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

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Abstract: In this talk we will deal with multiple ergodic averages having iterates a common integer-valued sequence that comes from appropriate classes of functions. In particular, we are dealing with the combination of a Hardy, a tempered, and a polynomial function satisfying some growth-rate restrictions to avoid local obstructions. Joint work with S. Donoso and W. Sun.

29 Sep 2022, 11:00, Room: A303
Speaker: Giorgos Chasapis (University of Crete)

Abstract: Let $\varepsilon_1,\varepsilon_2,\ldots$ be independent Rademacher random variables and $g_1,g_2,\ldots$ be independent standard Gaussian random variables. Pinelis has proved that there is an absolute constant $C>0$ such that for every $n\in\mathbb{N}$, real numbers $v_1,\ldots,v_n$ and any $t>0$, $$ \mathbb{P}\left(\left|\sum_{j=1}^n v_j\varepsilon_j\right|\gtrapprox t\right)\lessapprox C\cdot \mathbb{P}\left(\left|\sum_{j=1}^n v_jg_j\right|\gtrapprox t\right). $$ We extend this Rademacher-Gaussian tail comparison to the case of complex coefficients and discuss related open problems.

13 Oct 2022, 12:00, Room: A303
Speaker: Silouanos Brazitikos (University of Crete)

Abstract: We explain connections between classical geometric inequalities, like isoperimetric or Loomis-Whitney, or Kakeya type and their functional analogues. Joint work with D. Alonso, J. Bernues and A. Carbery

Abstract: 100 years ago, Friedmann and Kasner discovered the first exact cosmological solutions to Einstein’s equations, revealing the presence of a striking new phenomenon, namely, the Big Bang singularity. Since then, it has been the object of study in a great deal of research on general relativity. However, the nature of the ‘generic’ Big Bang singularity remains a mystery. Rivaling scenarios are abound (monotonicity, chaos, spikes) that make the classification of all solutions a very intricate problem. I will give a historic overview of the subject and describe recent progress that confirms a small part of the conjectural picture.

Abstract: Furstenberg’s dynamical proof of the Szemerédi theorem initiated a thorough examination of multiple ergodic averages, laying the grounds for a new subfield within ergodic theory. Special attention has been paid to averages of commuting transformations with polynomial iterates owing to their central role in Bergelson and Leibman’s proof of the polynomial Szemerédi theorem. Their norm convergence has been established in a celebrated paper of Walsh, but for a long time, little more has been known due to obstacles encountered by existing methods. Recently, there has been an outburst of research activity which sheds new light on their limiting behaviour. I will discuss a number of novel results, including new seminorm estimates and limit formulas for these averages. Additionally, I will talk about new criteria for joint ergodicity of general families of integer sequences whose potential utility reaches far beyond polynomial sequences. The talk will be based on two recent papers written jointly with Nikos Frantzikinakis.

Abstract: We will present a new algorithm on computing the covering radius of a polytope, whose vertices have rational coordinates. We will then apply this algorithm to solve the first unknown case of the shifted lonely runner conjecture (the case of four runners). For this purpose, we will use the geometric reformulation of this conjecture, which was given by the speaker and Schymura in 2017, and yields an inequality involving the covering radius of a special class of zonotopes.

Such an algorithm was first given by Kannan in 1992, but ours is much faster. Moreover, the geometric reformulation of the lonely runner conjecture leads to an improvement of Tao's result in 2017, concerning the number of cases one needs to consider in order to solve the lonely runner conjecture for $\leq n$ runners (work in progress).

Abstract: In relation to the Erdos similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are not universal in measure, i.e. they satisfy the above conjecture. These are symmetric Cantor sets $C$ which can be quite thin: the length of the $n$-th generation intervals defining the Cantor set is decreasing almost doubly exponentially. Further, we achieve to construct a set, not just of positive measure, but of full measure not containing any affine copy of $C$. Our method is probabilistic.

On the motion of a rigid body in a viscous fluid from the Functional Analysis point of view

Abstract: Some functional Sobolev spaces, which naturally arise when studying the motion of a rigid body inside a region with a viscous incompressible fluid, are investigated. New estimates are obtained, on the basis of which some conclusions about the behavior of the body near the container boundary are made.

Abstract: Let $\mathbb{A}$ be a discrete, unbounded, infinite set in $\mathbb{R}$. Can we find a ``large" measurable set $E\subset \mathbb{R}$ which does not contain any affine copy $x + t\mathbb{A}$ of $\mathbb{A}$ (for any $x\in \mathbb{R}$, $t > 0$)?

If $a_n$ is a real, nonnegative sequence that does not increase exponentially, then, for any $0\leq p < 1$, we construct a Lebesgue measurable set which has measure at least $p$ in any unit interval and which contains no affine copy of the given sequence. We generalize this to higher dimensions and also for some ``non-linear" copies of the sequence. Our method is probabilistic.

