ICTS

In these talks we present some basic definitions, examples and results required to understand complex analysis of several variables. Some topics to be covered are: elementary properties of holomorphic functions of several variables, Hormander's theorem in one variable, basics of complex vector bundles and connections on them.

In these talks we present some basic definitions, examples and results required to understand complex analysis of several variables. Some topics to be covered are: elementary properties of holomorphic functions of several variables, Hormander's theorem in one variable, basics of complex vector bundles and connections on them.

Quadrature domains are those on which integrable holomorphic functions satisfy a generalized mean value equality. The purpose of this series of talks will be to discuss several remarkable properties that these domains possess.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

In these talks we present some basic definitions, examples and results required to understand complex analysis of several variables. Some topics to be covered are: elementary properties of holomorphic functions of several variables, Hormander's theorem in one variable, basics of complex vector bundles and connections on them.

Quadrature domains are those on which integrable holomorphic functions satisfy a generalized mean value equality. The purpose of this series of talks will be to discuss several remarkable properties that these domains possess.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry.  After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds.  We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu.  The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi.  Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles.  An application is a proof of a special case of the L^2 extension theorem but with optimal constant.  We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

In these talks we present some basic definitions, examples and results required to understand complex analysis of several variables. Some topics to be covered are: elementary properties of holomorphic functions of several variables, Hormander's theorem in one variable, basics of complex vector bundles and connections on them.

Quadrature domains are those on which integrable holomorphic functions satisfy a generalized mean value equality. The purpose of this series of talks will be to discuss several remarkable properties that these domains possess.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

In these talks we present some basic definitions, examples and results required to understand complex analysis of several variables. Some topics to be covered are: elementary properties of holomorphic functions of several variables, Hormander's theorem in one variable, basics of complex vector bundles and connections on them.

Quadrature domains are those on which integrable holomorphic functions satisfy a generalized mean value equality. The purpose of this series of talks will be to discuss several remarkable properties that these domains possess.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

We illustrate the use of a simple normal families argument to prove that the space of noncon stant holomorphic mappings is finite under certain scenarios. In particular, this argument can be used to give a straightforward proof of a classical theorem of de Franchis that states that there are only finitely many non constant holomorphic mappings between two compact hyperbolic Riemann surfaces.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

Existence of non-wandering Fatou component for rational maps in one complex variable is a celebrated result, known due to Sullivan. In this talk, I will discuss the analog questions in higher dimensions, i.e., in $C^2$ for Henon maps and polynomial maps. Next, I will discuss on a non-wandering phenomenon of certain Henon maps, that can be extended to higher dimensions, i.e., on $C^3$ and higher dimensions for the class of shift-like maps.

This series of lectures provides an introduction to L^2 estimates for dbar and their use in complex analysis and analytic geometry. After a review of basic complex geometry we will discuss Hörmander's Theorem on compact, and more generally complete, Kähler manifolds. We shall then develop the twisted estimates of Donnelly-Feffernan-Ohsawa, following an approach that we first learned from Siu. The twisted methods will be applied to prove an L^2 extension theorem that generalizes the classical theorem of Ohsawa and Takegoshi. Finally we will discuss a theorem of Berndtsson regarding the positivity of the curvature of certain vector bundles obtained as L^2 sections of the fibers of a family of holomorphic vector bundles. An application is a proof of a special case of the L^2 extension theorem but with optimal constant. We will end by showing that the latter theorem is actually equivalent to Berndtsson's Theorem.

These lectures will give an introduction to compactness in the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory and then compactness of the $\overline{\partial}$--Neumann operator will be treated in some detaisl. I hope to present various open problems. Prerequisites are a solid background in basic complex and functional analysis, some facility with differential forms, and perhaps at least a fleeting acquaintance with elliptic partial differential equations. Some knowledge in several complex variables will be helpful, if only for motivation.

