Mini-course by Xiaojun Huang

Title: On Several Extension Problems in Several Complex Variables

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Lecture Notes

Part 1: Lewy extension theorem and method of Bishop disks
Part 2: Kernel’s extension theory for multiple-valued holomorphic functions
Part 3: Calabi extension theory and Bergman metrics

Mini-course by Dror Varolin

Title: Introduction to L^2 methods in complex analysis and geometry

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Lecture Notes

Part 1: Introduction to the geometry of holomorphic vector bundles: Introduce the notion of holomorphic vector bundle, Hermitian metrics, Chern connection, Kahler Manifolds; examples.
Part 2: Two fundamental PDE in complex analytic geometry: Statement of the Hodge Theorem, idea of proof from Lichnerowicz formula, the Bochner-Kodaira Identity, Hormander’s theorem (including versions by Skoda and Demailly), a word about singular Hermitian metrics.
Part 3: Applications to complex geometry: Kodaira Embedding Theorem (explanation and proof), Stein manifolds and the Levi Problem (ideas of proof).

Mini-course by Dmitri Zaitsev

Title: Geometry of finite types and tower multitype.

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Part 1: Introduction to geometric tools in proving global regularity of the d-bar-Neumann operator. Generalized stratifications with convexity properties and potential-theoretic implications. Towers and tower multitype for real hypersurfaces and their main structure properties. Application to construction of generalized stratifications for finite Levi types.
Part 2: Finite commutator types. Special subbundles. Real and complex formal orbits. Formal submanifolds. Complex-tangential property for real orbits. Applications to finite tower multitype.
Part 3: Orders of contact. Regular and singular contact types. Formal CR submanifolds and their intrinsic complexifications. Formal Huang-Yin condition. A formal variant of a result of Diederich-Fornaess. Application to finite regular contact types.
Part 4: Proof of the formal variant of a Diederich-Fornaess result. Approximation of formal maps and vector fields by smooth ones modulo suitable ideals. Relative jets and contact orders. Even relative contact orders for pseudoconvex hypersurfaces. Supertangent vector fields. A fundamental Lie algebra property for supertangent vector fields and completion of the proof.