| Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
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Course: Several Complex Variables Instructor: Nilanjan Das Room: G25 Level: Postgraduate Time: Currently offered |
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| Syllabus Past Exams Syllabus: • d-bar operator on C, solution of d-bar equation for smooth data with compact support in C. Harmonic functions in C, harmonic conjugates, mean-value property, Poisson kernel and solution of the Dirichlet problem for the unit disc. Subharmonic functions in C, maximum principle, various properties of subharmonic functions. Approximating subharmonic functions by smooth subharmonic functions. Definition of holomorphic functions and mappings of several complex variables. dbar operators in C n. Power series in several complex variables. Cauchy integral formula for holomorphic functions on a polydisc, local power series representation. Uniqueness principle. d-bar equation on C n: 1-forms, (1, 0)-forms and (0, 1)-forms. Solution of the d-bar equation for smooth data with compact support in Cn, application to proving Hartogs phenomenon. Domains of convergence of power series, Reinhardt domains, logarithmically convex domains. Convergence of Taylor series on Reinhardt domains. Convexity with respect to a family of functions. Convex domains in R n, convex exhaustion functions, defining functions for strongly convex domains, continuity principle for convex domains. Definitions and basic properties of plurisubharmonic functions in C n. Definition of pseudoconvex domains in C n. Various equivalent characterizations of pseudoconvexity. The Levi form and Levi pseudoconvexity for domains with C 2-boundary, equivalence with pseudoconvexity. Defining functions for strictly Levi pseudoconvex domains. Holomorphic convexity. Domains of holomorphy. Domain holomorphically convex if and only if it is a domain of holomorphy. Domains of holomorphy are pseudoconvex. Hormander’s solution of the d-bar equation on pseudoconvex domains. Unbounded operators on Hilbert spaces, adjoints, closed operators. Weak partial derivatives of locally integrable functions on R n. d-bar operators on functions and 1-forms. Weighted L 2-spaces of functions, (0, 1)-forms and (0, 2)-forms. Abstract criterion for existence of weak solutions to d-bar problem. Approximation in the graph norm of L 2(0, 1)-forms by smooth (0, 1)-forms with compact support. Existence of weak[’6cf solutions to d-bar problem on pseudoconvex domains. Sobolev spaces, Sobolev embedding, and regularity of solutions to dbar problem on pseudoconvex domains. Solution of the Levi Problem:any pseudoconvex domain is a domain of holomorphy. Suggested Texts : (a) Hormander: An Introduction to Complex Analysis in Several Variables. (b) Krantz: Function Theory of Several Complex Variables. Evaluation:
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[ Semester Schedule ][ SMU ] [Indian Statistical Institute] |