For unpublished articles see Math. Arxiv . If you do not want to bother yourself with all these articles and you want to have a flavor of what I have done from 2000 to 2010 then read this . Two main reasons why I am defined over complex numbers 1. see the article of Ali Enayat in Akhbar 1375 of IPM 2. I did a Complex Dynamics course with Dr. Siavash Shashahani right before my departure to Brazil.  I am now mainly busy with:

The project GMCD: Gauss-Manin Connection in Disguise.



  1. Gauss-Manin connection in disguise: Jacobi forms of index zero ,(with Jin Cao and Roberto Villaflor) to be published by IMRN, 2023.
  2. Periods of families of curves in threefolds , Preprint. Here, two letters of P. Deligne. Letter 1 and Letter 2
  3. Smooth Points of the Space of Plane Foliations with a Center (with Lubomir Gavrilov)) , to be published by IMRN, 2022.
  4. Algebraic curves and foliations (with Cesar Camacho), To be published by Bulletin Of The London Mathematical Society, 2022.
  5. On elliptic modular foliations II , Moscow Mathematical Journal, Volume 22, Number 1, 2022, Pages 103–12.
  6. A Course in Hodge Theory: With Emphasis on Multiple Integrals,  Somerville, MA: International Press Boston, 2021.
  7. Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific, To be published November 2021
  8. A Course in Hodge Theory: Periods of Algebraic cycles (Together with Roberto Villaflor) 33 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, Brazil,2021. This has been originated from my lecture notes An Advanced course in Hodge Theory and Roberto's thesis
  9. Convolution of Picard-Fuchs equations (with Stefan Reiter), Michigan Mathematical Journal, Volume 71, Issue 4, 2022.
  10. Gauss-Manin connection in disguise: genus two curves, (with Jin Cao and S.-T. Yau) Advances in Mathematics Volume 383, 2021.
  11. On reconstructing subvarieties from their periods (Together with Emre Sertoz), To appear in Rendiconti del Circolo Matematico di Palermo Series 2, 2020.
  12. Special Components of Noether-Lefschetz loci , Rendiconti del Circolo Matematico di Palermo Series 2 volume 70, pages 861–874 (2021).
  13. Product formulas for weight two new forms , (together with Younes Nikdelan), Cadernos de IME, No. 14, pages 1-7, 2020
  14. Gauss-Manin connection in disguise: Dwork family, (together with Younes Nikdelan), Journal of Differential Geometry,119 (2021) 73-98
  15. Periods of linear algebraic cycles, (together Roberto villaflor), Pure and Applied Mathematics Quartely, Vol. 14, No. 3-4, 563-577, 2018.
  16. Integral Hodge conjecture for Fermat varieties, (together with Enzo Aljovin and Roberto Villaflor), Journal of Symbolic Computation Volume 95, 2019, Pages 177-184 .    
  17. Manifolds ways to Darboux-Halphen system, (together with J. A. C. Morales, Y. Nikdelan, R. Roychowdhury, M. A. C. Torres), SIGMA, 14, (2018), 003.    
  18. Calabi-Yau modular forms in limit: Elliptic fibrations, (together with B. Haghighat and S.-T. Yau), Communications in Number Theory and Physics, Vol. 11, Number 4, 879-912, 2017    
  19. Limit cycles, Abelian Integrals and Hilbert's 16th Problem (Book), (together with Marco Uribe), 31 Colo'quio Brasileiro de Matema'tica, 2017    
  20. Gauss-Manin connection in disguise: Noether-Lefschetz and Hodge loci, Asian Journal of Mathematics,  Vol.21, No. 3, pp. 463-482, 2017.   
  21. 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.   
  22. Gauss-Manin connection in disguise: Calabi-Yau threefolds (together with Murad Alim, Emanuel Scheidegger, Shing-Tung Yau),    Comm. Math. Phys.  344, (2016), no. 3, 889-914.  
  23. Remarks on a theorem of Perron, (together with C. Camacho),  J. Diff. Equ. 260, 1465-1471, (2016).   
  24. Humbert surfaces and the moduli of lattice polarized K3 surfaces (with Charles Doran, Ursula Whitcher and Andrew Harder), String-Math 2014,  Proceedings of Symposia in Pure Mathematics, Vol. 93, 124-153, (2016). see here for the singular code.      
