Monday, 15 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Debraj Chakrabarti Background material on the Cauchy-Riemann equations

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.

12:30 to 14:00 Break Lunch
14:00 to 15:30 Problem/Discussion Session Problem/Discussion Session
15:30 to 16:00 Break Tea/coffee
16:00 to 17:00 Kaushal Verma Introduction to quadrature domains

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.

Tuesday, 16 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Debraj Chakrabarti Background material on the Cauchy-Riemann equations

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.

12:30 to 14:00 Break Lunch
14:00 to 15:30 Problem/Discussion Session Problem/Discussion Session
15:30 to 16:00 Break Tea/coffee
16:00 to 17:00 Kaushal Verma Introduction to quadrature domains

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.

Wednesday, 17 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Debraj Chakrabarti Background material on the Cauchy-Riemann equations

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.

12:30 to 14:00 Break Lunch
14:00 to 15:30 Problem/Discussion Session Problem/Discussion Session
15:30 to 16:00 Break Tea/coffee
16:00 to 17:00 Kaushal Verma Introduction to quadrature domains

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.

Thursday, 18 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Debraj Chakrabarti Background material on the Cauchy-Riemann equations

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.

12:30 to 14:00 Break Lunch
14:00 to 15:30 Problem/Discussion Session Problem/Discussion Session
15:30 to 16:00 Break Tea/coffee
16:00 to 17:00 Kaushal Verma Introduction to quadrature domains

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.

Friday, 19 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Debraj Chakrabarti Background material on the Cauchy-Riemann equations

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.

12:30 to 14:00 Break Lunch
14:00 to 15:30 Problem/Discussion Session Problem/Discussion Session
15:30 to 16:00 Break Tea/coffee
16:00 to 17:00 Kaushal Verma Introduction to quadrature domains

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.

Monday, 22 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 12:30 Problem/Discussion Session Problem/Discussion Session
12:30 to 14:30 Break Lunch
14:30 to 15:30 Vamsi Pingali Applications of the theory of the ̄∂ equation.

I shall talk about two applications of the theory of the ̄∂ equation - the Kodaira embedding theorem and the theory of L^2 interpolation. The former deals with the question "Which compact complex manifolds can be embedded in projective space ?" and the latter with "When can a holomorphic function from a collection of points in C be extended in an L^2-controlled manner to an entire function ?" If time permits, I may discuss higher dimensional versions of interpolation.

15:30 to 16:00 Break Tea/coffee
16:00 to 16:30 G.P. Balakumar Remarks on the higher dimensional Suita conjecture

To study the analog of Suita's conjecture for domains D $\mathbb{C}^n for n \geq 2$, Blocki introduced the biholomorphic invariant $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of D along the diagonal, $I^k_D(z)$ is the Kobayashi indicatrix, and $\lambda$ the Lebesgue measure. The behaviour of the invariant function $F^k_D(z)$ on strongly pseudconvex domains and more generally on `Levi corank one domains' (which includes the class of all smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2)$ was studied in a recent joint work with D. Borah, P. Mahajan and K. Verma and to render a brief report of this work, will be the aim of this talk.

16:30 to 17:00 Jaikrishnan Janardhanan Finiteness theorems for the space of holomorphic mappings.

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.

Tuesday, 23 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 12:30 Problem/Discussion Session Problem/Discussion Session
12:30 to 14:30 Break Lunch
14:30 to 15:30 Harish Seshadri Complex geometry of Teichmuller domains

It was conjectured by Y. T. Siu that Teichmuller space cannot be biholomorphic to a convex domain. This was proved recently by V. Markovic. I will outline the background to this conjecture and Markovic's proof. I will also discuss a different approach to this problem (joint work with S. Gupta).

15:30 to 16:00 Break Tea/coffee
16:00 to 16:30 Prachi Mahajan A comparison of two biholomorphic invariants

The Fridman invariant, which is a biholomorphic invariant on Kobayashi hyperbolic manifolds, can be seen as the dual of the much studied squeezing function. In this talk, we compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behaviour is apriori known. This is joint work with Kaushal Verma.

Wednesday, 24 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 12:30 Problem/Discussion Session Problem/Discussion Session
12:30 to 14:30 Break Lunch
Thursday, 25 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 12:30 Problem/Discussion Session Problem/Discussion Session
12:30 to 14:30 Break Lunch
14:30 to 15:30 Vamsi Pingali Applications of the theory of the ̄∂ equation.

I shall talk about two applications of the theory of the ̄∂ equation - the Kodaira embedding theorem and the theory of L^2 interpolation. The former deals with the question "Which compact complex manifolds can be embedded in projective space ?" and the latter with "When can a holomorphic function from a collection of points in C be extended in an L^2-controlled manner to an entire function ?" If time permits, I may discuss higher dimensional versions of interpolation.

15:30 to 16:00 Break Tea/coffee
16:00 to 16:30 Sayani Bera Short C^k's and its properties.

The Short Ck 's were first construcred (by Fornaess) as the basin of attraction of a non-autonomous dynamical system of Hènon like maps around an attracting fixed point. However, they indeed arise from the dynamics of a single Hènon map as the sub-level set of the positive (or negative) Green function. The goal of this talk would be to explore some properties of these short C^k's, arising both from autonomous and non- autonomous setting and link it to existing properties of Hènon maps.

