Seminars and Colloquia by Series

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

Laplacian Eigenfunctions and the Hot Spots conjecture

Series
Analysis Seminar
Time
Wednesday, November 10, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ZOOM (see abstract for link)
Speaker
Stefan SteinerbergerUniversity of Washington

The Hot Spots conjecture (due to J. Rauch from the 1970s) is one of the most interesting open problems in elementary PDEs: it basically says that if we run the heat equation in an insulated domain for a long period of time, then the hottest and the coldest spot will be on the boundary. What makes things more difficult is that the statement is actually false but that it's extremely nontrivial to construct counterexamples. The statement is widely expected to be true for convex domains but even triangles in the plane were only proven recently. We discuss the problem, show some recent pictures of counterexample domains and discuss some philosophically related results: (1) the hottest and the coldest spots are at least very far away from each other and (2) whenever the hottest spot is inside the domain, it is not that much hotter than the hottest spot on the boundary. Many of these questions should have analogues on combinatorial graphs and we mention some results in that direction as well.

The seminar will be held on Zoom and can be found at the link

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

Signal Reconstruction, Operator Representations of Frames, and Open Problems in Dynamical Sampling

Series
Analysis Seminar
Time
Wednesday, November 3, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ZOOM (see abstract for link)
Speaker
Victor BaileyGeorgia Tech

Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know  $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. In this setting, we can obtain conditions on $\Omega, A, l_i$ that allow the stable reconstruction of $f$. Dynamical Sampling is closely related to frame theory and has applications to wireless sensor networks among other areas. In this talk, we will discuss the Dynamical Sampling problem, its motivation, related problems inspired by it, current/future work, and open problems. 

The seminar will be held on Zoom and can be found at the link

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

A Non-commutative Take on F. and M. Riesz

Series
Analysis Seminar
Time
Wednesday, October 20, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ZOOM
Speaker
Edward TimkoGeorgia Tech

In this talk, we present an operator theoretic analogue of the F. and M. Riesz Theorem. We first recast the classical theorem in operator theoretic terms. We then establish an analogous result in the context of representations of the Cuntz algebra, highlighting notable differences from the classical setting. At the end, we will discuss some extensions of these ideas. This is joint work with R. Clouâtre and R. Martin.

Zoom Link:  

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

Absolute continuity and the Banach-Zaretsky Theorem

Series
Analysis Seminar
Time
Wednesday, September 29, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE (Zoom link in abstract)
Speaker
Chris HeilGeorgia Tech

This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday.  Years ago, John wrote a text "Real Variable and Integration", published in 1976.  This was not the text that I first learned real analysis from, but it became an important reference for me.  A later revision and expansion by John and Wojtek Czaja appeared in 2009.  Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory.  My goal was that each proof was to be both rigorous and enlightening.  I failed (in the chapters on differentiation and absolute continuity).  I will discuss the real analysis theorem whose proof I find the most difficult and unenlightening.  But I will also present the Banach-Zaretsky Theorem, which I first learned from John's text.  This is an elegant but often overlooked result, and by using it I (re)discovered enlightening proofs of theorems whose standard proofs are technical and difficult.  This talk will be a tour of the absolutely fundamental concept of absolute continuity from the viewpoint of the Banach-Zaretsky Theorem.

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

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