Complex Analysis Seminar


Thursday 4pm-5pm at (place: FH 1900)

To give a talk in this seminar please write to
Sonmez Sahutoglu (Sonmez.Sahutoglu "at" utoledo.edu) or
Zeljko Cuckovic (Zeljko.Cuckovic "at" utoledo.edu)

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SPRING 2012
April 26, 2012
Amila Appuhamy (University Toledo)
A Brown-Halmos type theorem for weighted Bergman space Toeplitz operators

Abstract: We denote the weighted Bergman space on the unit disc $D$ in the complex plane by $A_{\alpha}^2$ and the Toeplitz operator on $A_{\alpha}^2$ with bounded symbol $\phi$ by by $T_{\phi}$. Let $f,g$ be bounded harmonic functions on $D$ and assume that $h$ is a bounded smooth enough function on $D$. For $\alpha$ any positive integer we will show that $T_fT_g=T_h$ if and only if $f$ is co-analytic or $g$ is analytic and in either case $h=fg$. This result is similar to what Patrick Ahern and Zeljko Cuckovic showed in their famous paper in 2001 for unweighted case.
April 20, 2012 at UH 4010 (special time and place)
N.V. Rao (University Toledo)
Local conformal maps of unit disk onto itself or the plane onto itself that are not obvious

Abstract: The question we ask is the following: Does there exist a locally conformal map of the unit disk onto itself which is not an automorphism? Over the years many people asked me this question and I thought about it off and on without success. Apparently even in Germany this question is asked but slightly differently. They also ask, "Does there exist a locally conformal map of $\mathbb{C}$ onto itself which is not an automorphism?'' The last was answered in the affirmative by Rainer Brucke in the Journal of Complex Variables and Elliptic Equations. In this talk I will explain their approach and provide a proof for the case of the unit disk.
April 6, 2012 at UH 3008 (special time and place)
Bhupendra Paudyal (University Toledo)
Eigenvalues of composition and weighted composition operators

Abstract: Starting from eigenvalues of composition operator $C_{\phi}$ and weighted composition operator $ u C_{\phi}$ on $ H(D)$, I will introduce eigenvalue problem in $\alpha$-Bloch spaces and show some of my recent progress to find sufficient conditions to have all the eigenfunctions (as in $ H(D)$) in $\alpha$-Bloch Spaces.
April 5, 2012

Abstract: Wind energy has great potential to supplement energy needs of the nation. Wind energy also affects wildlife especially nocturnally active birds/bats. Fatalities to birds and others have been reported near wind turbines. This research will develop radar/IR camera/acoustic monitoring based system for monitoring, detection of wildlife near on-shore and offshore wind turbine farms. The research will help in identifying behavior of nocturnally active birds and others. This work can be employed by wildlife biologist for developing mitigation techniques for both on-shore/off-shore wind farm applications.
March 22, 2012
Pratibha Ghatage (Cleveland State University)
Closed-range composition operators

Abstract: If $\phi$ is an analytic self-map of the open unit disk $D$, then the composition operator $C_{\phi}$ maps most commonly studied spaces of analytic functions into themselves in a one-to-one and bounded fashion. We delineate a set of conditions on $\phi$ so $C_{\phi}$ is bounded below on the Bloch and Bergman spaces. In discussing the Bergman space, we relate behaviour of composition operators to that of Toeplitz operators and use the connection to get necessary and sufficient conditions on $\phi$. We conclude with an open question on Toeplitz operators.
March 15, 2012
David Redett (Indiana University-Purdue University Fort Wayne)
An introduction to weakly stationary processes

Abstract: I will begin by defining what it means for a stochastic process to be weakly stationary. We will then consider some examples. After identifying the spectral measure for a weakly stationary process, we will see how properties of this measure are reflected in the structure of the weakly stationary process. This talk is introductory in nature. Students with a background in measure theory should be able to follow the talk.
February 23, 2012

