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Analysis Seminar in Crete (2023-24)

Σεμιναριο Αναλυσης

In chronological ordering

29 Aug 2023, 10:15, Room: A303
Speaker: Nir Lev (Bar Ilan University)

Tiling by translates of a function

Abstract: I will discuss tilings of the real line by translates of a function $f$, that is, systems $\{f(x - \lambda), \lambda \in \Lambda\}$ of translates of $f$ that form a partition of unity. Which functions $f$ can tile by translations, and what can be the structure of the translation set $\Lambda$? I will survey the subject and present some recent results.

29 Aug 2023, 11:30, Room: A303
Speaker: Oleksiy Klurman (University of Bristol)

An update on Fekete polynomials

Abstract: Extremal properties of Littlewood polynomials (with coefficients $\pm 1$) have been extensively studied throughout the past century. Among special classes of Littlewood polynomials, particular attention has been given to so-called "Fekete polynomials" (with coefficients being Legendre symbols). Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials and their extremal properties have attracted considerable attention, particularly due to their intimate connection with the putative Siegel zero and the small class number problem. The goal of this talk is to discuss a general approach to understanding the behaviour of such polynomials, resolving several open problems. This is based on joint work with Y. Lamzouri and M. Munsch.

29 Aug 2023, 12:45, Room: A303
Speaker: Mate Matolcsi (Renyi Institute and BME (Budapest))

A new upper bound on the density of unit-avoiding sets in the plane

Abstract: A set W in the plane is unit-avoiding if no two points of W are unit distance away. Erdős conjectured that the maximal upper density of such a set is less than 1/4. In a recent joint work with G. Ambrus, A. Csiszarik, D. Varga, P. Zsamboki, we have proved this conjecture. The proof is an interesting mix of Fourier analysis, geometry, combinatorics and computer science.

Host comment: See related article on Quantamagazine: Mathematicians solve long standing coloring problem.

26 Oct 2023, 11:15, Room: A303
Speaker: Konstantinos Tsinas (University of Crete)

Multiple ergodic averages along primes

Abstract:

We discuss convergence (in $L^2$) results for multiple ergodic averages along sequences of polynomial growth evaluated at primes. Building on the work of Frantzikinakis, Host, and Kra who showed that polynomial ergodic averages along primes converge, we generalize their results to other sequences with polynomial growth. Combining our results with Furstenberg's correspondence principle, we derive several applications in combinatorics. The most interesting application is that positive density subsets of $\mathbb{N}$ contain arbitrarily long arithmetic progressions with common difference of the form $\lfloor{p^c}\rfloor$, where $c$ is a positive non-integer and $p$ is a prime number. The main tools in the proof are a recent result of Matomäki, Shao, Tao, and Teräväinen on the uniformity of the von Mangoldt function in short intervals, a polynomial approximation of our sequences with good equidistribution properties, and a lifting trick that allows us to replace $\mathbb{Z}$-actions on a probability space by $\mathbb{R}$-actions on an extension of the original system.

Joint work with A. Koutsogiannis.

16 Nov 2023, 11:15, Room: A303
Speaker: Nikos Frantzikinakis (University of Crete)

Partition regularity of Pythagorean pairs.

Abstract: An algebraic equation is partition regular if every finite coloring of the integers has monochromatic solutions. Necessary and sufficient conditions for partition regularity of linear equations were given by Radó in 1933. However, very little is currently known about homogeneous quadratic equations in three variables; the most notable problem is the partition regularity of Pythagorean triples. We will prove that Pythagorean pairs are partition regular, i.e., for any finite coloring of the integers we can find $x, y$ of the same color such that $x^2+y^2=z^2$ for some integer $z$ (and a similar statement with the roles of $y$ and $z$ swapped). Our method combines Gowers uniformity properties of bounded multiplicative functions and a new approach based on concentration estimates of pretentious multiplicative functions.

This is joint work with Oleksiy Klurman and Joel Moreira.

07 Dec 2023, 11:15, Room: A303
Speaker: Mihalis Kolountzakis (University of Crete)

Connectifying counterexamples

Abstract: Recently Greenfeld and Tao found an example of a finite subset in $\mathbb{Z}^d$ (for some large $d$) which tiles $\mathbb{Z}^d$ by translations but only aperiodically, thus disproving the so-called Periodic Tiling Conjecture in high enough dimension.

Roughly 20 years ago the Fuglede (or Spectral set) conjecture was disproved by Tao (in the spectral implies tiling direction) and by Kolountzakis and Matolcsi (in the tiling implies spectral direction). In this problem the dimension $d$ eventually got down to 3 for both directions.

In both these problems (aperiodicity and Fuglede conjecture) the examples found are highly dispersed subsets of $\mathbb{Z}^d$. In this work we show how to modify these examples to obtain (pathwise) connected subsets of $\mathbb{Z}^d$ as examples by increasing the dimension $d$.

This is joint work with Rachel Greenfeld.

15 Dec 2023, 13:15 PM, Room: A303

Roth's theorem and the Hardy-Littlewood majorant problem for thin subsets of Primes

Abstract: We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many non-trivial three-term arithmetic progressions. We also prove that the Hardy-Littlewood majorant property holds for these sets. Notably, our considerations recover the results for the Piatetski--Shapiro primes for exponents close to 1, which are primes of the form $\lfloor n^c\rfloor$ for a fixed $c>1$.

22 Feb 2024, 11:15, Room: A303
Speaker: Silouanos Brazitikos (University of Crete)

Isoperimetric Type Problems for Parallelotopes

Abstract: We investigate isoperimetric type problems restricted to some classes of parallelotopes. Discrete and continuous versions are considered.

28 Mar 2024, 11:15, Room: Α303
Speaker: Vassili Nestoridis (University of Athens)

Complex approximation of functions and all their derivatives in compact sets

Abstract: In Mergelyan type theorems we uniformly approximate functions in compact sets $K$ by polynomials, rational functions or holomorphic functions in varying open sets containing $K$. We improve uniform approximation in $K$ by approximating as well all derivatives. The case of one complex variable is contained in the article arXiv:2006.02389 by Armeniakos-Kotsovolis-Nestoridis, published in Monatchefte fur Mathematik (2022). In the present talk, we will present generalizations for several complex variables, based on a collaboration with P. M. Gauthier.

06 Jun 2024, 11:15, Room: Α303
Speaker: Dimitris-Marios Liakopoulos (University of Crete)

Reverse Brascamp-Lieb inequality and dual Bollobas-Thomason

Abstract: We will discuss inequalities that compare the volume of a set with the volume of lower dimensional sections of the set. Such inequalities can be derived from Brascamp-Lieb and its reverse.

All seminars

Seminar organizer for 2023-24: Silouanos Brazitikos

Page maintained by Mihalis Kolountzakis.