| Abstract: In this talk, I will present a part of main theorem of the paper "Compact operators via the Berezin transform" written by professors Sheldon Axler and Dechao Zheng. This paper mainly discusses if $S$ equals a finite sum of finite product of Toeplitz operators, then $S$ is compact iff Berezin transform of $S$ equals to 0 on boundary of the unit disk. |
April 21, 2011
| Abstract: The first nonlocal function algebra was constructed by Eva Kallin in 1963. We construct nonlocal function algebras generated by smooth functions on manifolds and investigate the minimum dimension of a manifold on which there exists a nonlocal function algebra. |
April 15, 2011 Joint with Shoemaker Lecture Series
Siqi Fu
(Rutgers University-Camden)
Eigenvalue distribution of the $\overline{\partial}$-Neumann Laplacian
Eigenvalue distribution of the $\overline{\partial}$-Neumann Laplacian
| Abstract: In this lecture, we continue our discussion of spectral theory of the $\overline{\partial}$-Neumann Laplacian. Here the focus is on pure discreteness of the spectrum and asymptotic distribution of eigenvalues. We explain how this is related to problems in quantum mechanics and how the compactly supported wavelets constructed by Daubechies and Lemarié-Meyer play a role in the subject. Part of this talk is based on previous joint work with E. Straube and with M. Christ. |
April 7, 2011
Alessandro Arsie
(University of Toledo)
Holomorphic embeddings of small codimension in complex projective spaces
Holomorphic embeddings of small codimension in complex projective spaces
| Abstract: After having reviewed some basic constructions related to holomorphic embeddings of compact Riemann surfaces and (algebraic) complex manifolds, I will introduce the long standing conjecture of Hartshorne about small codimension embeddings and its relation with splitting of vector bundles and projective geometry. According to Hartshone's conjecture, a complex submanifold $X^k$ of dimension $k$ in $\mathbb{P}^n(\mathbb{C})$ of small codimension (how small depends on $n$ and $k$) is always a complete intersection, namely the homogenous ideal $I(X^k)$ of $X^k$ is generated by $n-k$ homogeneous polynomials. From the differential geometric point of view, being a complete intersection entails that it is possible to find $n-k$ hypersurfaces, possibly singular, such that: 1) on $X^k$ they are smooth, 2) their set-theoretic intersection is $X^k$ and 3) they meet transversally at $X^k$. Complete intersections are very well understood, and from this point of view Hartshorne's conjecture states that the only complex submanifolds of small codimension in $\mathbb{P}^n(\mathbb{C})$ are the trivial ones. I will also present my past contributions to this problem and I will briefly review the various approaches that have been tried so far to settle this question. |
March 31, 2011
Mehmet Celik
(University of North Texas at Dallas)
Compactness of the $\overline\partial$-Neumann problem and Hankel operators
Compactness of the $\overline\partial$-Neumann problem and Hankel operators
| Abstract: It is known that on a bounded pseudoconvex domain, the compactness of the $\overline\partial$-Neumann operator implies the compactness of the commutators between the Bergamn projections and multiplication operators, those commutators are actually the Hankel operators. The interest is to see to what extend properties of the commutators can be used to characterize compactness properties of the $\overline\partial$-Neumann problem. In this talk we will present some observations between the compactness properties of these operators on a smooth bounded domain. This is joint work with Sonmez Sahutoglu. |
March 29, 2011 Special day: Tuesday, and place:FH 1900
| Abstract: In this talk, I will present a part of the paper "On limits of sequences of holomorphic functions" written by Steven G. Krantz. The paper mainly discusses the pointwise limit of a sequence of holomorphic functions in one complex variable, and will present some example related to that. |
March 24, 2011
Adam Coffman
(Indiana University-Purdue University Fort Wayne)
Smooth counterexamples to strong unique continuation for a Beltrami system in $\mathbb{C}^2$
Smooth counterexamples to strong unique continuation for a Beltrami system in $\mathbb{C}^2$
