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Series: Analysis Seminar

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

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

The theory of two variable orthogonal polynomials is not very well
developed. I will discuss some recent results on two variable orthogonal
polynomials on the bicircle and time permitting on the square associate
with orthogonality measures that are one over a trigonometric
polynomial. Such measures have come to be called Bernstein-Szego
measures.
This is joint work with Plamen Iliev and Greg Knese.