Complex Analysis Seminar at the University of Toledo

Abstract: Let $\Omega_1, \Omega_2$ be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szegö kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.

April 10, 2014

Yifei Pan (Indiana University - Purdue University Fort Wayne)
On some almost complex structures in $\mathbb{C}^2$ and their entire $J$-holomorphic curves

Abstract: In this talk, we are going to discuss existence of entire $J$-holomorphic curves for some almost complex structure in $\mathbb{C}^2$ (i.e., defined in the whole complex plane $\mathbb{C}$) by reducing it to problems of existence of quasiconformal mappings and solutions of nonlinear $\partial$ equation obtained by Chirka-Rosay.

April 3, 2014

Zeljko Cuckovic (University of Toledo)
Spectral properties of Toeplitz operators

Abstract: We will survey some know results about the spectral properties of Toeplitz operators on the Hardy and Bergman spaces without giving proofs. In the Bergman space case, the results are partial, and there are many open questions. The purpose of the talk is to introduce this important topic to the graduate students with the hope they may work on these kinds of problems.

February 27, 2014

Akaki Tikaradze (University of Toledo)
On commutants of Toeplitz operators 3

Abstract: I will discuss some of the results/problems related to commutants of Toeplitz operators on the Bergman spaces of certain domains. The talk will be of introductory nature and will be accessible for students.

February 20, 2014

Zeljko Cuckovic (University of Toledo)
Survey of some "ancient" results about commutants of analytic Toeplitz operators on the Hardy space 2

Abstract: The purpose of this talk is to introduce the graduate students to certain operator theory problems on the Hardy space which are related to the previous talks this semester.

February 13, 2014

Zeljko Cuckovic (University of Toledo)
Survey of some "ancient" results about commutants of analytic Toeplitz operators on the Hardy space 1

Abstract: The purpose of this talk is to introduce the graduate students to certain operator theory problems on the Hardy space which are related to the previous talks this semester.

February 6, 2014

Akaki Tikaradze (University of Toledo)
On commutants of Toeplitz operators 2

January 30, 2014

Akaki Tikaradze (University of Toledo)
On commutants of Toeplitz operators 1

FALL 2013

November 21, 2013

Michael Dabkowski (University of Michigan)
Well-posedness of slightly supercritical active scalar equations

Abstract: We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.

November 14, 2013

Yuan Zhang (Indiana University - Purdue University at Fort Wayne)
On the existence of solutions to nonlinear systems of higher order Poisson type

Abstract: In this talk, we discuss the existence of higher order Poisson type nonlinear systems. We prove a Residue type phenomenon for the fundamental solution of Laplacian in $\mathbb{R}^n, n\geq 3$. This is analogous to the Residue theorem for the Cauchy kernel in $\mathbb{C}$. With the aid of the Residue type formula for the fundamental solution, we derive the higher order derivative formula for the Newtonian potential and obtain its appropriate $\mathcal{C}^{k, \alpha}$ estimates. The existence of solutions to higher order Poisson type nonlinear systems is concluded as an application of the fixed point theorem. This is a joint work with Yifei Pan.

November 7, 2013

Alan Wiggin (University of Michigan at Dearborn)
Results and problems on perturbations of II$_1$ factors

Abstract: Let $A$ and $B$ be C$^*$-algebras represented on the same Hilbert space $H$ and let \[d(A,B)=\max\{\sup_{x\in A}\inf_{y\in B}\| x-y\|, \sup_{x\in B}\inf_{y\in A}\| x-y\|\}.\] Kadison and Kastler asked whether there is a universal constant $c>0$ such that if $d(A,B)<{}c$, then $A$ is $*$-isomorphic to $B$. This problem is open for separable C$^*$-algebras and for separably-acting von Neumann algebras. In particular, a general answer is unknown for II$_1$ factors. After recalling the relevant definitions and giving examples, we will describe our recent joint work with Jan Cameron (Vassar), Erik Christensen (Copenhagen), Allan Sinclair (Edinburgh), Roger Smith (Texas A&M), and Stuart White (Glasgow) providing new examples of separable II$_1$ factors $M$ satisfying the above property for any $N$ with $d(M,N)$ sufficiently small. We'll conclude with some open problems in the area of perturbations of II$_1$ factors.

