Seminars and Colloquia by Series

Uncertainty principles and Schrodinger operators on fractals

Series
Analysis Seminar
Time
Wednesday, October 23, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kasso Okoudjou – University of Maryland and M.I.T. –

In the first part of this talk, I will give an overview of a theory of harmonic analysis on a class of fractals that includes the Sierpinski gasket. The starting point of the theory is the introduction by J. Kigami of a Laplacian operator on these fractals. After reviewing the construction of this fractal Laplacian, I will survey some of the properties of its spectrum. In the second part of the talk, I will discuss the fractal analogs of the Heisenberg uncertainty principle, and the spectral properties a class of  Schr\"odinger operators.  

A random walk through sub-riemanian geometry

Series
Analysis Seminar
Time
Wednesday, October 9, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Masha Gordina – University of Connecticut –

A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors.  Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators.  We will  construct a random walk on M which converges to a process whose infinitesimal generator  is  one of the natural sub-elliptic  Laplacian  operators.  We will also describe these  Laplacians geometrically and discuss the difficulty of defining one which is canonical.   Examples will be provided.  This is a joint work with Tom Laetsch.

Variants of the Christ-Kiselev lemma and an application to the maximal Fourier restriction

Series
Analysis Seminar
Time
Wednesday, September 25, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vjekoslav Kovac – University of Zagreb

Back in the year 2000, Christ and Kiselev introduced a useful "maximal trick" in their study of spectral properties of Schro edinger operators.
The trick was completely abstract and only at the level of basic functional analysis and measure theory. Over the years it was reproven,
generalized, and reused by many authors. We will present its recent application in the theory of restriction of the Fourier transform to
surfaces in the Euclidean space.

A complex analytic approach to mixed spectral problems

Series
Analysis Seminar
Time
Wednesday, September 18, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Speaker
Burak Hatinoğlu – Texas A&M

This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional regular Schroedinger operators, we will discuss complex analytic methods for the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential?

\ell^p improving and sparse inequalities for averages along the square integers

Series
Analysis Seminar
Time
Wednesday, September 11, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rui Han – Georgia Tech

Let $f$ be defined on $\mathbb{Z}$. Let $A_N f$ be the average of $f$ along the square integers. 

We show that $A_N$ satisfies a local scale-free $\ell^{p}$-improving estimate, for $3/2

This parameter range is sharp up to the endpoint. We will also talk about sparse bounds for the maximal function 
$A f =\sup _{N\geq 1} |A_Nf|$. This work is based on a joint work with Michael T. Lacey and Fan Yang.

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 Kolountzakis – University 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 Lacey – Georgia 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 Park – Georgia 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.

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