My primary research area is Several Complex Variables and CR 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.
CR geometry is a modern branch of mathematics which roughly speaking studies real submanifolds in complex spaces and functions on them that satisfy the tangential Cauchy-Riemann equations. In particular, it relates the geometry of the boundary of a domain in a complex space to the function theory on the domain. These considerations lead to a deep understanding of pseudoconvexity, a complex variables analogue of convexity, and to the study of the tangential Cauchy-Riemann equations.
- [ArXiv] D. Chakrabarti and R. Shafikov. Distributional boundary values of holomorphic functions on product domains Submitted
- [ArXiv] R. Shafikov, A. Sukhov. Discs in hulls of real immersions to Stein manifolds. Submitted
- [ArXiv] R. Shafikov, A. Sukhov. Approximation on real surfaces and rational convexity of Lagrangian inclusions. Submitted
- [ArXiv] R. Shafikov, A. Sukhov. Lagrangian inclusion with an open Whitney umbrella is rationally convex. Submitted
- [ArXiv] R. Shafikov and A. Sukhov. Germs of singular Levi-flat hypersurfaces and holomorphic foliations. To appear in Commentarii Mathematici Helvetici.
- [ArXiv] I. Kossovskiy and R. Shafikov. Divergent CR-Equivalences and Meromorphic Differential Equations. To appear in J. of European Math. Soc. (JEMS).
- [ArXiv] I. Kossovskiy and R. Shafikov. Analytic Differential Equations and Spherical Real Hypersurfaces To appear in J. of Diff. Geometry.
- [PDF] S. Pinchuk and R. Shafikov. Critical sets of proper holomorphic mappings. To appear in Proceedings of AMS.
- [PDF] R. Shafikov and A. Sukhov. Polynomially convex hulls of singular real manifolds. To appear in Transactions of AMS.
- [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
- [PDF] R. Shafikov and A. Sukhov. Local Polynomial Convexity of the Unfolded Whitney Umbreall in $\mathbb C^2$. Int. Math. Res. Not. IMRN 2013, no. 22, 5148–5195.
- [PDF] Adamus, J., Randriambololona, S., Shafikov, R. Tameness of complex dimension in real analytic sets. Canadian J. Math., 65 (2013), no. 4, 721–739.
- [PDF] Shafikov, R., Verma, K. Holomorphic mappings between domains in $\mathbb C^2$. Canad. J. Math. 64(2), 2012, pp. 429--454.
- [PDF] Adamus, J., Shafikov, R. On the holomorphic closure dimension of real analytic sets. Trans. Amer. Math. Soc. 363 (2011), no 11, 5761-5772.
- [PDF] Chakrabarti, D., Shafikov, R. CR functions on Subanalytic Hypersurfaces. Indiana Univ. Math. J. 59 No. 2 (2010), 459–494
- [PDF] Lárusson F., Shafikov, R. Schlicht envelopes of holomorphy and foliations by lines. J. Geom. Anal. 19 (2009), no. 2, 373--389.
- [PDF] Chakrabarti, D., Shafikov, R. Holomorphic Extension of CR Functions from Quadratic Cones. Math. Ann. 341 (2008), 543-573.
- [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
- [PDF] Nemirovki, S., Shafikov, R. Conjectures of Cheng and Ramadanov. Russian Math. Surveys, 61 (4) (2006), 780-782.
- [PDF] Shafikov, R. Real Analytic Sets in Complex Spaces and CR Maps. Math. Z. 256 (2007), no. 4, 757--767.
- [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.
- [PDF] Nemirovski, S., Shafikov, R. Uniformization of strictly pseudoconvex domains, II. Izvestiya: Mathematics 69:6 (2005) p. 1203-1210.
- [PDF] Nemirovski, S., Shafikov, R. Uniformization of strictly pseudoconvex domains, I. Izvestiya: Mathematics 69:6 (2005) p. 1189-1202.
- [PDF] Hill, C. Denson, Shafikov, R. Holomorphic correspondences between CR manifolds Indiana Univ. Math. J. 54 No. 2 (2005), 417-442.
- [PDF] Shafikov, R., Wolf, C. Stable sets, hyperbolicity and dimension Discrete Contin. Dynam. Systems. 12 no 3 (2005), 403-412.
- [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.
- [PDF] Shafikov, R. Analytic Continuation of Holomorphic Correspondences and Equivalence of Domains in $\mathbb C^n$. Invent. Math. 152 (2003), 665 - 682.
- [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.
- [PDF] Shafikov, R. On Boundary Regularity of Proper Holomorphic Mappings. Math. Z. 242 (2002), 517-528.
- [PDF] Shafikov, R. Analytic Continuation of Germs of Holomorphic Mappings Between Real Hypersurfaces in $\mathbb C^n$. Mich. Math. J. 47 (2000), 133-149.