## Seminars and Colloquia by Series

Wednesday, October 25, 2017 - 13:55 , Location: Skiles 005 , Michael Greenblatt , University of Illinois, Chicago , Organizer: Michael Lacey
A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.
Wednesday, October 18, 2017 - 13:55 , Location: Skiles 005 , Alex Yosevich , University of Rochester , Organizer: Michael Lacey
We are going to prove that indicator functions of convex sets with a smooth boundary cannot serve as window functions for orthogonal Gabor bases.
Wednesday, October 11, 2017 - 13:55 , Location: Skiles 005 , Akram Aldroubi , Vanderbilt University , Organizer: Shahaf Nitzan
Dynamical sampling is the problem of recovering an unknown function from a set of space-time samples. This problem has many connections to problems in frame theory, operator theory and functional analysis.  In this talk, we will state the problem and discuss its relations to various areas of functional analysis and operator theory, and  we will give a brief review of previous results and present several new ones.
Wednesday, October 4, 2017 - 13:55 , Location: Skiles 005 , , Institute of Mathematics, Yerevan Armenia , Organizer: Michael Lacey
We introduce a class of operators on abstract measurable spaces, which unifies variety of operators in Harmonic Analysis. We prove that such operators can be dominated by simple sparse operators. Those domination theorems imply some new estimations for Calderón-Zygmund operators,  martingale transforms and Carleson operators.
Wednesday, September 27, 2017 - 13:55 , Location: Skiles 005 , Michael Northington , Georgia Tech , Organizer: Shahaf Nitzan
The Gabor system of a function is the set of all of its integer translations and modulations.  The Balian-Low Theorem states that the Gabor system of a function which is well localized in both time and frequency cannot form an Riesz basis for $L^2(\mathbb{R})$.  An important tool in the proof is a characterization of the Riesz basis property in terms of the boundedness of the Zak transform of the function.  In this talk, we will discuss results showing that weaker basis-type properties also correspond to boundedness of the Zak transform, but in the sense of Fourier multipliers.  We will also discuss using these results to prove generalizations of the Balian-Low theorem for Gabor systems with weaker basis properties, as well as for shift-invariant spaces with multiple generators and in higher dimensions.
Friday, September 22, 2017 - 12:05 , Location: Skiles 006 , Francesco Di Plinio , University of Virginia , Organizer: Amalia Culiuc
It is a conjecture of Zygmund that the averages of a square integrable function over line segments oriented along a Lipschitz vector field on the plane converge pointwise almost everywhere. This statement is equivalent to the weak L^2 boundedness of the directional maximal operator along the vector field. A related conjecture, attributed to Stein, is the weak L^2 boundedness of the directional Hilbert transform taken along a Lipschitz vector field. In this talk, we will discuss recent partial progress towards Stein’s conjecture obtained in collaboration with I. Parissis, and separately with S. Guo, C. Thiele and P. Zorin-Kranich. In particular, I will discuss the recently obtained sharp bound for the Hilbert transform along finite order lacunary sets in two dimensions and possible higher dimensional generalization
Wednesday, September 20, 2017 - 13:55 , Location: Skiles 005 , Robert Kesler , Georgia Tech , Organizer: Shahaf Nitzan
Magyar, Stein, and Wainger proved a discrete variant in Zd of the continuous spherical maximal theorem in Rd for all d ≥ 5. Their argument proceeded via the celebrated “circle method” of Hardy, Littlewood, and Ramanujan and relied on estimates for continuous spherical maximal averages via a general transference principle. In this talk, we introduce a range of sparse bounds for discrete spherical maximal averages and discuss some ideas needed to obtain satisfactory control on the major and minor arcs. No sparse bounds were previously known in this setting.
Wednesday, September 13, 2017 - 13:55 , Location: Skiles 005 , Michael Lacey , Georgia Tech , Organizer: Shahaf Nitzan
A sparse bound is a novel method to bound a bilinear form. Such a bound gives effortless weighted inequalities, which are also easy to quantify.  The range of forms which admit a sparse bound is broad.  This short survey of the subject will include the case of spherical averages, which has a remarkably easy proof.
Wednesday, September 6, 2017 - 01:55 , Location: Skiles 005 , Shahaf Nitzan , Georgia Tech , Organizer: Shahaf Nitzan
The classical Balian-Low theorem states that if both a function and it's Fourier transform decay too fast then the Gabor system generated by this function (i.e. the system obtained from this function by taking integer translations and integer modulations) cannot be an orthonormal basis or a Riesz basis.Though it provides for an excellent `thumbs--rule' in time-frequency analysis, the Balian--Low theorem is not adaptable to many applications. This is due to the fact that in realistic situations information about a signal is given by a finite dimensional vector rather then by a function over the real line. In this work we obtain an analog of the Balian--Low theorem in the finite dimensional setting, as well as analogs to some of its extensions. Moreover, we will note that the classical Balian--Low theorem can be derived from these finite dimensional analogs.
Wednesday, August 23, 2017 - 14:05 , Location: Skiles 005 , Joey Iverson , University of Maryland , Organizer: Shahaf Nitzan
Abstract: Shift-invariant (SI) spaces play a prominent role in the study of wavelets, Gabor systems, and other group frames. Working in the setting of LCA groups, we use a variant of the Zak transform to classify SI spaces, and to simultaneously describe families of vectors whose shifts form frames for the SI spaces they generate.