Abstract: We study the Berezin transform of operators acting on polyanalytic Bergman spaces on the unit disk. We proceed by studying algebraic properties of Toeplitz operators acting on these spaces and obtain results on finite rank commutators and semi-commutators of Toeplitz operators with harmonic symbols. (joint work with Trieu Le)
February 16, 2012

Abstract: This will be an informal talk on the use of operator theory in solving the Nevanlinna-Pick interpolation problem. This approach was pioneered by D. Sarason in the sixties.
February 9, 2012
Dusty Grundmeier (University of Michigan)
Rigidity of CR Mappings for Hyperquadrics

Abstract: This is joint work with Jiri Lebl and Liz Vivas. We prove that the rank of a Hermitian form on the space of holomorphic polynomials can be bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a result along the lines of the Baouendi-Huang and Baouendi-Ebenfelt-Huang rigidity theorems for CR mappings between hyperquadrics. If we have a real-analytic CR mapping of a hyperquadric not equivalent to a sphere to another hyperquadric Q(A,B), then either the image of the mapping is contained in a complex affine subspace or A is bounded by a constant depending only on B.
February 2, 2012
Malgorzata Marciniak (University of Toledo)
Compactly supported cohomology groups of toric surfaces 3

Abstract: The details of inverse images of the sheaf of germs of holomorphic functions onto various curves in a Hirzebruch surface will be discussed. Then I will formulate a general formula for compactly supported cohomology groups of an arbitrary toric surface.
January 26, 2012

Abstract: I will present and discuss some results of joint work with Alex Schuster. In the generalized Bargmann-Fock space $H^p(e^{-p\phi})$ of entire holomorphic functions in $C^n$ that are $L^p$ with respect to a weight $e^{-p\phi}$ satisfying $c dd^c|z|^2 < dd^c \phi \le Cdd^c |z|^2$, one can consider, for a measure $m$, the operator $$T_mf (z) = \int _{C ^n} f(x) K(z,x) e^{-2\phi(x)} d\mu (x),$$ where $K$ is the Bergman kernel, i.e., the kernel of the orthogonal projection $L^2(e^{-\phi}) \to H^2(e^{-\phi})$. We showed that $T_m:H^p(e^{-p\phi}) \to H^p(e^{-p\phi})$ is (well-defined and) bounded if and only if for some $q \ge 1$ the inclusion $H^q(e^{-q\phi}) \to H^q(e^{-q\phi}dm)$ is bounded. (Such measures m are called Carleson measures, and they have a geometric characterization that showing that the Carleson property is independent of q.) The boundedness of $T_m$ when $m$ is Carleson relies on strong decay properties of the Bergman kernel $K$ established by Christ in the case n=1 and by Delin in all dimensions. In the unit ball there is an analogous notion of weighted Bergman spaces, and though the Bergman kernels of these spaces do decay, they seem to have much worse decay properties; in particular the corresponding results for Toeplitz operators are not known.
January 19, 2012
Malgorzata Marciniak (University of Toledo)
Compactly supported cohomology groups of toric surfaces 2

Abstract: During my talk I will define compactly supported Dolbeaut cohomologies and present an "additive" property used to obtain the result mentioned in the title. This property contains an inverse image of a sheaf of germs of holomorphic function which will be explained briefly and intuitively.
FALL 2011
December 08, 2011
Malgorzata Marciniak (University of Toledo)
Compactly supported cohomology groups of toric surfaces 1

Abstract: During my talk I will define compactly supported Dolbeaut cohomologies and present an "additive" property used to obtain the result mentioned in the title. This property contains an inverse image of a sheaf of germs of holomorphic function which will be explained briefly and intuitively.
December 1, 2011

Abstract: For a smooth domain in the complex plane, the Cauchy integral gives a projection to the Hardy space that via the Kerzman-Stein equation can be seen to approximate the Szegö projection. The error in the approximation is captured by the Kerzman-Stein operator, a compact skew-Hermitian operator given by integration against a smooth kernel. In this talk we report on recent progress toward addressing a problem of Norberto Kerzman that seeks to relate the (imaginary) spectrum of the Kerzman-Stein operator to the geometry of the underlying domain.
November 10, 2011

