Seminars and Colloquia by Series

Zak transform analysis of shift-invariant subspaces

Series
Analysis Seminar
Time
Wednesday, August 23, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joey IversonUniversity of Maryland
Abstract: Shift-invariant (SI) spaces play a prominent role in the study of wavelets, Gabor systems, and other group frames. Working in the setting of LCA groups, we use a variant of the Zak transform to classify SI spaces, and to simultaneously describe families of vectors whose shifts form frames for the SI spaces they generate.

Self-similar tilings of General Fractal Blow-ups and Anderson Putnam Theory

Series
Analysis Seminar
Time
Wednesday, June 21, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael F. BarnsleyAustralian National University
In this seminar I will discuss current work, joint with AndrewVince and Alex Grant. The goal is to tie together several related areas, namelytiling theory, IFS theory, and NCG, in terms most familiar to fractal geometers.Our focus is on the underlying code space structure. Ideas and a conjecture willbe illustrated using the Golden b tilings of Robert Ammann

Density theorem for continuous frames and the uncertainty principle

Series
Analysis Seminar
Time
Wednesday, April 19, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mishko MitkovskiiClemson University
A well-known elementary linear algebra fact says that any linear independent set of vectors in a finite-dimensional vector space cannot have more elements than any spanning set. One way to obtain an analog of this result in the infinite dimensional setting is by replacing the comparison of cardinalities with a more suitable concept - which is the concept of densities. Basically one needs to compare the cardinalities locally everywhere and then take the appropriate limits. We provide a rigorous way to do this and obtain a universal density theorem that generalizes many classical density results. I will also discuss the connection between this result and the uncertainty principle in harmonic analysis.

Falconer type theorems for simplices

Series
Analysis Seminar
Time
Wednesday, April 12, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eyvi PalssonVirginia Tech
Finding and understanding patterns in data sets is of significant importance in many applications. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will present recent progress on Falconer type problems for simplices. The main techniques used come from analysis and geometric measure theory.

Bounding marginals of product measures

Series
Analysis Seminar
Time
Wednesday, April 5, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Tech
It was shown by Keith Ball that the maximal section of an n-dimensional cube is \sqrt{2}. We show the analogous sharp bound for a maximal marginal of a product measure with bounded density. We also show an optimal bound for all k-codimensional marginals in this setting, conjectured by Rudelson and Vershynin. This bound yields a sharp small ball inequality for the length of a projection of a random vector. This talk is based on the joint work with G. Paouris and P. Pivovarov.

Persistence as a spectral property

Series
Analysis Seminar
Time
Wednesday, March 29, 2017 - 02:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shahaf NitzanGeorgia Tech
A Gaussian stationary sequence is a random function f: Z --> R, for which any vector (f(x_1), ..., f(x_n)) has a centered multi-normal distribution and whose distribution is invariant to shifts. Persistence is the event of such a random function to remain positive on a long interval [0,N]. Estimating the probability of this event has important implications in engineering , physics, and probability. However, though active efforts to understand persistence were made in the last 50 years, until recently, only specific examples and very general bounds were obtained. In the last few years, a new point of view simplifies the study of persistence, namely - relating it to the spectral measure of the process. In this talk we will use this point of view to study the persistence in cases where the spectral measure is 'small' or 'big' near zero. This talk is based on Joint work with Naomi Feldheim and Ohad Feldheim.

Means and powers of convex bodies

Series
Analysis Seminar
Time
Wednesday, March 15, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Liran RotemUniversity of Minnesota
In this talk we will discuss several ways to construct new convex bodies out of old ones. We will start by defining various methods of "averaging" convex bodies, both old and new. We will explain the relationships between the various definitions and their connections to basic conjectures in convex geometry. We will then discuss the power operation, and explain for example why every convex body has a square root, but not every convex body has a square. If time permits, we will briefly discuss more complicated constructions such as logarithms. The talk is based on joint work with Vitali Milman.

Sparse operators and the sparse T1 Theorem

Series
Analysis Seminar
Time
Wednesday, March 8, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dario MenaGeorgia Tech
We impose standard $T1$-type assumptions on a Calderón-Zygmund operator $T$, and deduce that for bounded compactly supported functions $f,g$ there is a sparse bilinear form $\Lambda$ so that $$ |\langle T f, g \rangle | \lesssim \Lambda (f,g). $$ The proof is short and elementary. The sparse bound quickly implies all the standard mapping properties of a Calderón-Zygmund on a (weighted) $L^p$ space.

Do Minkowski averages get progressively more convex?

Series
Analysis Seminar
Time
Wednesday, March 1, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Artem ZvavitchKent State University
For a compact subset $A$ of $R^n$ , let $A(k)$ be the Minkowski sum of $k$ copies of $A$, scaled by $1/k$. It is well known that $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to infinity. A few years ago, Bobkov, Madiman and Wang conjectured that the volume of $A(k)$ is non-decreasing in $k$, or in other words, that when the volume deficit between the convex hull of $A$ and $A(k)$ goes to $0$, it actually does so monotonically. While this conjecture holds true in dimension $1$, we show that it fails in dimension $12$ or greater. Then we consider whether one can have monotonicity of convergence of $A(k)$ when its non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of $A(k)$ converges monotonically to $0$ as $k$ increases; even the convergence does not seem to have been known before. We also obtain some results for the Hausdorff distance to the convex hull, along the way clarifying various properties of these notions of non-convexity that may be of independent interest.Joint work with Mokshay Madiman, Matthieu Fradelizi and Arnaud Marsiglietti.

Interpolation sets and arithmetic progressions

Series
Analysis Seminar
Time
Wednesday, February 8, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Itay LondnerTel-Aviv University
Given a set S of positive measure on the unit circle, a set of integers K is an interpolation set (IS) for S if for any data {c(k)} in l^2(K) there exists a function f in L^2(S) such that its Fourier coefficients satisfy f^(k)=c(k) for all k in K. In the talk I will discuss the relationship between the concept of IS and the existence of arbitrarily long arithmetic progressions with specified lengths and step sizes in K. Multidimensional analogues of this subject will also be considered.This talk is based on joint work with Alexander Olevskii.

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