The plan of the lectures is as follows:

Lecture 1: The basic $\mathcal{L}^{2}$ theory of $\overline{\partial}$ on domains in $\mathbb{C}^{n}$
Lecture 2 : Complex Laplacian and $\overline{\partial}$--Neumann operator.
Lecture 3 : Compactness of the $\overline{\partial}$--Neumann operator.
Lecture 4 : Hartogs domains in $\mathbb{C}^{n}$
Lecture 5 : Obstructions to compactness.

The title of this talk is taken from a paper by Charles Fefferman in 1976, where he initiated the geometry of strictly pseudoconvex domains based on Einstein equation. This paper is elementary but contains many ideas developed later. In 1974, Fefferman proved that two strictly pseudo convex domains are biholomorphic if and only if the boundaries are diffeomorphic by a map preserving the tangential Caucy-Riemann equations — the geometry of the latter setting is called CR geometry. The paper in 1976 aimed to give explicit between of the biholomorphic and CR geometries. Our first goal is to describe the boundary behavior of the Bergman kernel in terms of the solution of the Monge–Ampère equation associated with the domain; the second goal is to construct global invariants of the domain by integrating the curvature of the Einstein metric given by the Monge–Ampère solution. Fefferman’s contribution is summarized in his lecture notes: Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in C^n, Bull. Amer. Math. Soc. (1983), 125-322. We plan to include many progresses after that.

In the first of two talks, I will describe an effective version of the Kodaira embedding theorem, proved by Simon Donaldson and Song Sun. In particular, a Fano manifold with a unit volume Kahler-Einstein metric and uniformly bounded scalar curvature can be embedded in a projective space of uniformly bounded dimension. The proof of Donaldson-Sun combines L^2 methods with the structure theory of Gromov-Hausdorff limits of Riemannian manifolds. In the second talk, I will give an overview of the work of Chen-Donaldson-Sun on the applications of these techniques to the characterisation of Fano manifolds admitting Kahler-Einstein metrics.

Quadrature domains are those domains for which the integral of every function in the Bergman class of functions can be written as a fixed quadrature identity. In this talk, we explore the classical results known about quadrature domains in the planar case and discuss the development of theory in higher dimensions. We shall see a density theorem for quadrature domains in a particular class of product domains. We shall also discuss a recent result disproving a conjecture by Steven Bell on one point one degree quadrature domains.

The title of this talk is taken from a paper by Charles Fefferman in 1976, where he initiated the geometry of strictly pseudoconvex domains based on Einstein equation. This paper is elementary but contains many ideas developed later. In 1974, Fefferman proved that two strictly pseudo convex domains are biholomorphic if and only if the boundaries are diffeomorphic by a map preserving the tangential Caucy-Riemann equations — the geometry of the latter setting is called CR geometry. The paper in 1976 aimed to give explicit between of the biholomorphic and CR geometries. Our first goal is to describe the boundary behavior of the Bergman kernel in terms of the solution of the Monge–Ampère equation associated with the domain; the second goal is to construct global invariants of the domain by integrating the curvature of the Einstein metric given by the Monge–Ampère solution. Fefferman’s contribution is summarized in his lecture notes: Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in C^n, Bull. Amer. Math. Soc. (1983), 125-322. We plan to include many progresses after that.

These lectures will give an introduction to compactness in the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory and then compactness of the $\overline{\partial}$--Neumann operator will be treated in some detaisl. I hope to present various open problems. Prerequisites are a solid background in basic complex and functional analysis, some facility with differential forms, and perhaps at least a fleeting acquaintance with elliptic partial differential equations. Some knowledge in several complex variables will be helpful, if only for motivation. The plan of the lectures is as follows: Lecture 1: The basic $\mathcal{L}^{2}$ theory of $\overline{\partial}$ on domains in $\mathbb{C}^{n}$ Lecture 2 : Complex Laplacian and $\overline{\partial}$--Neumann operator. Lecture 3 : Compactness of the $\overline{\partial}$--Neumann operator. Lecture 4 : Hartogs domains in $\mathbb{C}^{n}$ Lecture 5 : Obstructions to compactness.

