Complex Analysis Seminar

Abstract: A classical result by G. D. Birkhoff states that there exists a universal entire function whose set of translates can approximate any given entire function as accurately as desired and on any compact subset of the plane. We consider implications of this example on the dynamic behaviour of composition operators acting on the space of holomorphic functions on a simply connected plane domain and endowed with the compactopen topology 
Abstract: The boundaryvalues of a holomorphic functions singular at the boundary can be represented as generalized functions. On an infinitely differentiable boundary one has distributional boundary values, and on a realanalytic boundary one has boundary values in the sense of Sato hyperfunctions. We discuss how one can generalize this notion to boundaries which are only piecewise smooth. We also give a characterization of the distributional boundary values of holomorphic functions on products of smoothly bounded domains in complex manifolds. This is joint work with Rasul Shafikov. 
Abstract: The unit disc in the plane is called a quadrature domain because the average of an analytic function over the disc with respect to area measure yields the value of the function at the origin. That is and is not as special as it sounds. 
Abstract: Riesz(Hilbert) transforms maybe the most fundamental example of fourier multipliers on $n$dimensional Euclidean spaces. Their $L^p$ bounedness has been a central subject in classical and modern analysis. In this talk, I will introduce P. A. Meyer and D. Bakry's beautiful work on Riesz transforms associated with diffusion Markov semigroups. At the end, I will explain how "cocycles" and "noncommutative" $L^p$ spaces are helpful in understanding the nondiffusion ones, such as Riesz transforms associated with Poisson semigroups on $\mathbb{R}^n$. This is based on a recent joint work with M. Junge and J. Parcet. 
Abstract: The Bergman projection is initially defined on $L^2$ space. However, its behavior on other function spaces is of considerable interest. In this talk, we focus on variants of the Hartogs triangle. We study the $L^p$ mapping properties of the weighted Bergman projections on these domains. As an application, we obtain the weighted $L^p$ Sobolev regularity of the ordinary Bergman projection on the corresponding domain. 
Abstract: First I will show, by a counterexample, that positivity of a Toeplitz operator on the Bergman space does not guarantee positivity of its symbol on the unit disk. Then, I will prove that positivity of the Berezin transform of a Toeplitz operator on the Bergman space of the unit disk does not imply positivity of the Toeplitz operator. 
Abstract: If one is interested in studying holomorphic functions it is a good idea to look at the inhomogeneous CauchyRiemann Equations. We will take a look at these equations in one and in two complex variables. The seminar will be accessible to first year graduate students. 
Abstract: Let $\Omega$ be a bounded domain in $\mathbb{C}^n$ and let $\lambda$ be a continuous function on $\overline{\Omega}$. We consider $\lambda$ as a weight function and look at the corresponding weighted Bergman projection operator $\mathbf{B}_{\Omega}^{\lambda}$. Analytic properties of $\mathbf{B}_{\Omega}^{\lambda}$ depend on the properties of the domainweight pair $(\Omega,\lambda)$. In this talk, we present results about $L^p$ mapping properties of $\mathbf{B}_{\Omega}^{\lambda}$. In particular, we explain how the geometry of $\Omega$ and the zeros of $\lambda$ control the $L^p$ boundedness range. 
Abstract: In this talk, I will present some results about invertibility of Toeplitz operators via Berezin transforms. 
Abstract: In this talk, I will present some results about invertibility of Toeplitz operators via Berezin transforms. 
Abstract: In 1979, Fefferman introduced a measure on the boundary of an arbitrary smooth pseudoconvex domain, which, in contrast to the Euclidean surface area measure, transforms 'well' under biholomorphisms. In particular, this measure is invariant under volumepreserving biholomophisms. In this talk, we will discuss two alternate characterizations  inspired by methods in convex geometry  of Fefferman's measure on strongly pseudoconvex domains. We will also give examples to illustrate how these characterizations can be made sense of in some nonsmooth cases. 
Abstract: I will be presenting a paper by Trieu Le titled "Compact Hankel operators on generalized Bergman spaces of the polydisc". In this talk we will prove that if $f$ is continuous up to the boundary of $\mathbb{D}^2$ and $H_f$ is compact on $A^2(\mathbb{D}^2)$ then $f=h+g$ where $h\in A^2(\mathbb{D}^2)\cap C(\overline{\mathbb{D}^2})$ and $g\in C(\overline{\mathbb{D}^2})$ such that $g(z)=0$ on $\partial \mathbb{D}$. 
Abstract: I will be presenting a paper by Trieu Le titled "Compact Hankel operators on generalized Bergman spaces of the polydisc". In this talk we will prove that if $f$ is continuous up to the boundary of $\mathbb{D}^2$ and $H_f$ is compact on $A^2(\mathbb{D}^2)$ then $f=h+g$ where $h\in A^2(\mathbb{D}^2)\cap C(\overline{\mathbb{D}^2})$ and $g\in C(\overline{\mathbb{D}^2})$ such that $g(z)=0$ on $\partial \mathbb{D}$. 
