Seminars and Colloquia by Series

Bispectrality and superintegrability

Series
Analysis Seminar
Time
Wednesday, November 1, 2017 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Plamen Iliev – Georgia Tech
The bispectral problem concerns the construction and the classification of operators possessing a symmetry between the space and spectral variables. Different versions of this problem can be solved using techniques from integrable systems, algebraic geometry, representation theory, classical orthogonal polynomials, etc. I will review the problem and some of these connections and then discuss new results related to the generic quantum superintegrable system on the sphere.

Maximal averages and Radon transforms for two-dimensional hypersurfaces

Series
Analysis Seminar
Time
Wednesday, October 25, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Greenblatt – University of Illinois, Chicago
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.

Gabor bases and convexity

Series
Analysis Seminar
Time
Wednesday, October 18, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex Yosevich – University of Rochester
We are going to prove that indicator functions of convex sets with a smooth boundary cannot serve as window functions for orthogonal Gabor bases.

Dynamical sampling and connections to operator theory and functional analysis

Series
Analysis Seminar
Time
Wednesday, October 11, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Akram Aldroubi – Vanderbilt University
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.

On sparse domination of some operators in Harmonic Analysis

Series
Analysis Seminar
Time
Wednesday, October 4, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Grigori Karagulyan – Institute of Mathematics, Yerevan Armenia
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.

Bounded Fourier multipliers with applications to Balian-Low type theorems

Series
Analysis Seminar
Time
Wednesday, September 27, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Northington – Georgia Tech
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.

Maximal averages and singular integrals along vector fields in higher dimension

Series
Analysis Seminar
Time
Friday, September 22, 2017 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Francesco Di Plinio – University of Virginia
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

Sparse Bounds for Discrete Spherical Maximal Averages

Series
Analysis Seminar
Time
Wednesday, September 20, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Kesler – Georgia Tech
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.

Some Recent Sparse Bounds

Series
Analysis Seminar
Time
Wednesday, September 13, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Lacey – Georgia Tech
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.

Finite dimension Balian-Low type theorems

Series
Analysis Seminar
Time
Wednesday, September 6, 2017 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shahaf Nitzan – Georgia Tech
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.

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