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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.

Series: Analysis Seminar

We show that multilinear dyadic paraproducts and Haar multipliers, as well as their commutators with locally integrable functions, can be pointwise dominated by multilinear sparse operators. These results lead to various quantitative weighted norm inequalities for these operators. In particular, we introduce multilinear analog of Bloom's inequality, and prove it for the commutators of the multilinear Haar multipliers.

Series: Analysis Seminar

The Linear Independence of Time-Frequency Translates Conjecture,
also known as the HRT conjecture, states that any finite set of
time-frequency translates of a given $L^2$ function must be linearly
independent. This conjecture, which was first stated in print in 1996,
remains open today. We will discuss this conjecture, its relation to
the Zero Divisor Conjecture in abstract algebra, and the (frustratingly
few) partial results that are currently available.

Series: Analysis Seminar

We show that the multiwavelets, introduced by Alpert in 1993, are
related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and
for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm,
using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the
matrices in the scaling relation for any size of the multiplicity of the
multiwavelets.Based on joint work with J.S. Geronimo and P. Iliev