Abstract.

Abstract. I will take two radial symbols and will show that positivity of Berezin transform of product of the symbols is not enough to show the positivity of product of two Toeplitz operators with these symbols.

Abstract. A connection in the plane between the Bergman projection and the Khavinson-Shapiro conjecture will be stated. Then, using an argument reminiscent of a Fischer decomposition, the following projections will be shown to map polynomials to polynomials: the Bergman projection of an ellipsoid in complex space, the polyharmonic Bergman projections of an ellipsoid in real space, and a weighted Szego projection of a planar ellipse.

Abstract. We will consider rational function series of the form $$\sum_{k=1}^{\infty}r_{n_k}(z)$$ where $r_{n_k}$ is a rational function of exact degree $n_k$, and $z$ is an extended complex variable. Using a bivariate logarithmic potential we will formulate the regions in which the series converges. We will see how the asymptotic behaviors of the poles and zeros of $r_{n_k}$ shape the regions of convergence. This is a generalization of the well-known case of the power series, where the regions of convergence are disks, while the poles are at infinity and the zeros at the origin. We will also consider some applications in interpolation and orthogonal expansion.

Abstract. It was once hoped that whenever a compact set in complex Euclidean space has a nontrivial polynomially convex hull, there must be an analytic disc in the hull. This hope was shattered by a counterexample given by Stolzenberg in 1963. Given that analytic discs do not always exist, it is natural to ask whether weaker semblances of analyticity must be present such as nonzero point derivations or nontrivial Gleason parts. A recent example of Cole, Ghosh, and the speaker shows that these can fail to exist as well. However, other examples of the speaker show that it is also possible for a hull containing no analytic discs to contain a large Gleason part and support many nonzero bounded point derivations.

Abstract. When self-maps of the unit disk exhibit what we call property UCI - meaning that they converge uniformly under iteration on the entire open disk to the Denjoy-Wolff point - much more can be said than usual about their spectra. We show that this strong assumption still leads to a wide, interesting class of maps that induce composition operators on $H^2$, and use this assumption to find the spectra of any associated weighted composition operators with bounded weight.

Abstract. In this talk, I will present various characterizations of compactness of some canonical operators on domains in $\mathbb{C}^n$. I will highlight how complex geometry of the boundary of the domain plays a role in these characterizations. In particular, I will prove that on smooth bounded pseudoconvex Hartogs domains in $\mathbb{C}^2$ compactness of the $\overline{\partial}$-Neumann operator is equivalent to compactness of all Hankel operators with symbols smooth on the closure of the domain. The talk is based on recent joint projects with Zeljko Cuckovic and Sonmez Sahutoglu.

Abstract. I will survey some results related to the Khavinson-Shapiro conjecture.

Abstract. Special relativity as a geometric structure on the Grassmannian (which for space-time is projective space). The localization of this structure in space-time geometry, and the role of gravity in determining space time geometry.

Abstract. Special relativity as a geometric structure on the Grassmannian (which for space-time is projective space). The localization of this structure in space-time geometry, and the role of gravity in determining space time geometry.

Abstract. I will talk about the notion of positive definite kernel functions and their associated Hilbert spaces, originated from the work of Aronszajn in the fifties. Familiar examples are reproducing kernels of the Hardy and Bergman spaces. I will discuss basic properties of kernel functions and their applications.

Abstract. I will talk about the notion of positive definite kernel functions and their associated Hilbert spaces, originated from the work of Aronszajn in the fifties. Familiar examples are reproducing kernels of the Hardy and Bergman spaces. I will discuss basic properties of kernel functions and their applications.

Abstract. I will prove a characterization of the invertibility of Toeplitz operators with nonnegative, bounded symbols on the weighted disc in the complex plane. I will then do the same for Toeplitz operators with Carleson measure symbols. If there is a time, I will partially investigate to what degree being Carleson is a local property for finite, positive, Borel measures.

Abstract. I will prove a characterization of the invertibility of Toeplitz operators with nonnegative, bounded symbols on the weighted disc in the complex plane. I will then do the same for Toeplitz operators with Carleson measure symbols. If there is a time, I will partially investigate to what degree being Carleson is a local property for finite, positive, Borel measures.

Abstract. In this talk I will give an elementary introduction to the Khavinson-Shapiro conjecture which concerns solving the Dirichlet problem with polynomial data. No background will be necessary and no new results will be given.

Abstract. Let $\Omega$ be an arbitrary bounded complete Reinhardt domain in $\mathbb{C}^n$. We show that for $n\geq 2$, if a Hankel operator with a conjugate holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^2(\Omega)$, then it must equal zero. This has previously been proved only for strongly pseudoconvex domains and for certain pseudoconvex domains.

Abstract. Let $T$ denote the $n$-shift plus certain weighted Volterra operator on the Bergman space. Inspired by Sarason's results, we show that the operator $T$ is similar to $M_{z^{n}}$ on some Hilbert space $S_{n}^{2}(\mathbb{D})$. Then the closed invariant subspaces of $T$ are characterized. This is joint work with Professor Cuckovic.

Abstract. Let $T$ denote the $n$-shift plus certain weighted Volterra operator on the Bergman space. Inspired by Sarason's results, we show that the operator $T$ is similar to $M_{z^{n}}$ on some Hilbert space $S_{n}^{2}(\mathbb{D})$. Then the closed invariant subspaces of $T$ are characterized. This is joint work with Professor Cuckovic.

Abstract. Essential norm of an operator is its distance to compact operators. In this talk we give estimates for the essential norm of the $\overline{\partial}$-Neumann operator on convex domains and on worm domains. This is joint work with Zeljko Cuckovic.

Abstract. We introduce a class of non-smooth, pseudoconvex domains which generalize the Hartogs triangle. After obtaining the explicit formula for the Bergman kernel of each domain, these formulas are used to understand the mapping properties of the Bergman projection. We discover a restricted range of p for which the Bergman projection is a bounded operator on $L^p$ and show that this range is sharp. We conclude by observing surprising and intriguing behavior of the bounded $L^p$ mapping range with respect to limit domains. This is joint work with Jeff McNeal.

Abstract. Let $\Omega\subset\mathbb{C}^2$ be a bounded, convex, and complete Reinhardt domain with a piecewise smooth boundary. Let $A^2(\Omega)$ denote the Bergman space of $\Omega$. Suppose $\phi\in C(\overline{\Omega})$ is such that the Hankel operator $H_{\phi}: A^2(\Omega)\rightarrow L^2(\Omega)$ is compact. I will present a proof that characterizes the behaviour of $\phi$ along analytic disks in $b\Omega$. Namely, if $f:\mathbb{D}\rightarrow b\Omega$ is an analytic disk, then $\phi\circ f$ is holomorphic on $\mathbb{D}$.

Abstract. Let $\Omega\subset\mathbb{C}^2$ be a bounded, convex, and complete Reinhardt domain with a piecewise smooth boundary. Let $A^2(\Omega)$ denote the Bergman space of $\Omega$. Suppose $\phi\in C(\overline{\Omega})$ is such that the Hankel operator $H_{\phi}: A^2(\Omega)\rightarrow L^2(\Omega)$ is compact. I will present a proof that characterizes the behaviour of $\phi$ along analytic disks in $b\Omega$. Namely, if $f:\mathbb{D}\rightarrow b\Omega$ is an analytic disk, then $\phi\circ f$ is holomorphic on $\mathbb{D}$.