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RESEARCH

My research area is several complex variables and complex geometry. The fundamental objects of complex analysis are complex manifolds, holomorphic functions on them, and holomorphic maps between them. Holomorphic functions can be defined in three equivalent ways as complex-differentiable functions, as convergent power series, and as solutions of the homogeneous Cauchy-Riemann equations. Thus, the very nature of differentiability over the complex numbers gives complex analysis its distinctive character and is the ultimate reason why it is linked to so many areas of mathematics.

More specifically I am interested in polynomial and rational convexity of real submanifolds in complex spaces, geometric properties of holomorphic mappings and functions, holomorphic foliations on Levi-flat hypersurfaces, and other related problems.

PAPERS

Submitted or Preprints

  1. [ArXiv] S. Pinchuk, R. Shafikov, and A. Sukhov. Segre envelopes of singular Levi-flat sets
  2. [ArXiv] S. Pinchuk, R. Shafikov, and A. Sukhov. Dicritical singularities and laminar currents on Levi-flat hypersurfaces. Submitted.
  3. [ArXiv] P. Gupta and R. Shafikov. Rational and Polynomial Density on Compact Real Manifolds. Submitted.
  4. [ArXiv] R. Shafikov, A. Sukhov. Discs in hulls of real immersions to Stein manifolds.
  5. [ArXiv] R. Shafikov, A. Sukhov. Approximation on real surfaces and rational convexity of Lagrangian inclusions. Submitted

Published or Accepted

  1. [ArXiv] D. Chakrabarti and R. Shafikov. Distributional boundary values of holomorphic functions on product domains To appear in Math. Z.
  2. [ArXiv] O. Mitrea and R. Shafikov. Open Whitney umbrellas are locally polynomially convex To appear in Proc. of AMS.
  3. [ArXiv] D. Chakrabarti and R. Shafikov. Distributional Boundary Values: Some New Perspectives. To appear in Contemporary Mathematics.
  4. [ArXiv] R. Shafikov, A. Sukhov. Lagrangian inclusion with an open Whitney umbrella is rationally convex. To appear in Contemporary Mathematics.
  5. [PDF] R. Shafikov and A. Sukhov. Germs of singular Levi-flat hypersurfaces and holomorphic foliations. Comment. Math. Helv. 90 (2015), no. 2, 479 - 502.
  6. [ArXiv] I. Kossovskiy and R. Shafikov. Divergent CR-Equivalences and Meromorphic Differential Equations. To appear in J. of European Math. Soc. (JEMS).
  7. [PDF] I. Kossovskiy and R. Shafikov. Analytic Differential Equations and Spherical Real Hypersurfaces J. Differential Geom. 102 (2016), no. 1, 67–126.
  8. [PDF] S. Pinchuk and R. Shafikov. Critical sets of proper holomorphic mappings. Proc. Amer. Math. Soc. 143 (2015), no. 10, 4335–4345.
  9. [PDF] R. Shafikov and A. Sukhov. Polynomially convex hulls of singular real manifolds. Trans. Amer. Math. Soc. 368 (2016), no. 4, 2469–2496.
  10. [PDF] I. Kossovskiy and R. Shafikov. Analytic Continuation of Holomorphic Mappings From Non-minimal Hypersurfaces. Indiana Univ. Math. J. 62 (2013), no. 6, 1891–1916
  11. [PDF] R. Shafikov and A. Sukhov. Local Polynomial Convexity of the Unfolded Whitney Umbrella in $\mathbb C^2$. Int. Math. Res. Not. IMRN 2013, no. 22, 5148–5195.
  12. [PDF] Adamus, J., Randriambololona, S., Shafikov, R. Tameness of complex dimension in real analytic sets. Canadian J. Math., 65 (2013), no. 4, 721–739.
  13. [PDF] Shafikov, R., Verma, K. Holomorphic mappings between domains in $\mathbb C^2$. Canad. J. Math. 64(2), 2012, pp. 429--454.
  14. [PDF] Adamus, J., Shafikov, R. On the holomorphic closure dimension of real analytic sets. Trans. Amer. Math. Soc. 363 (2011), no 11, 5761-5772.
  15. [PDF] Chakrabarti, D., Shafikov, R. CR functions on Subanalytic Hypersurfaces. Indiana Univ. Math. J. 59 No. 2 (2010), 459–494
  16. [PDF] Lárusson F., Shafikov, R. Schlicht envelopes of holomorphy and foliations by lines. J. Geom. Anal. 19 (2009), no. 2, 373--389.
  17. [PDF] Chakrabarti, D., Shafikov, R. Holomorphic Extension of CR Functions from Quadratic Cones. Math. Ann. 341 (2008), 543-573.
  18. [PDF] Shafikov, R., Verma, K. Extension of holomorphic maps between real hypersurfaces of different dimension. Annales de l'institut Fourier, 57 no. 6 (2007), p. 2063-2080
  19. [PDF] Nemirovki, S., Shafikov, R. Conjectures of Cheng and Ramadanov. Russian Math. Surveys, 61 (4) (2006), 780-782.
  20. [PDF] Shafikov, R. Real Analytic Sets in Complex Spaces and CR Maps. Math. Z. 256 (2007), no. 4, 757--767.
  21. [PDF] Shafikov, R., Verma, K. Boundary regularity of correspondences in $\mathbb C^n$. IAS. Proc. Indian Acad. Sci. (Math. Sci.) Vol. 116, No. 1, 2006, pp. 1-12.
  22. [PDF] Nemirovski, S., Shafikov, R. Uniformization of strictly pseudoconvex domains, II. Izvestiya: Mathematics 69:6 (2005) p. 1203-1210.
  23. [PDF] Nemirovski, S., Shafikov, R. Uniformization of strictly pseudoconvex domains, I. Izvestiya: Mathematics 69:6 (2005) p. 1189-1202.
  24. [PDF] Hill, C. Denson, Shafikov, R. Holomorphic correspondences between CR manifolds Indiana Univ. Math. J. 54 No. 2 (2005), 417-442.
  25. [PDF] Shafikov, R., Wolf, C. Stable sets, hyperbolicity and dimension Discrete Contin. Dynam. Systems. 12 no 3 (2005), 403-412.
  26. [PDF] Shafikov, R., Verma, K. A Local Extension Theorem for Proper Holomorphic Mappings in $\mathbb C^2$. J. Geom. Anal. 13 (2003), no. 4, 697 - 714.
  27. [PDF] Shafikov, R. Analytic Continuation of Holomorphic Correspondences and Equivalence of Domains in $\mathbb C^n$. Invent. Math. 152 (2003), 665 - 682.
  28. [PDF] Shafikov, R., Wolf, C. Filtrations, hyperbolicity and dimension for polynomial automorphisms of $\mathbb C^n$. Michigan Math. J. 51 (2003), no. 3, 631--649.
  29. [PDF] Shafikov, R. On Boundary Regularity of Proper Holomorphic Mappings. Math. Z. 242 (2002), 517-528.
  30. [PDF] Shafikov, R. Analytic Continuation of Germs of Holomorphic Mappings Between Real Hypersurfaces in $\mathbb C^n$. Mich. Math. J. 47 (2000), 133-149.