  25. Quasi-modular forms attached to elliptic curves: Hecke operators. Journal of Number Theory, Volume 157, December 2015, Pages 424–441. The singular code related to the computations in this article are here
  26.  Modular-type functions attached to mirror quintic Calabi-Yau varieties , Mathematische Zeitscrift, Vol 281, Issue 3, pp. 907-929.
  27.  Automorphic forms for triangular groups: Integrality properties (Together with Khosro M. Shokri), Journal of Number Theory, 145:67–78, 2014.
  28.  Foliated neighborhoods of exceptional submanifolds (Together with Cesar Camacho)  Indiana University Mathematical Journal, Vol 63, No. 5, (2014).
  29.   Automorphic forms for triangular groups (Together with Charles Doran, Terry Gannon, Khosro M. Shokri). Communications in Number Theory and Physics,  Vol 7, No. 4, 689-737,2013.  For verification of the equalities in this article I have used this Singular code. I have also written the following html text for the q-expansions appearing in this text.  See also this.
  30.  Quasi-modular forms attached to Hodge structures I , Fields Communication Series, 67, 2013.   
  31.  On Ramanujan relations between Eisenstein series , Manuscripta Mathematicae, Vol. 139, Numbers 3-4 (2012), 495-514. The algorithms in this article are implemented here .  
  32. Quasi-modular forms attached to elliptic curves, I, Annales Math\E9matique Blaise Pascal,  v. 19, p. 307-377, 2012.   My lecture notes at Besse, France 2010 and this  workshop held at IMPA 2011. In 2008 I wrote some pages, which gave origin to the lecture notes. P. Deligne wrote some comments on that pages. 
  33. Eisenstein type series for Calabi-Yau varieties , Nuclear Physics B, 847, (2011) 460–484. You can obtain the Singular code of this article here (see also the older version). See also this for the result of calculations.
  34. Multiple Integrals and Modular Differential Equations, 28 Col\F3quio Brasileiro de Matem\E1tica, p. 168, 2011.  Older versions of this text were written under the title "Modular foliations and periods of hypersurfaces",Version 2008 and Version 2006. The algorithms of this text are implemented in a library of Singular which can be downloaded from this page. 
  35.  Differential modular forms and some analytic relations between Eisenstein series  Ramanujan Journal, 17:53-76, 2008.
  36.  Heun equations coming from geometry (with Stefan Reiter). Bull. Braz. Math. Soc. 43(3), 423-442, 2012.   See the corresponding 38  family  of elliptic curves with 4 singular fibers . See also the library painleve-heun.lib, and pianleve-heun-auxiliary.lib. We use this sequence of commands to calculate the table of the article 
  37.   Painlev\'e VI equations with algebraic solutions and families of curves (with Stefan Reiter), Journal of Experimental Mathematics, Vol. 19, Number 2, 2010, P. 161-173. We have used  painleve-heun.lib  and this sequence of commands to calculate the table of this article. For the original implementation of algorithms used in this article and also calculations of Fuchsian systems in Schlesinger form see here 
  38.  Moduli of polarized Hodge structures  Bull. Soc. Bras. Mat. 39(1), 81-107,2008.
  39.  Mixed Hodge structure of affine hypersurfaces, Ann. Inst. Fourier., Tome 57, (2007), fasicule 3, 775-801.                   
  40. Relative Cohomology with Respect to a Lefschetz Pencil. Jou. Reine und Angew. Math. (Crelle Journal), 594, 175-199, 2006.
  41. Hypergeometric series and Hodge cycles of four dimensional cubic hypersurfaces (Together with S. Reiter), Int. Jou. of Number Theory, Vol. 2, No. 6, 2006.
  42. Transfer operator for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,{\mathbb Z})$ (Together with J. Hilgert and D. Mayer) Math. Proceedings of the Cambridge Philosophical Society, Vol. 139 issue 1, 2005. (math.NT/03032).
  43.          
 


