Topics in several complex variables first USA-Uzbekistan Conference on Analysis and Mathematical Physics, May 20-23, 2014, California State University, Fullerton, California, CA

This volume contains the proceedings of the Special Session on Several Complex Variables, which was held during the first USA-Uzbekistan Conference on Analysis and Mathematical Physics from May 20–23, 2014, at California State University, Fullerton. This volume covers a wide variety of topics in pluripotential theory, symplectic geometry and almost complex structures, integral formulas, holomorphic extension, and complex dynamics. In particular, the reader will find articles on Lagrangian submanifolds and rational convexity, multidimensional residues, S-parabolic Stein manifolds, Segre varieties, and the theory of quasianalytic functions. Cover 1 Title page 4 Contents 6 Preface 8 Polynomials on parabolic manifolds 10 1. Introduction 10 2. Parabolic manifolds 12 3. Aspects of pluripotential theory on S-parabolic manifolds 15 4. Example 18 5. Regular parabolic manifolds 24 References 29 Pseudoholomorphic discs and symplectic structures in Hilbert space 32 1. Introduction 32 2. Almost complex structures on Hilbert spaces 34 3. Some properties of the Cauchy integral 39 4. Local existence and regularity of pseudoholomorphic discs 44 5. Boundary value problems for J-holomorphic discs 46 6. Non-squeezing for symplectic transformations 50 7. Appendix I: Morrey’s embedding 52 8. Appendix II: Proof of Theorem 3.2 53 References 56 Pluripolarity of graphs of quasianalytic functions of several complex variables 60 1. Introduction 60 2. Preliminaries 63 3. Pluripolarity of graphs of Gonchar quasianalytic functions of several variables 70 4. Pluripolarity of graphs of algebroid functions 72 5. Pluripolarity of graphs of Denjoy quasianalytic functions of several variables 73 References 76 Lagrangian inclusion with an open Whitney umbrella is rationally convex 80 1. Introduction 80 2. Proof of Theorem 1 81 References 83 On systems of non-algebraic equations in Cn 86 Introduction 86 1. Preliminaries 86 2. Calculation of residue integrals 87 3. Power sums 90 4. Finding the sums of some multivariate series 91 References 97 Integral formulas for some matrix domains 98 1. Introduction 98 2. Results 99 3. Open problems 104 References 104 Nevanlinna’s characteristic functions with complex Hessian potential 106 1. Introduction 106 2. m-sh functions 107 3. Properties of the order function 108 4. Upper bounds for k -order functions 110 References 113 Holomorphic continuation of functions in the domains with singular boundaries 116 References 124 Segre varieties and uniform estimates for harmonic functions 126 1. Introduction 126 2. Uniform estimates for harmonic functions 126 3. Proof of Theorem 1 128 References 129 Uniqueness property for Gonchar quasianalytic functions of several variables 130 1. Introduction 130 2. Main result 130 3. A theorem of Plesniak 134 Acknowledgement 137 References 137 Capacity dimension of the Perez-Marco set 140 1. Introduction 140 2. Main result 141 References 146 Analogue of the Lelong’s theorem for m-wsh functions 148 Introduction 148 1. m-wsh functions and their properties 149 2. A criteria of pluriharmonicity 150 References 153 Pluriregular compacts in Pn 154 Introduction 154 1. Quasiplurisubharmonic functions 155 2. Green function and pluriregularity in Pn 157 3. Appendix. Pluriregularity of the Julia set 163 References 164 Back Cover 168 Polynomials On Parabolic Manifolds / Alexandre Sukhov And Alexander Tumanov -- Pseudoholomorphic Discs And Symplectic Structures In Hilbert Space / Zafar Ibragimov -- Pluripolarity Of Graphs Of Quasianalytic Functions Of Several Complex Variables / Rasul Shafikov And Alexandre Sukhov -- Lagrangian Inclusion With An Open Whitney Umbrella Is Rationally Convex / Aleksandr M. Kytmanov And Olga V. Khodos -- On Systems Of Non-algebraic Equations In Cn / Gulmirza Khudayberganov, Uktam Rakhmonov And Zokir Matyakubov -- Integral Formulas For Some Matrix Domains / Bakhrom Abdullaev -- Nevanlinna's Characteristic Functions With Complex Hessian Potential / D. Kh. Djumabaev -- Holomorphic Continuation Of Functions In The Domains With Singular Boundaries / Sergey Pinchuk And Klas Diederich -- Segre Varieties And Uniform Estimates For Harmonic Functions / Sevdiyor Imomkulov And Zafar Ibragimov -- Uniqueness Property For Gonchar Quasianalytic Functions Of Several Variable / Karim Rakhimov -- Capacity Dimension Of The Perez-marco Set / B. I. Abdullaev, A. A. Atamuratov And M. D. Vaisova -- Analogue Of The Lelong's Theorem For M-wsh Functions / Azimbay Sadullaev. Zair Ibragimov [and Three Others], Editors. Includes Bibliographical References.