Abstract:  We will report on some recent joint work with I.Ruzsa and with
H.Furstenberg and B.Weiss which is exemplified by the following results
(the proofs of which use ergodic-theoretic techniques):
Theorem 1. If A is a set of positive upper density in Z^2, then
A-A contains a Cartesian square BxB, where B is a symmetric subset of Z
which has positive asymptotic density.
Theorem 2. There exists a set E in Z^3 which has positive upper density
such that E-E DOES NOT contain any Cartesian cube BxBxB with B having
positive upper density.
Theorem 3. If A, B, C are three subsets of Z with positive upper density
and one of them is syndetic, then A+B+C is a Bohr set. 

Abstract:

Abstract: The word "low-pass filter" refers to a particular kind of Fourier multiplier used to construct a scaling function for a wavelet basis on L^2(R). The characterization of polynomial low-pass filters was accomplished by A.Cohen and W.Lawton around 1990; subsequently, claims were made that their results held for the class of continuous periodic functions. With Dobric and Hitczenko, we showed that this result is false. The Cohen-Lawton theorem can be viewed as a type of Perron-Frobenius uniqueness theorem for eigenvectors/eigenvalues of a positive operator acting on the class of trigonometric polynomials. However, the special requirements imposed by the wavelet setting introduce an obstruction to this approach when the candidate filter is merely continuous. This talk will outline how probability and ergodic theory provides a solution to this problem.

Abstract: Mildly mixing transformations were introduced by Furstenberg and Weiss, who provided an intriguing characterization as multipliers of a certain class of mpt's on infinite measure spaces. This class has weak mixing but not strong mixing. Roughly speaking, no nonconstant function can be recurrent under powers of the mpt. The traditional classes: ergodic, weak mixing, strong mixing, are easily seen to be Borel sets (when the probability space is separable and one uses the strong topology of operators on Hilbert space.) For mild mixing this is no longer so. To see this one uses a method of descriptive set theory due to A. Kechris and R. Lyons, a perfect Kronecker set, and so-called normal processes. We also mention another class (akin to partially mixing) which is equally natural and may or may not be the same as mildly mixing.

Abstract: We will discuss some quantitative uniqueness theorems at infinity, for solutions to elliptic, parabolic and dispersive equations. These results have applications to disordered media, regularity of solutions and hopefully to control theory.

Abstract: Given a map $v$ from the plane into the unit circle, consider a truncated Hilbert transform computed on line segments determined by $v$
$$
H_v f(x)=p.v. \int_{-1}^{1} f(x-y v(x)) dy/y
$$
This is a particular Radon transform. We show that assuming only that $v$ has $1+\epsilon$ derivatives, that $H_v$ maps $L^2$ into $L^2$. As a corollary we derive Carleson's theorem on pointwise convergence of Fourier series.
The methods of proof invoke methods of Carleson's Theorem, appropriately adapted to the plane. We also use a variant of the Kakeya maximal function, specifically adapted to the choice of $v$. This is joint work with Xiaochun Li.

Abstract

Abstract: We use the Koosis theorem to find the asymptotics of orthogonal polynomials for the case when a Szego measure on the unit circumference is perturbed by an arbitrary finite measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk.

Abstract: We want to understand the pointwise almost everywhere convergence of convolutions by an approximate identity. For some approximate identities this is possible in general. For some, only some things are clear on some Lebesgue spaces. The known results and open questions in this area will be presented in this talk.

Abstract:

Abstract: The ideas and results I will present here come from recent work with Jack Feldman and Elon Lindenstrauss. Ergodic theorems are an extremely broad and rich field with wide application. My intention is to explore a kind of ergodic theorem that is somewhat different from the usual. I will begin with the central result in pointwise theory, the Birkhoff ergodic theorem. This classical result tells us that for $T$ a measure preserving transformation of a probability space, and $f$ in $L^1$ of that space, the averages $\frac 1n\sum_{i=0}^{n-1} f(T^i(x))$ converge almost everywhere. Following a very classical and successful approach, I will split this result into two parts. One first obtains a maximal lemma which shows that the set of $f$ for which the result holds is an $L^1$ closed subspace. Then one exhibits a dense family of functions for which the result holds. The only variation I make here is to use the Besikovitch covering lemma to obtain the maximal lemma. I will next take a rather different perspective on this result, one not so tied to the action $T$. One can regard the partitioning of the measure space into orbits as a foliation of the space by copies of $\mathbb Z$. The set of orbit points $x, T(x), ... , T^{n-1}(x)$ form a large ball in such an orbit. The measure $\mu$ can be disintigrated over such large balls as a family of conditional measures. In this case the disintigration is very simple, it is a uniform atomic measure putting mass $1/n$ on each atom. The averages that the Birkhoff theorem shows converge are nothing more than the averages with respect to these fiber measures over ever larger balls. From this persective one then can start to formulate a general picture. Suppose now $X$ is a Polish space, a manifold if you like, and suppose it is foliated by leaves that we will assume are topologically $\mathbb R^n$ or $\mathbb Z^n$. Further suppose you have a Borel probability measure $\mu$ on $X$. Under mild assumptions one can again decompose the measure $\mu$ onto balls of radius $N$ on the leaves of the foliation. This measure now though can be quite singular on the leaves, giving mass 0 to large regions of the foliation. None the less one can ask if a Birkhoff theorem holds, or under what conditions it will hold. I will show that in this very loose perspective one still gets a maximal lemma from the Besikovitch covering theorem and this reduces the ergodic theorem in this perspective to whether or not a dense family exists. I will discuss a few known applications of this perspective; an ergodic theorem for the Paterson-Sullivan measure on Horocycles for geometrically finite Fuchsian groups by myself and a pointwise ergodic theorem for nonsingular and recurrent commuting transformations by Jack Feldman. I will outline where we are currently working to move forward with this program.

Abstract: Pointwise, maximal and dominated ergodic theorem with "weighted" averages for Lamperti representations of amenable \sigma-compact locally compact groups in L^\alpha (1<\alpha<\infty) over a \sigma-finite measure space (\Omega,\mathcal F,m) are proved (in particular, these theorems imply ergodic theorems for positive group representations and group actions). We discuss various conditions on the "weights" under which these theorems hold.

Abstract: In the talk we consider the existence and uniqueness of non-linear Fourier transform that appears in questions of scattering of discrete Scroedinger operator. We show how recent results in Harmonic Analysis (including matrix Hunt-Muckenhoupt-Wheeden weights) naturally appear in the area. We also explain a very simple approach to prove scattering in many cases.

Abstract: These problems, which have been studied for about 15 years, are analogues of older, continuous problems. I hope to explain the motivation for the study of these problems. Work on these problems uses ideas from number theory. It is natural to wonder whether number theoretic techniques are "natural" for these problems. I hope to discuss this issue. If there is time I would like to discuss recent results due to Ionescu, Magyar, Stein and myself.

Abstract: We give a short review of the work of Marshall Ash. His work is extensive so it is necessary to restrict the areas covered. After visiting his interest areas we restrict our discussion to differentiation. We briefly review historical work and finally concentrate on Marshall's work in differentiation.

Abstract: In several papers with Roger Jones and some others, we investigated oscillatory behavior of ergodic averages, that is, we examined upcrossings, jumps, squarefunctions and related beasts. While we established some basic inequalities---such as the one which gives a direct relationship between martingales and ergodic averages---quite a few problems remained. Some of these problems seem fundamental, some are puzzling, some are surprising, and some are all three. In my talk, I shall describe a selection of these unsolved problems.