| Abstract: (joint work with Yifei Pan) We construct an example of a smooth map $\mathbb{C}\to\mathbb{C}^2$ which vanishes to infinite order at the origin, and such that the ratio of the norm of the $\bar z$ derivative to the norm of the $z$ derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation. |
March 03, 2011
| Abstract: Let $D^n$ be the polydisk in $\mathbb{C}^n$ and symbols $\phi, \psi\in C(\overline{D^n})$ such that $\phi$ and $\psi$ are pluriharmonic on any $(n-1)$-dimensional polydisk in the boundary of $D^n$. Then $H_{\psi}^* H_{\phi}$ is compact on $A^2(D^n)$ if and only if for every $1\le j,k\le n$ such that $j\ne k$ and any $(n-1)$-dimensional polydisk $D$, orthogonal to the $z_j$-axis in the boundary of $D^n$, either $\phi$ or $\psi$ is holomorphic in $z_k$ on $D$. I will present a part of the paper "Compactness of products of Hankel operators on the polydisk and some product domains" written by two professors Zeljko Cuckovic and Sonmez Sahutoglu. |
February 24, 2011
| Abstract: Some of the properties of parametric Toeplitz operators (PTO), on the Hilbert-Hardy space, are investigated and is shown that much of their behavior is similar to that of the classical Toeplitz operators. We also study some other operator equations. |
February 17, 2011
| Abstract: A linear continuous operator on a topological vector space is called hypercyclic if it has a dense orbit. This talk stems from Bourdon and Shapiro's characterization of hypercyclicity of linear fractional composition operators on the Hardy space. We will characterize disjoint hypercyclicity of finitely many linear fractional composition operators acting on spaces of holomorphic functions on the unit disc. |
February 10, 2011
Trieu Le
(University of Toledo)
Some algebraic properties of Toeplitz operators on the Segal-Bargmann space
Some algebraic properties of Toeplitz operators on the Segal-Bargmann space
| Abstract: Let $\mathcal{H}$ be the Segal-Bargmann space, which consists of Gaussian square-integrable entire functions on $\mathbb{C}^n$. For a bounded function $f$ on $\mathbb{C}^n$, $T_f$ denotes the Toeplitz operator with symbol $f$ acting on $\mathcal{H}$. Let $f_1$ and $f_2$ be two bounded radial functions, one of which is non-constant, we will discuss the necessary and sufficient conditions on a bounded function $g$ for which $T_{f_1}T_g=T_gT_{f_2}$. We then use this result to study the commuting and zero-product problems for Toeplitz operators on $\mathcal{H}$. This is joint work with W. Bauer. |
February 03, 2011
| Abstract: In this talk we will figure out all the eigenvalues of schörder's equation for holomorphic self maps of the disc having an interior fixed point and will discuss about the holomorphic solutions on hardy spaces. |
January 27, 2011
| Abstract: We prove the following localization for compactness of Hankel operators on Bergman spaces. Assume that $\Omega$ is a bounded pseudoconvex domain in $\mathbb{C}^n, p$ is a boundary point of $\Omega$, and $B( p, r )$ is a ball centered at $p$ with radius $r$ so that $U = \Omega\cap B( p, r )$ is a domain. We show that if the Hankel operator $H_{\phi}$ with symbol $\phi\in C^1(\overline\Omega)$ is compact on $A^2 (\Omega)$ then $H_{R(\phi)}$ is compact on $A^2 (U)$ where $R(\phi)$ denotes the restriction of $\phi$ on $U$, and $A^2 (\Omega)$ and $A^2 (U )$ denote the Bergman spaces on $\Omega$ and $U,$ respectively. |
FALL 2010
December 9, 2010
| Abstract: By a well-known theorem of Beurling, invariant subspaces of the shift operator on the Hardy space can be described in terms of inner functions. Thus, it is natural to try to generalize this theorem to a wider class of operators. I will discuss some known results in this direction (due to Richter, Shimorin, Olofsson...). |
December 2, 2010
Mehdi Nikpour
(University of Toledo)
On the adjoint of a weighted composition operator on some functional Hilbert spaces
On the adjoint of a weighted composition operator on some functional Hilbert spaces
| Abstract: In this short talk, the straightforward extensions of some results, obtained by M. J. Martìn and D. Vukotić on the adjoint of a composition operator, to the weighted case are presented. |
November 18, 2010
| Abstract: I will show the definition and present few examples of toric varieties. |
November 4, 2010
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Abstract:
I will show the definition and present few examples of toric varieties.
