Seminars and Colloquia by Series

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 $L$.   

 

Calibrations and energy-minimizing maps of rank-1 symmetric spaces

Series
Analysis Seminar
Time
Wednesday, March 16, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joeseph HoisungtonUniversity of Georgia

We will prove lower bounds for energy functionals of mappings of the real, complex and quaternionic projective spaces with their canonical Riemannian metrics.  For real and complex projective spaces, these results are sharp, and we will characterize the family of energy-minimizing mappings which occur in these results.  For complex projective spaces, these results extend to all Kahler metrics.  We will discuss the connections between these results and several theorems and questions in systolic geometry.

L^2-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

Series
Analysis Seminar
Time
Wednesday, March 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Carmelo PuliattiUniversity of the Basque Country, Spain

We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix  $A$ with real, bounded, and possibly non-symmetric coefficients. If a proper {$L^1$-mean oscillation} of the coefficients of $A$ satisfies suitable Dini-type assumptions, we prove the following: if $\mu$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$,   and

$$T_\mu f(x)=\int \nabla_x\Gamma_A (x,y)f(y)\, d\mu(y)$$

denotes the gradient of the single layer potential associated with $L_A$, then

$$1+ \|T_\mu\|_{L^2(\mu)\to L^2(\mu)}\approx 1+ \|\mathcal R_\mu\|_{L^2(\mu)\to L^2(\mu)},$$

where $\mathcal R_\mu$ indicates the $n$-dimensional Riesz transform. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for $\mathcal R_\mu$, which were recently extended to  $T_\mu$ under a H\"older continuity assumption on the coefficients of the matrix $A$.

This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

Measure theoretic Rogers-Shephard and Zhang type inequalities

Series
Analysis Seminar
Time
Wednesday, February 9, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Michael RoysdonTel Aviv University

This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures, the first of which is a reverse form of the Brunn-Minkowsk inequality, and the second of which can be seen to be a reverse affine isoperimetric inequality; the feature of both inequalities is that they each provide a classification of the n-dimensional simplex in the volume case. The covariogram of a measure plays an essential role in the proofs of each of these inequalities. In particular, we will discuss a variational formula concerning the covariogram resulting in a measure theoretic version of the projection body, an object which has recently gained a lot of attention--these objects were previously studied by Livshyts in her analysis of the Shephard problem for general measure.

 

The talk will be on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

 

Persistence Exponents for Gaussian stationary functions

Series
Analysis Seminar
Time
Wednesday, February 2, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Naomi FeldheimBar-Ilan University

Let f be a real-valued Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.

What is the probability that f remains above a certain fixed line for a long period of time?

We give simple spectral(and almost tight) conditions under which this probability is asymptotically exponential, that is, that the limit of log P(f>a on [0,T])/ T, as T approaches infinity, exists.

This limit defines "the persistence exponent", and we further show it is continuous in the level a, in the spectral measure corresponding to f (in an appropriate sense), and is unaffected by the singular part of the spectral measure.

Proofs rely on tools from harmonic analysis.

Joint work with Ohad Feldheim and Sumit Mukherjee, arXiv:2112.04820.

The talk will be on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

An adaptation of Kohler-Jobin rearrangement technique with fixed torsional rigidity to the Gaussian space

Series
Analysis Seminar
Time
Wednesday, January 26, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Orli HerscoviciGeorgia Tech

Please Note:

In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a ``modified''  torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure.

We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.
 

The seminar will be held on Zoom via the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

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