Abstract. In the second talk we will derive the Bezout formulation of the Corona Problem. One of Carleson's powerful ideas in his proof of the Corona Problem is the notion of Carleson measure. We will discuss the Carleson Embedding Theorem which gives an integral characterization of Carleson measures.

Abstract. I will develop a necessary background to introduce the Corona Problem. Its solution involved some of the greatest Analysts of the 20th century: Lennart Carleson, Lars Hormander and Tom Wolff. The talk is aimed at graduate students and non-experts who would like to learn about this classical problem that is still open in higher dimensions.

Abstract. One major difference between complex analysis in one and several variables is the lack of a true analogue to the one-variable Cauchy transform, $\mathbf{C}$. However, by looking at domains satisfying a convexity condition, we are able to construct the Leray transform, $\mathbb{L}$, which shares many of $\mathbf{C}$'s familiar properties. A significant amount of recent work has been done to study the mapping properties of $\mathbb{L}$ in various settings. This talk will focus on a family of model domains in $\mathbb{C}^2$, and discuss new techniques used in the analysis of the Leray operator. These models can be used to locally approximate a very general class of domains, and it is expected that the theorems in the model case will carry over to the general case. I will also discuss what these results mean in terms of dual CR structures on hypersurfaces in projective space. This is joint work with David Barrett.

Abstract. Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt. This is joint work with Nihat Gökhan Göğüş.

Abstract. Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt. This is joint work with Nihat Gökhan Göğüş.

Abstract. Click here for the abstract.

Abstract. Let $f$ and $g$ be polynomials in $z$. We will prove necessary and sufficient conditions for the product $T_f T_{\bar{g}}$ to be a finite rank perturbation of a Toeplitz operator.

Abstract. Let $f$ and $g$ be polynomials in $z$. We will prove necessary and sufficient conditions for the product $T_f T_{\bar{g}}$ to be a finite rank perturbation of a Toeplitz operator.

Abstract. A few years ago, J-F. Olsen and M.C. Reguera obtained a curious sharp estimate for the norm of Hankel operators with conjugate holomorphic symbols on the Bergman space over the unit disk: $$\|H_{\bar{\psi}}\| \leq \dfrac{\|\psi'\|_{2}}{\sqrt{2}}.$$ Here, $\psi$ is a holomorphic function on the unit disk. The equality occurs when $\psi(z)=z$. Their approach is elementary but computational. I shall discuss their proof with the hope of generalizing it to more general settings.

Abstract. Two classical distinguished Hermitian metrics on a complex domain are the Bergman metric, coming from the reproducing kernel of the space of square-integrable holomorphic functions, and the Poincare metric, i.e. a Kähler-Einstein metric with prescribed (natural) behaviour at the boundary. In the setting of compact Kähler manifolds rather than domains, the so-called balanced metrics were introduced some time ago by S. Donaldson, building on earlier works on S.T. Yau and G. Tian. The talk will discuss the questions of existence and uniqueness of balanced metrics on (noncompact) complex domains, where some answers are yet unknown nowadays even for the simplest case of the unit disc.

Abstract. In this talk, I will give a short survey of several recent results on reducing subspaces of Toeplitz operators on Hardy, Bergman and Dirichlet spaces in one and several variables. Then I will focus on the reducibility of the $N$-th power of a truncated Toeplitz operator (or so-called a model operator). We form a conjecture and try to prove the conjecture for small $N$.