Papers on holomorphic foliations, Hilbert 16-th problem, limit cycles, center problem and topology of fibrations (2000-2010):


  1. On the topology of integrable foliations. , Bol. Soc. Bras. Mat., Vol. 31, No. 3 (2000), 305-336. (math.GT/0206075).
  2. On deformation of foliations with a center in the projective space. An. Acad. Brasil. Ci\^enc. 73 (2001), no. 2, 191--196.
  3.  Abelian integrals in holomorphic foliations Rev. Mat. Iberoamericana, Vol. 20, issue 1 (2004), (math.AG/0203062)  
  4. Center conditions: Rigidity of logarithmic differential equations Journal of Differential Equations, 197 (2004) 197-217 \\ (math.AG/0205068).
  5. Center conditions for polynomial differential equations: discussion of some problems, Kokyuroku 1447, Research Institute for Mathematical Sciences (RIMS), Kyoto University, 2005.
  6. Meromorphic connections on $\pl$ and the multiplicity of Abelian integrals. Journal of Dynamical and Control Systems, Vol. 12, No. 2, April 2006, 217-228.
  7. Calculation of Mixed Hodge structures, Gauss-Manin connections and Picard-Fuchs equations. Real and Complex Singularities, Trends in Mathematics, 247-262, 2006, Birkhauser (proceeding of Sao Carlos Conference in Singularities, CIRM, 2004). (math.AG/0412235). The calculations at the end of this article are done using the following version of foliation1.12.lib . For the changes done after this version see the head of the last version . 
  8. The infinitesimal 16th Hilbert problem in dimension zero (Together with L. Gavrilov) Bull. Sci. Math, 131(2007) 242-257.  
  9. Commuting holonomies and rigidity of holomorphic foliations (with Isao Nakai) Bull. London Math. Soc. 40(2008), 473-478,
  10. On the non-persistence of Hamiltonian identity cycles, (together with L. Gavrilov and I. Nakai), Journal of Differential Equations, Vol. 246, 2706-2723.
  11. Projective limit cycles (Together with Evilson da Silva), Moscow Mathematical Journal, Vol. 9, Number 4, p. 855-866.
  12. On elliptic modular foliations Indagationes Mathematicae19 (2008) No.2 263-286.             
     

Papers on negatively embedded varieties (after H. Grauert 1962):

 
  1. Fibered neighborhoods of curves in surfaces. (Together with C. Camacho and P. Sad) J. Geom. Anal. 13 (2003), no. 1, 55--66. (first page of the published version )
  2. Neighborhoods of analytic varieties. (Together with C. Camacho) Monograf\'\i as del Instituto de Matem\'atica y Ciencias Afines, 35. Instituto de Matem\'atica y Cienc\'\i as Afines, IMCA, Lima; Pontificia Universidad Cat\'olica del Per\'u, Lima, 2003.
  3. The moduli of quasi-homogeneous Stein surface singularities (Together with C. Camacho and B. Scardua), Journal of Geometric Analysis (2009) 19:244-260 
  4. Embedded curves and foliations (Together with Paulo Sad), Publ. Mat. 55 (2011), 401–411.
  5. Foliated neighborhoods of exceptional submanifolds (Together with Cesar Camacho)  Indiana University Mathematical Journal, Vol 63, No. 5, (2014).
  6.  All the works above are going to be collected in the book  Topics in Complex Geometry and Holomorphic foliations 
 

Unpublished manuscripts:

  1. Algebro-Geometric Approach to Abelian Integrals in Differential Equations Papers contributed in honor of Siavash Shahshahani's 60th birthday, {\tt http://sharif.edu/~shahyad/papers.html}
  2. Picard-Lefschetz Theorie der Anordnungen. This is an elementary text written in German. I wrote it in the second year (2002) of my stay at Germany. At that time I was learning German and so it may contain many grammatical errors (more than my texts in english :-) ).
  3. De Rham cohomologies associated to iterated integrals. My talk at "Topology Symposium" Osaka, August 2006.
  4. MPHS I: Differential modular forms,Elliptic curves and Ramanujan foliation The content of this article is ellaborated more and it is divided in two pieces: "Elliptic modular foliations" and "Differential modular forms and some analytic relations between Eisenstein series".
  5. Around Hilbert 16-th problem, The lecture notes of a mini-course given in Santiago-Chile, 17-25 July.
  6. Iterated integrals in holomorphic foliations This is the first version of the article Commuting holonomies and rigidity of holomorphic foliations It contains an exposition of iterated integrals in holomorphic foliations. The referee asked me to cut the expository piece and I did so (otherwise it could not be published :-) )
  7. Lie elements, shuffle products and the $\pi_1$ de Rham theorem (Together with L. Gavrilov and I. Nakai)
  8. A short brief on my my research