23 Mar 2023, 11:00, Room: A303
Speaker: Silouanos Brazitikos (University of Crete)

Abstract: Let $\mu$ be a log-concave probability measure on $\mathbb{R}^n$ and for any $N > n$ consider the random polytope $K_N = \mathrm{conv}\,\{X_1, . . . , X_N \}$, where $X_1, X_2, \ldots$ are independent random points in $\mathbb{R}^n$ distributed according to $\mu$. We study the question if there exists a threshold for the expected measure of $K_N$.

06 Apr 2023, 11:00, Room: A303
Speaker: Giorgos Chasapis (University of Crete)

Sharp moment comparison for sums of rotationally invariant random vectors and geometric applications

Abstract: Let $\xi_1,\xi_2,\ldots$ be i.i.d. random vectors uniformly distributed on the Euclidean unit sphere $S^{d-1}$ of $\mathbb{R}^d$ and for any $a=(a_1,\ldots,a_n)\in\mathbb{R}^n$ let $X_a=\sum_{j=1}^n a_j\xi_j$. What is the infimal value of $\|X_a\|_q$, $-(d-1) < q < 2$, over all $n\in\mathbb{N}$ and unit vectors $a\in\mathbb{R}^n$? We review some well-studied instances of this far-reaching extension of the classical Khinchin inequality, including several recent results and their connection to problems in the geometry of normed spaces.

Abstract: Green and Tao famously showed that the von Mangoldt function is Gowers uniform and used this to give an asymptotic formula for the number of solutions to any linear system of equations (of finite complexity) in the primes. Their theorem however gave no quantitative rate of decay for the Gowers uniformity norms (of degree greater than 3). In this talk, I will discuss a quantification of their result and applications to e.g. Szemerédi's theorem with shifted prime differences. This is based on joint work with Terence Tao.

Abstract: Let $\Omega \subseteq \mathbb{R}^d$ be a measurable set. We call it spectral if there exists a set $\Lambda \subseteq \mathbb{R}^d$ (the spectrum) such that the characters $$ e_\lambda(x) := e^{2\pi \lambda \cdot x},\ \ \lambda \in \Lambda, $$ form an orthogonal basis for $L^2(\Omega)$. As an example, a Euclidean ball in $\mathbb{R}^d$ is not spectral while a cube is spectral ($\Lambda=\mathbb{Z}^d$ is a spectrum for $[0, 1]^d$ -- this is the usual Fourier series). The main question that interests us is which domains $\Omega$ are spectral. The Fuglede conjecture from the 1970s stated that $\Omega$ is spectral if and only if $\Omega$ can tile space by translations, that is, when there exists $T \subseteq \mathbb{R}^d$ with $$ \sum_{t \in T} \textbf{1}_\Omega(x-t) = 1,\ \ \ \text{ for almost every } x \in \mathbb{R}^d. $$ Since 2004 this is known to be false in both directions for $d \ge 3$. Still, the connections between spectrality and tiling have continued to be studied intensively for special classes of $\Omega$ as well as for abelian groups other than Euclidean space, where both spectrality and tiling can be defined analogously. A major development in recent years was the proof by N. Lev and M. Matolcsi that any spectral set $\Omega$ can weak-tile its complement $\Omega^c$ (weak-tiling is tiling with weighted copies of the set, with nonnegative weights). This led to the proof of the Fuglede Conjecture for the class of convex bodies in any dimension, a result that attracted researchers over at least two decades, work that had led to many partial results. In this talk we will describe some of the background and show even more applications of the weak-tiling idea to spectrality (e.g. of Cantor-type sets). We will also point out some open problems. This is mostly joint work with N. Lev and M. Matolcsi.

Abstract: We consider the Hilbert matrix $\big(\frac 1{n+m+1}\big)_{n\geq 0,\,m\geq 0}$ which defines an operator $\mathcal H$ acting on analytic functions $f(z)=\sum_{n=0}^{\infty}a_nz^n$ on the unit disc as follows: $$\mathcal H(f)(z)=\sum_{m=0}^{\infty}\big(\sum_{n=0}^{\infty}\frac{a_n}{n+m+1}\big)z^m.$$ In particular, we consider the space $K^p$ of such functions with norm defined by $\|f\|_{K^p}^p=\sum_{n=0}^{\infty}(n+1)^{p-2}|a_n|^p<+\infty$, and we prove, in a nontrivial way, that $\mathcal H$ is bounded on $K^p$ and that its norm is equal to the number $\frac{\pi}{\sin\frac{\pi}p}$.

This is joint work with V. Daskalogiannis and P. Galanopoulos, University of Thessaloniki.