Seminars and Colloquia by Series

Zeros of optimal polynomial approximants and spectra of Jacobi matrices

Series
Analysis Seminar
Time
Wednesday, November 29, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Catherine BeneteauUniversity of South Florida
In this talk, I will discuss some polynomials that are best approximants (in some sense!) to reciprocals of functions in some analytic function spaces of the unit disk. I will examine the extremal problem of finding a zero of minimal modulus, and will show how that extremal problem is related to the spectrum of a certain Jacobi matrix and real orthogonal polynomials on the real line.

t-Haar multipliers revisited

Series
Analysis Seminar
Time
Wednesday, November 15, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Cristina PereyraUniversity of New Mexico
t-Haar multipliers are examples of Haar multipliers were the symbol depends both on the frequency variable (dyadic intervals) and on the space variable, akin to pseudo differential operators. They were introduced more than 20 years ago, the corresponding multiplier when $t=1$ appeared first in connection to the resolvent of the dyadic paraproduct, the cases $t=\pm 1/2$ is intimately connected to direct and reverse inequalities for the dyadic square function in $L^2$, the case $t=1/p$ naturally appears in the study of weighted inequalities in $L^p$. Much has happened in the theory of weighted inequalities in the last two decades, highlights are the resolution of the $A_2$ conjecture (now theorem) by Hyt\"onen in 2012 and the resolution of the two weight problem for the Hilbert transform by Lacey, Sawyer, Shen and Uriarte Tuero in 2014. Among the competing methods used to prove these results were Bellman functions, corona decompositions, and domination by sparse operators. The later method has gained a lot of traction and is being widely used in contexts beyond what it was originally conceived for in work of Lerner, several of these new applications have originated here at Gatech. In this talk I would like to tell you what I know about t-Haar multipliers (some work goes back to my PhD thesis and joint work with Nets Katz and with my former students Daewon Chung, Jean Moraes, and Oleksandra Beznosova), and what we ought to know in terms of sparse domination.

A Tb Theorem for compactness and boundedness of Calderón-Zygmund operators

Series
Analysis Seminar
Time
Wednesday, November 8, 2017 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Francisco Villarroya UGA
In this talk I will introduce a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on L^p(R^n) by means of testing functions as general as possible. In the classical theory for boundedness, the testing functions satisfy a non-degeneracy property called accretivity, which essentially implies the existence of a positive lower bound for the absolute value of the averages of the testing functions over all dyadic cubes. However, in the setting of compact operators, due to their better properties, the hypothesis of accretivity can be relaxed to a large extend. As a by-product, the results also describe those Calderón-Zygmund operators whose boundedness can be checked with non-accretive testing functions.

Bispectrality and superintegrability

Series
Analysis Seminar
Time
Wednesday, November 1, 2017 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Plamen IlievGeorgia 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 GreenblattUniversity 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 YosevichUniversity 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 AldroubiVanderbilt 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 KaragulyanInstitute 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 NorthingtonGeorgia 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 PlinioUniversity 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

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