramanujan, "On certain arithmetical functions", Trans. Cambridge Philos. Soc., 22 (1916),159-184.

[R] J.C. Rebelo, "Singularités des flots holomorphes",Ann. Inst. Fourier (Grenoble), 46, 2 (1996), 411-428.

[RR] J.C. Rebelo and H. Reis, "Uniformizing complex ODEs and applications", Revista Matematica Iberoamericana, 30, 3 (2014), 799-874.