Abstract: Whenever a sequence fails to converge, it makes sense to ask if its sequence of averages converges. A bounded operator $T$ on $H^2$ is called mean weakly asymptotically Toeplitz whenever the sequence $M_n(T):=\frac{1}{n+1}\sum_{k=0}^{n}T_{z}^{\ast k}TT_{z}^{k}$ converges weakly on $H^2$. Nazarov-Shapiro proved that every composition operator is mean weakly asymptotically Toeplitz. In this talk, we improve their result.
November 3, 2011

Abstract: We prove the following theorem: Suppose that $f, g\in L^{\infty}(D),$ such that $f_n=g_n=0$ for all $n\gg0.$ If $f$ is harmonic and nonholomorphic and $[T_f, T_g]=0,$ then $g\in \mathbb{C}+\mathbb{C}f.$ Joint work with Trieu Le.
October 20, 2011

Abstract: Building on techniques developed by Cowen and Nazarov-Shapiro, it is shown that the adjoint of Composition operators, induced with unit disc-automorphisms, are not strongly asymptotically Toeplitz, the notion introduced by Barria and Halmos. This result sheds some (soft) light on the Nazarov-Shapiro's guess. Also, using the Bourdon-MacCluer's results, Toeplitzity of the product of a composition operator with its adjoint are studied.
October 13, 2011
Sonmez Sahutoglu (University of Toledo)
What is a pseudoconvex domain? Part 4

Abstract: This is the last part of a series of talks geared towards graduate students. We will discuss equivalent definitions of pseudoconvex domains.
October 06, 2011
Sonmez Sahutoglu (University of Toledo)
What is a pseudoconvex domain? Part 3

Abstract: This is the third part of a series of talks geared towards graduate students. We will discuss equivalent definitions of pseudoconvex domains.
September 29, 2011
Sonmez Sahutoglu (University of Toledo)
What is a pseudoconvex domain? Part 2

Abstract: This is the second part of a series of talks geared towards graduate students. We will discuss equivalent definitions of pseudoconvex domains.
September 22, 2011
Sonmez Sahutoglu (University of Toledo)
What is a pseudoconvex domain? Part 1

Abstract: This is the first part of a series of talks geared towards graduate students. We will discuss equivalent definitions of pseudoconvex domains.
September 15, 2011
Sivaguru Ravisankar (Ohio State University)
Lipschitz properties of harmonic and holomorphic functions

Abstract: We present two results, one concerning Lipschitz harmonic functions and the other concerning Lipschitz holomorphic functions. We show that a harmonic function, in a smoothly bounded domain $\Omega$ in $\mathbb{R}^n,$ that is Lipschitz-$\alpha$ ($0<\alpha<1$) along a family of curves transversal to $b\Omega$ is Lipschitz-$\alpha$ in $\Omega.$ We also show that a Lipschitz holomorphic function on a smoothly bounded domain $\Omega$ in $\mathbb{C}^n$ ($n>1$) has a Lipschitz gain in certain directions. This gain, or the lack thereof, in a fixed direction is determined by the relative size of complex discs (relative to the distance of its center to the boundary of the domain) that can be fit inside the domain in this direction.
September 8, 2011

Abstract: Cowen and MacCluer formulated the following several variables Schroeder's equation in their 2003 paper. Let $B^n$ be the unit ball in $\mathbb{C}^n$, and $\phi:B^n \to B^n$ analytic, fixing 0, having full rank near 0, and not unitary on any slice. Does there exist an analytic $F:B^n \to \mathbb{C}^n$ satisfying $$F\circ \phi = \phi'(0)F?$$ This talk will give necessary and sufficient conditions for a solution under the general hypothesis, and show that any formal power series solution represents an analytic function on the whole ball, $B^n$.
September 1, 2011
ORGANIZATIONAL MEETING
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