October 31, 2013

Yunus Zeytuncu (University of Michigan at Dearborn)
$L^p$ regularity of weighted Bergman projections

Abstract: Let $\mathbb{D}$ be the unit disc in $\mathbb{C}^1$ and $\mu$ be a weight on $\mathbb{D}$. We discuss the relation between the weight $\mu$ and $L^p$ regularity of the corresponding weighted Bergman projection $\mathbf{B}_{\mu}$. We also present some applications to unweighted Bergman projections in higher dimensions.

October 24, 2013

Yunus Zeytuncu (University of Michigan at Dearborn)
Regularity of canonical operators and the Nebenhülle of Hartogs domains

Abstract: Let $\mathbb{D}$ denote the unit disk in $\mathbb{C}$ and let $\phi(z)$ be a bounded subharmonic function on $\mathbb{D}$. We consider the pseudoconvex complete Hartogs domains in $\mathbb{C}^2$ of the form \[\Omega=\left\{(z,w)\in \mathbb{C}^2 :z\in \mathbb{D} \text{ and } |w|< {e}^{-\phi(z)}\right\}.\] Let $N_1$ denote the $\bar{\partial}$-Neumann operator on $L^2_{(0,1)}(\Omega)$ and $\mathbf{B}_{\Omega}$ denote the Bergman projection on $L^2(\Omega)$. In this talk, we relate the regularity properties of $N_1$ and $\mathbf{B}_{\Omega}$ to the Nebenhülle of $\Omega$ and the Stein neighborhood bases of $\Omega$.

October 17, 2013

Kit Chan (Bowling Green State University)
Zero-one laws in hypercyclicity

Abstract: An operator $T: X\rightarrow X$ on a Banach space $X$ is said to be hypercyclic if there is a vector $x$ in $X$ whose orbit $\{x, Tx, T^2x, \ldots\}$ is dense in $X$. While a hypercyclic operator cannot exist on a finite dimensional Banach space, hypercyclicity is an interesting phenomenon in the infinite dimensional case. For some properties, such as density of common hypercyclic vectors, density of hypercyclic operators in the operator algebra $B(X)$, and limits points of orbits, hypercyclicity exhibits interesting zero-one behaviors. In this talk, we explore some of the interesting properties, particularly for the shift operators and for the adjoint multiplication operators on the Bergman space.

October 10, 2013

Bhupendra Paudyal (University of Toledo)
Invariant subspace of Volterra operator plus multiplication by $z$ acting on Hardy space

Abstract: Let $H^2$ be Hardy space on the unit disk and \[(Tf)(z)= \int_0^z f(w) dw + z f(z)\] be an operator acting on $H^2$. Then I will discuss the lattice of closed invariant subspaces of the operator $T$.

October 3, 2013

Paul Reschke (University of Michigan)
Distinguished forms for complex surface automorphisms

Abstract: We will discuss some aspects of the interplay between complex algebraic geometry and complex analysis in the study of the dynamics of automorphisms of complex surfaces. In particular, we will show how a combination of tools from both arenas can be used to understand the measures of maximal entropy for automorphisms with positive entropy. We will also show how some of these ideas aid in the characterization of the possible entropies of complex surface automorphisms.

September 26, 2013

Zbigniew Blocki (UJagiellonian University, Poland)
Suita conjecture and the Ohsawa-Takegoshi extension theorem

Abstract: N. Suita in 1972 conjectured that for any bounded domain $D$ on the plane one has $c_D^2\leq\pi K_D$, where $c_D(z)$ is the logarithimc capacity w.r.t. $z$ of the complement of $D$ and $K_D$ is the Bergman kernel on the diagonal. As noticed by Ohsawa in 1995, this problem is closely related to the Ohsawa-Takegoshi extension theorem is several complex variables. The goal of the talk is to present the extension theorem with optimal constant which also gives the Suita Conjecture. The main tool is the $\overline{\partial}$-equation and Hormander's estimate.

September 19, 2013

Sonmez Sahutoglu (University Toledo)
On Axler-Zheng theorem in $\mathbb{C}^n$

Abstract: We prove a version of Axler-Zheng's Theorem on smooth bounded pseudoconvex domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-Neumann operator is compact. This is joint work with Zeljko Cuckovic.

August 29, 2013

ORGANIZATIONAL MEETING