Abstract: I will be presenting a paper by Trieu Le titled "Compact Hankel operators on generalized Bergman spaces of the polydisc". I will give two talks, one this semester, and the other next semester. During the first talk, I will consider Hankel operators $H_f(\phi)=(IP)M_{\phi}(f)$ with symbol $f$ continuous up to the boundary of $\mathbb{D}^2$ and how if $f$ admits a certain decomposition $f=h+g$ where $h\in A^2(\mathbb{D}^2)\cap C(\overline{\mathbb{D}^2})$ and $g\in C(\overline{\mathbb{D}^2})$ such that $g(z)=0$ on $\partial \mathbb{D}$, then $H_f$ is compact. 
Abstract: Integration with respect to Brownian motion gives an unifying representation for a large class of diffusion processes. And the diffusion processes represented this way exhibit a rich connection with the theory of partial differential equation. In the second part of this talk, we will introduce the basic concept of stochastic integration and its connection with partial differential equations. We will show an example of how the Brownian motion is linked with the Dirichlet's problem for the Laplace operator. 
Abstract: Integration with respect to Brownian motion gives an unifying representation for a large class of diffusion processes. And the diffusion processes represented this way exhibit a rich connection with the theory of partial differential equation. In the second part of this talk, we will introduce the basic concept of stochastic integration and its connection with partial differential equations. We will show an example of how the Brownian motion is linked with the Dirichlet's problem for the Laplace operator. 
Abstract: Brownian motion is the name given to the irregular movement of pollen, suspended in water, observed by the botanist Robert Brown in 1828. This random movement results in a diffusion of the pollen in the water. Thus the mathematical formulation of Brownian motion is closely connected with the heat equation and has evolved into a system of probabilistic methods for studying partial differential equations. In this talk, we will introduce the Brownian motion as of its historical role in physics and its mathematical formulation in the theory of stochastic processes. We will also study some representative properties of Brownian motion that are key to direct probabilistic methods for partial differential equations. 
Abstract:
There are many different definitions of weighted Hardy spaces.
In this talk we will use the definition that considers some
positive weight function on the boundary of the unit disk and
defines the weighted spaces as the spaces of holomorphic functions
from the Smirnov class such that the powers of their absolute values
are integrable with the given weight.
Among such spaces are the spaces introduced by M. Stessin and the speaker on hyperconvex domains in $\mathbb{C}^n$. Not too much is known about them. However, on the unit disk they are quite well studied by M. Alan, N. Gogus, S. Sahin and K. Shrestha. They have shown that these spaces have properties analogous to the properties of the classical spaces. But the main question remains: what are they good for? In our talk we will present some new results and demonstrate how the usage of such spaces can provide shortcuts for the proofs of classical results. 
Abstract: I will discuss some useful tricks with matrices and operators that I have learned and used in several papers. Some tricks are familiar but some are not so well known. The treat of the talk will be a proof of von Neumann's inequality: if $T$ is a bounded operator on a Hilbert space with $\T\\leq 1$ and $p$ is a polynomial, then $\p(T)\\leq \sup\{p(z): z=1\}$. 
Abstract: We investigate the existence of parabolic attracting domains for germs tangent to the identity, when there is a formal invariant curve associated to the given germ. Formal curves are algebraic objects that might have geometrical meaning. However this is not always the case. We review some classical results for holomorphic germs in one dimension and explain the corresponding results for holomorphic germs in several dimensions. This is joint work with Lorena LopezHernanz. 
Abstract: Let the Hankel operator $H_{\phi}$ be defined on the Bergman space of a $C^1$smooth bounded convex domain $\Omega$ in $\mathbb{C}^2$. With some restrictions on $\phi$ we will derive an essential norm estimate for $H_{\phi}$ in terms of the behavior of $\phi$ on the disks in the boundary of $\Omega$. This is joint work with Zeljko Cuckovic. 
Abstract: I will expand a bit on this important construction, since it is central to the study of surfaces. 
Abstract: I will survey some mysterious connections between the finite sporadic simple group $M_{24}$ and K3 surfaces, a family of compact complex surfaces. These connections are mysterious because the Mathieu group cannot act on a K3. The most recent connection is at this point conjectural, and is apparently motivated by results in quantum field theory. 
Abstract: We will survey recent results on subclasses of certain analytic function spaces recently introduced by E. A. Poletsky and M. I. Stessin. For a while everything will be explained when the underlying set is the unit disk. The construction of these spaces is based on potential theory. An effort will be given to make the talk reasonably selfcontained. In the first talk the construction of these spaces together with recent results with M. A. Alan will be explained. 
Abstract: We will survey recent results on subclasses of certain analytic function spaces recently introduced by E. A. Poletsky and M. I. Stessin. For a while everything will be explained when the underlying set is the unit disk. The construction of these spaces is based on potential theory. An effort will be given to make the talk reasonably selfcontained. In the first talk the construction of these spaces together with recent results with M. A. Alan will be explained. 
Abstract: We will survey recent results on subclasses of certain analytic function spaces recently introduced by E. A. Poletsky and M. I. Stessin. For a while everything will be explained when the underlying set is the unit disk. The construction of these spaces is based on potential theory. An effort will be given to make the talk reasonably selfcontained. In the first talk the construction of these spaces together with recent results with M. A. Alan will be explained. 