Lecture Notes |
October 28, 2010
Petko Marinov
(University of Toledo)
Stable Capillary surfaces with planar or spherical boundary in the absence of gravity
Stable Capillary surfaces with planar or spherical boundary in the absence of gravity
| Abstract: In this talk, I will present the following result: the only immersed stable capillary surfaces with boundary embedded in a plane are the spherical caps. There is no gravity assumed so these surfaces will have constant mean curvature and will make a constant contact angle with the plane. The case when the genus of the surface is zero the result is basically known. For the positive genus case, a special perturbation is constructed and its normal component makes the second variation of energy negative, which means the configuration is unstable. If time permits, I will discuss our advances towards a similar problem where the surface "sits" inside of a sphere. I will at least explain the construction of the vector field that we are using in this case, which turns out to be Killing on the hyperbolic ball. |
October 21, 2010
Mei-Chi Shaw
(University of Notre Dame)
The Cauchy-Riemann equations and $l^2$ Serre duality on complex manifolds
The Cauchy-Riemann equations and $l^2$ Serre duality on complex manifolds
| Abstract: In this talk we study the closed range property and boundary regularity of the Cauchy-Riemann equations on domains in complex manifolds. Recently, $L^2$ results for the Cauchy-Riemann equations on product domains in complex manifolds and an analogue of the classical Künneth formula have been obtained (joint work with Debraj Chakrabarti). We also discuss an $L^2$ version of the Serre duality on domains in complex manifolds with applications. |
October 7, 2010
| Abstract: Let $\sum$ be a hyperbolic Riemann surface, and $C$ is a closed subset of $\sum$. We study the spaces of square integrable and bounded holomorphic k-differentials on $\sum\setminus C$, where $k\geq2$ is an integer. These spaces are Banach spaces and for special norm we will have Hilbert spaces. Then the Poincaré series which is an important technique to construct $k$-differentials will be introduced. The main result will provide descriptions of the kernel of the Poincaré series map, which is a surjective, linear map between two spaces of integrable, holomorphic $k$-differentials. This is a joint work with T. Foth. |
September 30, 2010
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Abstract:
Recall that the Bergman space $A^2(B_n)$ is the space of all square integrable holomorphic functions on the unit ball in $C^n$ ($n\geq 1$). It is well known that $A^2(B_n)$ admits an orthonormal basis consisting of multiples of holomorphic monomials. I will discuss the following problems:
(a) Describe all bounded functions $f$ for which the Toeplitz operator $T_f$ is diagonal with respect to the above orthonormal basis? (b) Given a non-constant function $f$ satisfying the conditions in (a), describe all bounded functions $g$ such that $T_g$ commutes with $T_f$. |
September 23, 2010
Nikolai Vasilevski
(CINVESTAV del I.P.N., Mexico)
On compactness of commutators and semi-commutators of Toeplitz operators on the Bergman space
On compactness of commutators and semi-commutators of Toeplitz operators on the Bergman space
| Abstract: Given a $C^*$-subalgebra $A$ of algebra $L_{\infty}(D),$ denote by $T(A)$ the $C^*$-algebra generated by all Toeplitz operators with symbols in $A$ and acting on the Bergman space over the unit disk $D.$ We will discuss the compactness properties of commutators and semi-commutators of Toeplitz operators from $T(A)$ as well as the structural properties of $T(A)$ and other operator algebras related to the above compactness properties. |
September 16, 2010
| Abstract: In the matricial point of view, moving one step to the southeast, provides us a bounded operator-valued linear transformation on the $C^*$-algebra of all bounded linear operators on the Hilbert-Hardy space to itself, which enables us first to answer partially a question raised by Paul R. Halmos, and second to embed Toeplitz operators in an extended setting$.$ |
September 9, 2010
| Abstract: I will present the idea of obtaining vanishing of $L^{2}$ cohomology with an estimate and some applications of the estimate emphasizing its importance. I will also present two of the results from my thesis that extend McNeal's vanishing theorem on Kähler Convex Manifolds. One of the result from my thesis is a vanishing theorem of $L^{2}$ cohomology, whereas, the other theorem only guarantees finite dimensionality of the $L^{2}$ cohomology. |
September 2, 2010
ORGANIZATIONAL MEETING
