Seminars and Colloquia by Series

Bases of exponentials and tilings

Series
Analysis Seminar
Time
Wednesday, September 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mihalis KolountzakisUniversity of Crete

Mathematicians have long been trying to understand which domains admit an orthogonal (or, sometimes, not) basis of exponentials of the form , for some set of frequencies (this is the spectrum of the domain). It is well known that we can do so for the cube, for instance (just take ), but can we find such a basis for the ball? The answer is no, if we demand orthogonality, but this problem is still open when, instead of orthogonality, we demand just a Riesz basis of exponentials.

 
This question has a lot to do with tiling by translation (i.e., with filling up space with no overlaps by translating around an object). Fuglede originally conjectured that an orthogonal exponential basis exists if and only if the domain can tile space by translation. This has been disproved in its full generality but when one adds side conditions, such as, for instance, a lattice set of frequencies, or the space being a group of a specific type, or many other natural conditions, the answer is often unknown, and sometimes known to be positive or known to be negative. A major recent  development is the proof (2019) by Lev and Matolcsi of the truth of the Fuglede conjecture for convex bodies in all dimensions.
 
This is a broad area of research, branching out by varying the side conditions on the domain or the group in which the domain lives, or by relaxing the orthogonality condition or even allowing time-frequency translates of a given function to serve as basis elements (Gabor, or Weyl-Heisenberg, bases). When working with both exponential bases and tiling problems the crucial object of study turns out to be the zero set of the Fourier Transform of the indicator function of the domain we care about. In particular we want to know how large structured sets this zero set contains, for instance how large difference sets it contains or what kind of tempered distributions it can support.
 
In this talk I will try to show how these objects are tied together, what has been done recently, and indicate specific open problems.

Averages over Discrete Spheres

Series
Analysis Seminar
Time
Wednesday, August 28, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LaceyGeorgia Tech

Fine properties of spherical averages in the continuous setting include
$L^p$  improving estimates
and sparse bounds, interesting in the settings of a fixed radius, lacunary sets of radii, and the
full set of radii. There is a parallel theory in the setting of discrete spherical averages, as studied
by Elias Stein, Akos Magyar, and Stephen Wainger. We recall the continuous case, outline the
discrete case, and illustrate a unifying proof technique. Joint work with Robert Kesler, and
Dario Mena Arias.

Energy on Spheres and Discreteness of Minimizing Measures

Series
Analysis Seminar
Time
Wednesday, April 10, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josiah ParkGeorgia Tech

When equiangular tight frames (ETF's), a type of structured optimal packing of lines, exist and are of size $|\Phi|=N$, $\Phi\subset\mathbb{F}^d$ (where $\mathbb{F}=\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$), for $p > 2$ the so-called $p$-frame energy $E_p(\Phi)=\sum\limits_{i\neq j} |\langle \varphi_{i}, \varphi_{j} \rangle|^p$ achieves its minimum value on an ETF over all sized $N$ collections of unit vectors. These energies have potential functions which are not positive definite when $p$ is not even. For these cases the apparent complexity of the problem of describing minimizers of these energies presents itself. While there are several open questions about the structure of these sets for fixed $N$ and fixed $p$, we focus on another question:

What structural properties are expressed by minimizing probability measures for the quantity $I_{p}(\mu)=\int\limits_{\mathbb{S}_{\mathbb{F}}^{d-1}}\int\limits_{\mathbb{S}_{\mathbb{F}}^{d-1}} |\langle x, y \rangle|^p d\mu(x) d\mu(y)$?
We collect a number of surprising observations. Whenever a tight spherical or projective $t$-design exists for the sphere $\mathbb{S}_{\mathbb{F}}^d$, equally distributing mass over it gives a minimizer of the quantity $I_{p}$ for a range of $p$ between consecutive even integers associated with the strength $t$. We show existence of discrete minimizers for several related potential functions, along with conditions which guarantee emptiness of the interior of the support of minimizers for these energies. 
This talk is based on joint work with D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.

On some extremal problems for polynomials

Series
Analysis Seminar
Time
Wednesday, April 3, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex StokolosGeorgia Southern

In this talk we will discuss some some extremal problems for polynomials. Applications to the problems in discrete dynamical systems as well as in the geometric complex analysis will be suggested.

Energy minimization on the sphere.

Series
Analysis Seminar
Time
Wednesday, March 27, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitry BilykUniversity of Minnesota

Many problems of spherical discrete and metric geometry may be reformulated as energy minimization problems and require techniques that stem from harmonic analysis, potential theory, optimization etc. We shall discuss several such problems as well of applications of these ideas to combinatorial geometry, discrepancy theory, signal processing etc.

The Bishop-Phelps-Bolloba ́s Property for Numerical Radius in the space of summable sequnces

Series
Analysis Seminar
Time
Wednesday, March 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Olena KozhushkinaUrsinus college
The Bishop-Phelps-Bolloba ́s property for numerical radius says that if we have a point in the Banach space and an operator that almost attains its numerical radius at this point, then there exist another point close to the original point and another operator close to the original operator, such that the new operator attains its numerical radius at this new point. We will show that the set of bounded linear operators from a Banach space X to X has a Bishop-Phelps-Bolloba ́s property for numerical radius whenever X is l1 or c0. We will also discuss some constructive versions of the Bishop-Phelps- Bolloba ́s theorem for l1(C), which are an essential tool for the proof of this result.

A restriction estimate in $\mathbb{R}^3$

Series
Analysis Seminar
Time
Wednesday, March 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hong WangMIT

If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of f? Stein conjectured in the 1960s that for any $p>3$, $\|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}$.

We make a small progress toward this conjecture and show that it holds for $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao.

Schur multipliers in perturbation theory

Series
Analysis Seminar
Time
Wednesday, February 27, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Anna SkripkaUniversity of New Mexico
Linear Schur multipliers, which act on matrices by entrywisemultiplications, as well as their generalizations have been studiedfor over a century and successfully applied in perturbation theory. Inthis talk, we will discuss extensions of Schur multipliers tomultilinear infinite dimensional transformations and then look intoapplications of the latter to approximation of operator functions.

The symmetric Gaussian isoperimetric inequality

Series
Analysis Seminar
Time
Wednesday, February 20, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Steven HeilmanUSC
It is well known that a Euclidean set of fixed Euclidean volume with least Euclidean surface area is a ball. For applications to theoretical computer science and social choice, an analogue of this statement for the Gaussian density is most relevant. In such a setting, a Euclidean set with fixed Gaussian volume and least Gaussian surface area is a half space, i.e. the set of points lying on one side of a hyperplane. This statement is called the Gaussian Isoperimetric Inequality. In the Gaussian Isoperimetric Inequality, if we restrict to sets that are symmetric (A= -A), then the half space is eliminated from consideration. It was conjectured by Barthe in 2001 that round cylinders (or their complements) have smallest Gaussian surface area among symmetric sets of fixed Gaussian volume. We discuss our result that says this conjecture is true if an integral of the curvature of the boundary of the set is not close to 1. https://arxiv.org/abs/1705.06643 http://arxiv.org/abs/1901.03934

Some results for functionals of Aharanov-Bohm type

Series
Analysis Seminar
Time
Wednesday, February 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LossGeorgia Tech
In this talk I present some variational problems of Aharanov-Bohm type, i.e., they include a magnetic flux that is entirely concentrated at a point. This is maybe the simplest example of a variational problems for systems, the wave function being necessarily complex. The functional is rotationally invariant and the issue to be discussed is whether the optimizer have this symmetry or whether it is broken.

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