Dept. of Mathematics and Applied Math.: The Analysis Seminar in Crete -- Σεμινάριο Ανάλυσης

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We are looking at families of functions or measures on the torus (in dimension one and two) which are specified by a finite number of parameters $N$. The task, for a given family, is to look at a small number of Fourier coefficients of the object, at a set of locations that is predetermined and may depend only on $N$, and determine the object. We look at (a) the indicator functions of at most $N$ intervals of the torus and (b) at sums of at most $N$ complex point masses on the two-dimensional torus. In the first case we reprove a theorem of Courtney which says that the Fourier coefficients at the locations $0, 1, \ldots, N$ are sufficient to determine the function (the intervals). In the second case we produce a set of locations of size $O(N \log N)$ which suffices to determine the measure.

Abstract: We will give several geometric characterizations of conformal maps in weighted Bergman spaces. These characterizations involve certain conformal invariants, as well as certain Euclidean geometric quantities and they extend some known results for Hardy spaces to weighted Bergman spaces. Moreover, we shall discuss the notion of the Bergman number which is the analogue of the Hardy number for weighted Bergman spaces.

Abstract: In the problem of the distance set we are seeking a lower bound for the "size" of the set of all distances between points of a set $A$ as a function of the "size" of the set $A$. While all approaches are using the Fourier Transform, I will describe a naive geometric method, which, without giving optimal results, can give non-trivial information.

Old and new results will be described on the following question: if $T$ is a common set of coset representatives for the subgroups $G_i$ of the abelian group $G$, all of them of the same index, how large must $T$ be? This will be discussed under various assumptions and several notions of what it means to be large.

The $G_i$ are mostly lattices in Euclidean spaces, and $T$ is mostly assumed to be measurable, but not always. Several open questions, of varying levels of difficulty will be presented.

A characterization of the unit disk and the harmonic measure doubling condition

Abstract: Suppose $D$ is a bounded Jordan domain in the plane. A well known theorem by Jerison and Kenig states that the boundary of $D$ is a quasicircle if and only if both $D$ and its complement are doubling domains with respect to the harmonic measure. This theorem fails if we only assume that $D$ is a doubling domain. We show that if $D$ is a doubling domain with constant $c = 1$, then it must be a disk.

Abstract: Furstenberg systems are measure preserving systems that are used to model the statistical behavior of bounded sequences of complex numbers. I will give a variety of examples of such systems and briefly explain how their dynamical properties can be used to prove multiple recurrence results that for the moment do not seem to be attainable by more traditional techniques. To make this more interactive, feel free to bring your favorite sequence and we'll see if we can say something interesting about the corresponding Furstenberg system.

3/31/2022, 11:00, Room: A303
Speaker: Kostantinos Tsinas (University of Crete)

Abstract: Our problem is to determine the limiting behavior of multiple ergodic averages, where the iterates involve sequences that arise from smooth, well-behaved functions and which do not grow faster than polynomials. We show that under some suitable linear independence assumptions, the corresponding averages converge (in the $L^2$ sense ) to the product of the integrals of the involved functions in ergodic systems. Our approach relies on some recent joint ergodicity results and some new seminorm estimates for multiple ergodic averages in our setting.

Abstract: In this talk, we will describe a standard two-step procedure for establishing pointwise convergence of Ergodic Averages. We will apply that procedure for a wide class of non-conventional ergodic averages and we will use quantitative forms of pointwise convergence to carry out those two steps and establish pointwise convergence on $L^1$. (Notably, we will disprove a conjecture of Rosenblatt–Wierdl.)

Abstract: The polynomial Szemerédi theorem of Bergelson and Leibman is a central result at the interface between ergodic theory and additive combinatorics, extending earlier results of Szemerédi and Furstenberg on arithmetic progressions. It states that each dense subset of integers contains certain polynomial configurations. The theorem follows from an ergodic theoretic result on the convergence of multiple ergodic averages with polynomial iterates. The limiting behaviour of such averages has been an object of intensive study by ergodic theorists and additive combinatorists alike. In this talk, I will discuss new results in this direction. Specifically, I will characterise the limits of multiple ergodic averages related to certain families of polynomial configurations for which little has been known previously. While doing so, I will highlight an interesting connection between the form that these limits take and the algebraic relations satisfied by the polynomial configurations.

Abstract: Sets of recurrence (and variants thereof) form a class of sets that arises naturally in ergodic theory, while van der Corput sets (and variants thereof) arise in the theory of uniform distribution of sequences. Even though these classes look a-priori unrelated, it turns out that they are closely connected, and in fact it is difficult to produce examples showing that they are not the same class of sets. In this talk, we construct a set of strong recurrence (which is a natural strengthening of the notion of recurrence) which is not a van der Corput set. We will not assume any familiarity with those notions, and everything we need will be introduced.

7/21/2022, 11:00, Room: A303
Speaker: Valeria Fragkiadaki (Texas A&M University)

Abstract: Sparse operators are usually seen as upper bounds for the norms of certain operators like for example Calderon - Zygmund or paraproducts. In this talk we try to bound sparse operators from above. We introduce some “sparse BMO” functions and use these to express sparse operators as sums of paraproducts and martingale transforms as well as to obtain an equivalence of norms between sparse operators and compositions of paraproducts.

Abstract: Delasrte's LP bound is useful whenever one wants to give an upper bound on the cardinality of a set with prescribed differences (e.g. all Hamming distance larger than d for codewords of length $n$, or all differences are quadratic residues in a cyclic group, etc). I will describe a connection between Delsarte's LP bound and spectral sets and tiles in finite groups and $\RR^n$.