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.
- Debraj Chakrabarti (2006-2008). Currently an associate professor at Central Michigan University, USA
- Ilya Kossovskiy (2010-2013). Currently an associate professor at Masaryk University, Brno, Czech Republic.
- Purvi Gupta (2015-2017). Currently Hill assistant professor at Rugters University, USA.
Former and present PhD students:
- Octavian Mitrea (2015-2019). Currently post-doc at Oklahoma State University
- Luke Broemeling (2016-present).
- P. Gupta and R. Shafikov. Polynomially Convex Embeddings of Even-Dimensional Compact Manifolds.
Papers: published and accepted
- S. Pinchuk, R. Shafikov, and A. Sukhov. On dicritical singularities of Levi-flat set. To appear in Arkiv för Matematik.
- S. Nemirovski and R. Shafikov. Uniformization and Steinness. Canad. Math. Bull. 61 (2018), no. 3, 637–639.
- S. Pinchuk, R. Shafikov, A. Sukhov. Some aspects of holomorphic mappings: a survey. Proc. Steklov Inst. Math. 298 (2017), no. 1, 212–247.
- S. Pinchuk, R. Shafikov, and A. Sukhov. Dicritical singularities and laminar currents on Levi-flat hypersurfaces. Izv. Ross. Akad. Nauk Ser. Mat. 81 (2017), no. 5, 150–164; translation in Izv. Math. 81 (2017), no. 5, 1030–1043
- P. Gupta and R. Shafikov. Rational and Polynomial Density on Compact Real Manifolds. Internat. J. Math. Vol. 28, No. 5 (2017).
- R. Shafikov, A. Sukhov. Discs in hulls of real immersions to Stein manifolds. To appear in Proc. Steklov Inst. Math.
- R. Shafikov, A. Sukhov. Rational approximation and Lagrangian inclusions. Enseign. Math. 62 (2016), no. 3-4, 487–499.
- D. Chakrabarti and R. Shafikov. Distributional boundary values of holomorphic functions on product domains.
Math. Z. 286 (2017), no. 3-4, 1145–1171.
- O. Mitrea and R. Shafikov. Open Whitney umbrellas are locally polynomially convex. Proc. Amer. Math. Soc. 144 (2016), no. 12, 5319–5332.
- D. Chakrabarti and R. Shafikov. Distributional Boundary Values: Some New Perspectives. Analysis and geometry in several complex variables, 65–70, Contemp. Math., 681, Amer. Math. Soc., Providence, RI, 2017.
- R. Shafikov, A. Sukhov. Lagrangian inclusion with an open Whitney umbrella is rationally convex. Topics in several complex variables, 71–75, Contemp. Math., 662, Amer. Math. Soc., Providence, RI, 2016.
- R. Shafikov and A. Sukhov. Germs of singular Levi-flat hypersurfaces and holomorphic foliations. Comment. Math. Helv. 90 (2015), no. 2, 479 – 502.
- I. Kossovskiy and R. Shafikov. Divergent CR-Equivalences and Meromorphic Differential Equations. J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2785–2819.
- I. Kossovskiy and R. Shafikov. Analytic Differential Equations and Spherical Real Hypersurfaces. J. Differential Geom. 102 (2016), no. 1, 67–126.
- S. Pinchuk and R. Shafikov. Critical sets of proper holomorphic mappings. Proc. Amer. Math. Soc. 143 (2015), no. 10, 4335–4345.
- R. Shafikov and A. Sukhov. Polynomially convex hulls of singular real manifolds. Trans. Amer. Math. Soc. 368 (2016), no. 4, 2469–2496.
- I. Kossovskiy and R. Shafikov. Analytic Continuation of Holomorphic Mappings From Non-minimal Hypersurfaces.
Indiana Univ. Math. J. 62 (2013), no. 6, 1891–1916
- 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.
- Adamus, J., Randriambololona, S., Shafikov, R. Tameness of complex dimension in real analytic sets. Canadian J. Math., 65 (2013), no. 4, 721–739.
- Shafikov, ., Verma, K. Holomorphic mappings between domains in \(\mathbb C^2\). Canad. J. Math. 64(2), 2012, pp. 429–454.
- Adamus, J., Shafikov, R. On the holomorphic closure dimension of real analytic sets. Trans. Amer. Math. Soc. 363 (2011), no 11, 5761-5772.
- Chakrabarti, D., Shafikov, R. CR functions on Subanalytic Hypersurfaces. Indiana Univ. Math. J. 59 No. 2 (2010), 459–494
- Lárusson F., Shafikov, R. Schlicht envelopes of holomorphy and foliations by lines. J. Geom. Anal. 19 (2009), no. 2, 373–389.
- Chakrabarti, D., Shafikov, R. Holomorphic Extension of CR Functions from Quadratic Cones. Math. Ann. 341 (2008), 543-573.
- 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. NOTE: this paper contains errors.
- Nemirovki, S., Shafikov, R. Conjectures of Cheng and Ramadanov. Russian Math. Surveys, 61 (4) (2006), 780-782.
- Shafikov, R. Real Analytic Sets in Complex Spaces and CR Maps. Math. Z. 256 (2007), no. 4, 757–767.
- 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.
- Nemirovski, S., Shafikov, R. Uniformization of strictly pseudoconvex domains, II. Izvestiya: Mathematics 69:6 (2005) p. 1203-1210.
- Nemirovski, S., Shafikov, R. Uniformization of strictly pseudoconvex domains, I. Izvestiya: Mathematics 69:6 (2005) p. 1189-1202.
- Hill, C. Denson, Shafikov, R. Holomorphic correspondences between CR manifolds Indiana Univ. Math. J. 54 No. 2 (2005), 417-442.
- Shafikov, R., Wolf, C. Stable sets, hyperbolicity and dimension Discrete Contin. Dynam. Systems. 12 no 3 (2005), 403-412.
- 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.
- Shafikov, R. Analytic Continuation of Holomorphic Correspondences and Equivalence of Domains in \(\mathbb C^n\). Invent. Math. 152 (2003), 665 – 682.
- Shafikov, R., Wolf, C. Filtrations, hyperbolicity and dimension for polynomial automorphisms of \(\mathbb C^n\). Michigan Math. J. 51 (2003), no. 3, 631–649.
- Shafikov, R. On Boundary Regularity of Proper Holomorphic Mappings.
Math. Z. 242 (2002), 517-528.
- Shafikov, R. Analytic Continuation of Germs of Holomorphic Mappings Between Real Hypersurfaces in \(\mathbb C^n\). Mich. Math. J. 47 (2000), 133-149.