In the first of two talks, I will describe an effective version of the Kodaira embedding theorem, proved by Simon Donaldson and Song Sun. In particular, a Fano manifold with a unit volume Kahler-Einstein metric and uniformly bounded scalar curvature can be embedded in a projective space of uniformly bounded dimension. The proof of Donaldson-Sun combines L^2 methods with the structure theory of Gromov-Hausdorff limits of Riemannian manifolds. In the second talk, I will give an overview of the work of Chen-Donaldson-Sun on the applications of these techniques to the characterisation of Fano manifolds admitting Kahler-Einstein metrics.

In this talk, we will consider the class of surfaces with isolated degenerated CR singularity in C2 which can be pulled back locally by a proper holomorphic map from C to C2 as a union of three pairwise transverse totally real surfaces. After normalization these surfaces form a oneparameter family. For this class of surfaces, we propose a Bishoptype trichotomy: elliptic, parabolic and hyperbolic surfaces. Here we focus on non parabolic case only. We show that elliptic surfaces are not locally polynomially convex. In certain cases, the local polynomial hull contains a ball. We also discuss polynomial convexity when the surfaces are hyperbolic.

These lectures will give an introduction to compactness in the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory and then compactness of the $\overline{\partial}$--Neumann operator will be treated in some detaisl. I hope to present various open problems. Prerequisites are a solid background in basic complex and functional analysis, some facility with differential forms, and perhaps at least a fleeting acquaintance with elliptic partial differential equations. Some knowledge in several complex variables will be helpful, if only for motivation. The plan of the lectures is as follows: Lecture 1: The basic $\mathcal{L}^{2}$ theory of $\overline{\partial}$ on domains in $\mathbb{C}^{n}$ Lecture 2 : Complex Laplacian and $\overline{\partial}$--Neumann operator. Lecture 3 : Compactness of the $\overline{\partial}$--Neumann operator. Lecture 4 : Hartogs domains in $\mathbb{C}^{n}$ Lecture 5 : Obstructions to compactness.

The title of this talk is taken from a paper by Charles Fefferman in 1976, where he initiated the geometry of strictly pseudoconvex domains based on Einstein equation. This paper is elementary but contains many ideas developed later. In 1974, Fefferman proved that two strictly pseudo convex domains are biholomorphic if and only if the boundaries are diffeomorphic by a map preserving the tangential Caucy-Riemann equations — the geometry of the latter setting is called CR geometry. The paper in 1976 aimed to give explicit between of the biholomorphic and CR geometries. Our first goal is to describe the boundary behavior of the Bergman kernel in terms of the solution of the Monge–Ampère equation associated with the domain; the second goal is to construct global invariants of the domain by integrating the curvature of the Einstein metric given by the Monge–Ampère solution. Fefferman’s contribution is summarized in his lecture notes: Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in C^n, Bull. Amer. Math. Soc. (1983), 125-322. We plan to include many progresses after that.

The title of this talk is taken from a paper by Charles Fefferman in 1976, where he initiated the geometry of strictly pseudoconvex domains based on Einstein equation. This paper is elementary but contains many ideas developed later. In 1974, Fefferman proved that two strictly pseudo convex domains are biholomorphic if and only if the boundaries are diffeomorphic by a map preserving the tangential Caucy-Riemann equations — the geometry of the latter setting is called CR geometry. The paper in 1976 aimed to give explicit between of the biholomorphic and CR geometries. Our first goal is to describe the boundary behavior of the Bergman kernel in terms of the solution of the Monge–Ampère equation associated with the domain; the second goal is to construct global invariants of the domain by integrating the curvature of the Einstein metric given by the Monge–Ampère solution. Fefferman’s contribution is summarized in his lecture notes: Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in C^n, Bull. Amer. Math. Soc. (1983), 125-322. We plan to include many progresses after that.

