Seminars and Colloquia by Series

TBA Ruixiang Zhang

Series
Analysis Seminar
Time
Wednesday, March 29, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruixiang ZhangUC Berkeley

Remarks on mobile sampling for general surfaces

Series
Analysis Seminar
Time
Wednesday, January 25, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ben JayeGaTech

Mobile sampling concerns finding surfaces upon which any function with Fourier transform supported in a symmetric convex set must have some large values.   We shall describe a sharp sufficient for mobile sampling in terms of the surface density introduced by Unnikrishnan and Vetterli.  Joint work with Mishko Mitkovski and Manasa Vempati.

Weighted Inequalities for Singular Integral Operators

Series
Analysis Seminar
Time
Wednesday, January 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 268
Speaker
Manasa VempatiGeorgia Tech

Weighted inequalities for singular integral operators are central in the study of non-homogeneous harmonic analysis. Two weight inequalities for singular integral operators, in-particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE's. In this talk, I will discuss several results concerning the two weight inequalities for various Calder\'on-Zygmund operators in both Euclidean setting and in the more generic setting of spaces of homogeneous type in the sense of Coifman and Weiss.

The two-weight conjecture for singular integral operators T was first raised by Nazarov, Treil and Volberg on finding the real variable characterization of the two weights u and v so that T is bounded on the weighted $L^2$ spaces. This conjecture was only solved completely for the Hilbert transform on R until recently. In this talk, I will describe our result that resolves a part of this conjecture for any Calder\'on-Zygmund operator on the spaces of homogeneous type by providing a complete set of sufficient conditions on the pair of weights. We will also discuss the existence of similar analogues for multilinear Calder\'on-Zygmund operators.

Normalizable frames

Series
Analysis Seminar
Time
Wednesday, October 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pu-Ting YuGeorgia Tech

Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a Bessel sequence or a frame for $H$ which does not contain any zero elements. We say that $\{x_n\}$ is a normalizable Bessel sequence or normalizable frame if the normalized sequence $\{x_n/||x_n||\}$ remains a Bessel sequence or frame. In this talk, we will present characterizations of normalizable and non-normalizable frames . In particular, we prove that normalizable frames can only have two formulations.  Perturbation theorems tailored for normalizable frames will be also presented. Finally, we will talk about some open questions related to the normalizable frames.

Bounds on some classical exponential Riesz basis

Series
Analysis Seminar
Time
Wednesday, October 12, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Thibaud AlemanyGeorgia Tech

We estimate the  Riesz basis (RB) bounds obtained in Hruschev, Nikolskii and Pavlov' s classical characterization of exponential RB. As an application, we  improve previously known estimates of the RB bounds in some classical cases, such as RB obtained by an Avdonin type perturbation, or RB which are the zero-set of sine-type functions. This talk is based on joint work with S. Nitzan

Affine spheres over Polygons, Extremal length and a new classical minimal surface: a problem I can do and two I cannot

Series
Analysis Seminar
Time
Wednesday, September 21, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael WolfGeorgia Tech

In this introductory talk, we describe an older result (with David Dumas) that relates hyperbolic affine spheres over polygons to polynomial Pick differentials in the plane. All the definitions will be developed.  In the last few minutes, I will quickly introduce two analytic problems in other directions that I struggle with.

The HRT Conjecture for single perturbations of confi gurations

Series
Analysis Seminar
Time
Wednesday, April 20, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Klaus 1447
Speaker
Kasso OkoudjouTufts University

 In 1996, C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any  non-zero square integrable function $g$ and  subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved. 

 

In this talk,  I will then introduce an inductive approach to investigate the conjecture, by attempting to answer the following question. Suppose the HRT conjecture is true for a function $g$ and a fixed set of $N$ points $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ For what other point $(a, b)\in \mathbb{R}^2\setminus \Lambda$ will the HRT remain true for the same function $g$ and the new set of $N+1$ points $\Lambda'=\Lambda \cup \{(a, b)\}$?  I will report on a recent joint work with V.~Oussa in which we use this approach to prove the conjecture when the initial configuration  $\Lambda=\{(a_k, b_k)\}_{k=1}^N $  is either a subset of the unit lattice $\mathbb{Z}^2$ or a subset of a line $L$.   

 

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