Seminars and Colloquia by Series

Wednesday, March 27, 2013 - 14:00 , Location: Skiles 005 , Debendra Banjade , University of Alabama , Organizer: Brett Wick
In 1980, T. M. Wolff has given the following version of the ideal membership for finitely generated ideals in $H^{\infty}(\mathbb{D})$: \[\ensuremath{\mbox{If \,\,}\left\{ f_{j}\right\} _{j=1}^{n}}\subset H^{\infty}(\mathbb{D}),\, h\in H^{\infty}(\mathbb{D})\,\,\mbox{and }\]\[\vert h(z)\vert\leq\left(\underset{j=1}{\overset{n}{\sum}}\vert f_{j}(z)\vert^{2}\right)^{\frac{1}{2}}\,\mbox{for all \ensuremath{z\in\mathbb{D},}}\]then \[h^{3}\in\mathcal{I}\left(\left\{ f_{j}\right\} _{j=1}^{n}\right),\,\,\mbox{the ideal generated by \ensuremath{\left\{ f_{j}\right\} _{j=1}^{n}}in \ensuremath{H^{\infty}}\ensuremath{(\mathbb{D})}. }\]In this talk, we will give an analogue of the Wolff's ideal problem in the multiplier algebra on weighted Dirichlet space. Also, we will give a characterization for radical ideal membership.
Wednesday, March 13, 2013 - 14:00 , Location: Skiles 005 , Gagik Amirkhanyan , Georgia Tech , Organizer: Brett Wick
For dimensions n greater than or equal to 3, and integers  N greater than 1, there is a distribution of points P in a unit cube [0,1]^{n}, of cardinality N, for which the discrepancy function D_N associated with P has an optimal Exponential Orlicz norm.  In particular the same distribution will have optimal L^p norms, for 1 < p < \infty.  The collection P is a random digit shift of the examples of  W.L. Chen and M. Skriganov.
Wednesday, March 6, 2013 - 14:00 , Location: Skiles 005 , Dechao Zheng , Vanderbilt University , Organizer: Brett Wick
On the Hardy space, by means of an elegant and ingenious argument, Widom showed that the spectrum of a bounded Toeplitz operator is always connected and Douglas showed that the essential spectrum of a bounded Toeplitz operator is also connected. On the Bergman space, in 1979, G. McDonald and the C. Sundberg showed that the essential spectrum of a Toeplitz operator with bounded harmonic symbol is connected if the symbol is either real or piecewise continuous on the boundary. They asked whether the essential spectrum of a Toeplitz operator on the Bergman space with bounded harmonic symbol is connected.  In this talk,  we will show an example that the spectrum and the essential spectrum of a Toeplitz operator with bounded harmonic symbol is disconnected. This is a joint work with Carl Sundberg.
Wednesday, February 27, 2013 - 14:00 , Location: Skiles 005 , Theresa Anderson , Brown University , Organizer: Michael Lacey
A recent conjecture in harmonic analysis that was exploredin the past 20 years was the A_2 conjecture, that is the sharp bound onthe A_p weight characteristic of a Calderon-Zygmund singular integraloperator on weighted L_p space.  The non-sharp bound had been knownsince the 1970's, but interest in the sharpness was spurred recentlyby connections to quasiconformal mappings and PDE.  Finally solved infull by Hytonen, the proof is complex, intricate and lengthy.  A new  "simple" approach using local mean oscillation and positive operatorbounds was published by Lerner.  We discuss this and some recent progress in the area, including our new proof for spaces of homogeneoustype, in the style of Lerner (Joint work with Armen Vagharshakyan).
Wednesday, February 20, 2013 - 14:00 , Location: Skiles 005 , Brett Wick , Georgia Tech , Organizer: Brett Wick
In this talk, we will characterize the compact operators on Bergman spaces of the ball and polydisc.  The main result we will discuss shows that an operator on the Bergman space is compact if and only if its Berezin transform vanishes on the boundary and additionally this operator belongs to the Toeplitz algebra.  We additionally will comment about how to extend these results to bounded symmetric domains, and for "Bergman-type" function spaces.
Wednesday, February 6, 2013 - 14:00 , Location: Skiles 005 , Alexander Reznikov , Michigan State University , Organizer: Michael Lacey
We consider a so-called "One sided bump conjecture", which gives asufficient condition for two weight boundedness of a Calderon-Zygmundoperator. The proof will essentially use the Corona decomposition, which isa main tool for a first proof of $A_2$ (also, $A_p$ and $A_p-A_\infty$)conjecture. We will focus on main difficulty, that does not allow to get afull proof of our one sided bump conjecture.
Wednesday, January 30, 2013 - 14:00 , Location: Skiles 005 , James Scurry , Georgia Tech , Organizer: Brett Wick
We will study one and two weight inequalities for several different operators from harmonic analysis, with an emphasis on vector-valued operators. A large portion of current research in the area of one weight inequalities is devoted to estimating a given operators' norm in terms of a weight's A_p characteristic; we consider some related problems and the extension of several results to the vector-valued setting. In the two weight setting we consider some of the difficulties of characterizing a two weight inequality through Sawyer-type testing conditions.
Wednesday, January 16, 2013 - 14:00 , Location: Skiles 005 , Josh Isralowitz , SUNY Albany , Organizer: Brett Wick
Weighted norm inequalities for singular integral operators acting on scalar weighted L^p is a classical topic that goes back to the 70's with the seminal work of R. Hunt, B. Muckenhoupt, and R. Wheeden.  On the other hand, weighted norm inequalities for singular integral operators with matrix valued kernels acting on matrix weighted L^p are poorly understood and results (obtained by F. Nazarov, S. Treil, and A. Volberg in the late 90's) are only known for the situation when the kernel is essentially scalar valued.In this talk, we discuss matrix weighted norm inequalities for matrix valued dyadic paraproducts and discuss the possibility of using our results and a recent result of T. Hytonen to obtain concrete weighted norm inequalities for singular integral operators with matrix kernels acting on matrix weighted L^p. This is joint work with Hyun-Kyoung Kwon and Sandra Pott.
Wednesday, November 28, 2012 - 14:00 , Location: Skiles 005 , Alden Waters , Jyvaskyla University , Organizer: Michael Lacey
We show how to construct frames for square integrable functionsout of modulated Gaussians. Using the frame representation of the Cauchydata, we show that we can build a suitable approximation to the solutionfor low regularity, time dependent wave equations. The talk will highlightthe relationship of the construction to harmonic analysis and will explorethe differences of the new construction to the standard Gaussian beamansatz.
Wednesday, November 14, 2012 - 14:00 , Location: Skiles 005 , Ionel Popescu , Georgia Tech , ipopescu@math.gatech.edu , Organizer: Brett Wick
This formula of Haagerup gives an expression of the log|x-y| in terms of Chebyshev polynomials of the first kind.   This is very useful for problems involving the logarithmic potentials which plays a prominent role in random matrices, free probability, orthogonal polynomials and other areas.  We will show how one can go from this to several things, for example the counting problems of planar diagrams and functional inequalities in free probability in particular an intriguing Poincare inequality and some related other inequalities.   If time allows I will also talk about a conjecture related to the Hilbert transform, semicircular and arcsine distribution.   Parts of this was with Stavros Garoufalidis and some other parts with Michel Ledoux.

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