These lectures will give an introduction to compactness in the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory and then compactness of the $\overline{\partial}$--Neumann operator will be treated in some detaisl. I hope to present various open problems. Prerequisites are a solid background in basic complex and functional analysis, some facility with differential forms, and perhaps at least a fleeting acquaintance with elliptic partial differential equations. Some knowledge in several complex variables will be helpful, if only for motivation. The plan of the lectures is as follows: Lecture 1: The basic $\mathcal{L}^{2}$ theory of $\overline{\partial}$ on domains in $\mathbb{C}^{n}$ Lecture 2 : Complex Laplacian and $\overline{\partial}$--Neumann operator. Lecture 3 : Compactness of the $\overline{\partial}$--Neumann operator. Lecture 4 : Hartogs domains in $\mathbb{C}^{n}$ Lecture 5 : Obstructions to compactness.

Given a smoothly bounded pseudoconvex domain, being able to construct a holomorphic peak function associated to each of its boundary points has a number of important consequences. Such constructions are, in general, very difficult on weakly pseudoconvex domains. Typically, they involve two steps:

1) The construction of a local peak function, which involves a delicate analysis of the geometry of the boundary around the point in question

2) The extension of the local peak function to the whole of the given domain. The second step involves the solution of an inhomogeneous Cauchy--Riemann equation.

In talk, we shall discuss a principle for globally extending local holomorphic peak functions for a family of domains that includes most classes of pseudoconvex domains for which peak-function constructions are of interest. Time permitting, we shall also take a look at some problems whose solutions rely on the ability to construct holomorphic peak functions.

These lectures will give an introduction to compactness in the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory and then compactness of the $\overline{\partial}$--Neumann operator will be treated in some detaisl. I hope to present various open problems. Prerequisites are a solid background in basic complex and functional analysis, some facility with differential forms, and perhaps at least a fleeting acquaintance with elliptic partial differential equations. Some knowledge in several complex variables will be helpful, if only for motivation. The plan of the lectures is as follows: Lecture 1: The basic $\mathcal{L}^{2}$ theory of $\overline{\partial}$ on domains in $\mathbb{C}^{n}$ Lecture 2 : Complex Laplacian and $\overline{\partial}$--Neumann operator. Lecture 3 : Compactness of the $\overline{\partial}$--Neumann operator. Lecture 4 : Hartogs domains in $\mathbb{C}^{n}$ Lecture 5 : Obstructions to compactness.

The title of this talk is taken from a paper by Charles Fefferman in 1976, where he initiated the geometry of strictly pseudoconvex domains based on Einstein equation. This paper is elementary but contains many ideas developed later. In 1974, Fefferman proved that two strictly pseudo convex domains are biholomorphic if and only if the boundaries are diffeomorphic by a map preserving the tangential Caucy-Riemann equations — the geometry of the latter setting is called CR geometry. The paper in 1976 aimed to give explicit between of the biholomorphic and CR geometries. Our first goal is to describe the boundary behavior of the Bergman kernel in terms of the solution of the Monge–Ampère equation associated with the domain; the second goal is to construct global invariants of the domain by integrating the curvature of the Einstein metric given by the Monge–Ampère solution. Fefferman’s contribution is summarized in his lecture notes: Michael Beals, Charles Fefferman, and Robert Grossman, Strictly pseudoconvex domains in C^n, Bull. Amer. Math. Soc. (1983), 125-322. We plan to include many progresses after that.

Given a smoothly bounded pseudoconvex domain, being able to construct a holomorphic peak function associated to each of its boundary points has a number of important consequences. Such constructions are, in general, very difficult on weakly pseudoconvex domains. Typically, they involve two steps:

1) The construction of a local peak function, which involves a delicate analysis of the geometry of the boundary around the point in question

2) The extension of the local peak function to the whole of the given domain. The second step involves the solution of an inhomogeneous Cauchy--Riemann equation.

In talk, we shall discuss a principle for globally extending local holomorphic peak functions for a family of domains that includes most classes of pseudoconvex domains for which peak-function constructions are of interest. Time permitting, we shall also take a look at some problems whose solutions rely on the ability to construct holomorphic peak functions.