Friday, 26 July 2019
Time Speaker Title Resources
10:00 to 11:00 Dror Varolin Hilbert Space Techniques in Complex Analysis and Geometry

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.

11:00 to 12:30 Problem/Discussion Session Problem/Discussion Session
12:30 to 14:30 Break Lunch
14:30 to 15:30 Harish Seshadri Complex geometry of Teichmuller domains

It was conjectured by Y. T. Siu that Teichmuller space cannot be biholomorphic to a convex domain. This was proved recently by V. Markovic. I will outline the background to this conjecture and Markovic's proof. I will also discuss a different approach to this problem (joint work with S. Gupta).

15:30 to 16:00 Break Tea/coffee
16:00 to 16:30 Vikramjeet Singh Chandel The 3-point spectral Pick interpolation problem

unit ball. The spectral unit ball is the set of all those n n complex matrices that have spectral radius less than 1. The interest in the interpolation problem arises from problems in Control Theory. The spectral unit ball is a non-convex, unbounded domain for which the group of bi-holomorphic transformations does not act transitively. Because of this latter fact, the well known Schur-algorithm could not be applied to address the interpolation problem. Bercovici?Foias?Tannenbaum, from an operator-theoretic viewpoint, gave a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler?Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc, leading to a necessary condition for the existence of an interpolant. This condition is difficult to check since it involves checking the positive definiteness of infinitely many matrices. In this talk, I shall present a couple of results that are based on one of my recent articles on the interpolation problem. The principal result is a necessary condition for the existence of a 3-point holomorphic interpolant F : D →?n, n ≥ 2. This condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma.

Monday, 29 July 2019
Time Speaker Title Resources
10:00 to 11:00 Emil Straube Lectures on the ̄∂–Neumann problem
These lectures will give an introduction to (aspects of) the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory. Next, compactness of the $\overline{\partial}$--Neumann operator will be treated, followed by global regularity. 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.
11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Kengo Hirachi Ampère equations, the Bergman kernel, and geometry of pseudoconvex 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.

12:30 to 14:30 Break Lunch
14:30 to 15:30 Ved Datar L^2 methods, projective embeddings and Kahler-Einstein metrics on Fano manifolds

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.

15:30 to 16:30 Break Tea/coffee
16:30 to 17:30 Pranav Haridas Quadrature domains in higher dimensions

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.

Tuesday, 30 July 2019
Time Speaker Title Resources
10:00 to 11:00 Kengo Hirachi Ampère equations, the Bergman kernel, and geometry of pseudoconvex 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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Emil Straube Lectures on the ̄∂–Neumann problem
These lectures will give an introduction to (aspects of) the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory. Next, compactness of the $\overline{\partial}$--Neumann operator will be treated, followed by global regularity. 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.
12:30 to 14:30 Break Lunch
14:30 to 15:30 Ved Datar L^2 methods, projective embeddings and Kahler-Einstein metrics on Fano manifolds

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.

15:30 to 16:30 Break Tea/coffee
16:30 to 17:00 Sushil Gorai Local polynomial convexity of a certain class of surfaces with isolated degenerated CR singularity.

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.

Wednesday, 31 July 2019
Time Speaker Title Resources
10:00 to 11:00 Emil Straube Lectures on the ̄∂–Neumann problem
These lectures will give an introduction to (aspects of) the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory. Next, compactness of the $\overline{\partial}$--Neumann operator will be treated, followed by global regularity. 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.
11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Kengo Hirachi Ampère equations, the Bergman kernel, and geometry of pseudoconvex 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.

12:30 to 14:30 Break Lunch
Thursday, 01 August 2019
Time Speaker Title Resources
10:00 to 11:00 Kengo Hirachi Ampère equations, the Bergman kernel, and geometry of pseudoconvex 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.

11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Emil Straube Lectures on the ̄∂–Neumann problem
These lectures will give an introduction to (aspects of) the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory. Next, compactness of the $\overline{\partial}$--Neumann operator will be treated, followed by global regularity. 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.
12:30 to 14:30 Break Lunch
14:30 to 15:30 Gautam Bharali From local to global holomorphic peak functions

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.

15:30 to 16:30 Break Tea/coffee
Friday, 02 August 2019
Time Speaker Title Resources
10:00 to 11:00 Emil Straube Lectures on the ̄∂–Neumann problem
These lectures will give an introduction to (aspects of) the $\overline{\partial}$--Neumann problem on pseudoconvex domains in $\mathbb{C}^{n}$. I will first discuss the basic $\mathcal{L}^{2}$ theory. Next, compactness of the $\overline{\partial}$--Neumann operator will be treated, followed by global regularity. 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.
11:00 to 11:30 Break Tea/coffee
11:30 to 12:30 Kengo Hirachi Ampère equations, the Bergman kernel, and geometry of pseudoconvex 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.

12:30 to 14:30 Break Lunch
14:30 to 15:30 Gautam Bharali From local to global holomorphic peak functions

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.

15:30 to 16:30 Break Tea/coffee