Analysis Seminar at the Rényi Institute
The seminar runs Thursdays from 12:15 to 13:15. Schedule for Fall of 2024:
September 19: Antti Käenmäki (Rényi)
Title: Do self-conformal measures on the real line resonate?Abstract
In the 60’s, Furstenberg conjectured that if $X,Y \subset [0,1]$ are closed sets invariant under multiplication by integers $p$ and $q \mod 1$, respectively, then for any $s \ne 0$, the resonance inequality $\dim_H(X+sY) < \min{ 1,\dim_H(X) + \dim_H(Y) }$ implies that $\log(p)/\log(q)$ is a rational number. Intuitively, if $\log(p)/\log(q)$ is irrational, then $X$ and $Y$ dissonate by which we mean that the expansions of $X$ and $Y$ in bases $p$ and $q$, respectively, have no common structure. A similar question can be asked for convolutions of two measures. In this talk, which is based on a recent work with B. Bárány, A. Pyörälä, and M. Wu, we answer whether two self-conformal measures in the real line resonate or dissonate.
September 26: Mouna Chegaar (BME)
Title: Trace inequality with Bessel convolutionAbstract
Considering potentials defined by the Bessel kernel with Bessel convolution, a Kerman-Sawyer type characterization of the trace inequality is given. As an application, an estimate on the least eigenvalue of Schrödinger-Bessel operators is derived.
October 3: Patrícia Szokol (Debrecen)
Title: Characterization of quasi-arithmetic means without regularity conditionAbstract
We focus on the characterization theorem of János Aczél (1948) on quasi-arithmetic means. He proved that every continuous, bisymmetric, symmetric, reflexive, strictly monotonic binary map on a proper interval is a quasi-arithmetic mean. Our first goal is to show a somewhat surprising result, namely, the purely algebraic bisymmetry property has a regularity improving feature. More precisely, our first theorem shows that the continuity property in the original theorem is redundant. As a continuation of this result, we demonstrate that it can be refined in the way that the symmetry condition can be weakened by assuming symmetry only for a pair of distinct points of an interval. The presentation is based on joint papers with Pál Burai and Gergely Kiss.
October 10: Michał Rams (IMPAN)
Title: Projections of fractal percolationsAbstract
Fractal percolations, introduced by Mandelbrot in relation to turbulent flows, are a very nice and natural random fractal constructions. They play in particular an important role in statistical physics, serving as one of only two easy-to-investigate model of phase transitions (the other one being the Ising model of ferromagnetic in dimension 2). In the talk I will present results, joint with prof. Simon Karoly from BME, about projections of percolations. In particular, we will be interested in the question whether the projection of a percolation contains an interval. I plan to present the full proof of the simplest case, together with a short tour about the directions this result was generalized later.
October 17: Gergely Kiss (Rényi)
Title: The discrete Pompeiu problem on Euclidean spacesAbstract
Let $K$ be a nonempty finite subset of the Euclidean space $\mathbb{R}^k$ ($k\geq 2$). In this talk we discuss the solution of the following so-called discrete Pompeiu problem: Is it true that whenever a function $f : \mathbb{R}^k \to \mathbb{C}$ is such that the sum of $f$ on every congruent copy of $K$ is zero, then $f$ vanishes everywhere? Some important consequences of the result will also be presented such as every finite subset of $\mathbb{R}^k$ having at least two elements is a Jackson set. This is a joint work with Miklós Laczkovich.
October 24: Richárd Balka (Rényi)
Title: Lipschitz images and dimensionsAbstract
We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that if $A$ and $B$ are compact metric spaces and the Hausdorff dimension of $A$ is bigger than the upper box dimension of $B$, then there exist a compact set $A’\subset A$ and a Lipschitz onto map $f\colon A’\to B$. As a corollary we prove that any `natural’ dimension in $\mathbb{R}^n$ must be between the Hausdorff and upper box dimensions. We show that if $A$ and $B$ are self-similar sets with the strong separation condition with equal Hausdorff dimension and $A$ is homogeneous, then $A$ can be mapped onto $B$ by a Lipschitz map if and only if $A$ and $B$ are bilipschitz equivalent. For given $\alpha>0$ we also give a characterization of those compact metric spaces that can be obtained as an $\alpha$-Hölder image of a compact subset of $\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension. This is a joint work with Tamás Keleti.
October 31: Alex Rutar (University of Jyväskylä)
Title: Non-convex optimization in fractal geometryAbstract
What does non-convex optimization have to do with fractal geometry? I will discuss two such applications: the first, more well known, in multifractal analysis; and the second, more recent, in the dimension theory of dynamically invariant sets. In these two settings, fractal geometric problems can often be reduced to an abstract optimization problem over some measure space. A technical difficulty is that the objective functions which arise are often non-convex and non-smooth. To understand how to work around these difficulties, a key perspective is the parametric geometry of Lagrange multipliers. The new results are based on joint work with (different subsets of) Amlan Banaji, Jonathan Fraser, Thomas Jordan and István Kolossváry.
November 21: András Kroó (Rényi)
Title: Homogeneous polynomials on convex bodies: Christoffel function, $L^p$ Markov inequalities, Marcinkiewicz-Zygmund type discretizationAbstract
In this talk we shall consider several extremal problems related to multivariate homogeneous polynomials. The space of real homogeneous polynomials of $d\geq 2$ variables and degree $n$ appears naturally as a fundamental tool in problems related to neural networks and approximation by ridge functions. Our main goal is to study the asymptotic behaviour of Christoffel functions and $L^p$ Markov type estimates for homogeneous polynomials on convex bodies. In order to tackle these questions homogeneous ”needle” polynomials attaining value $1$ at a given point and rapidly decreasing moving away from this point are introduced. These homogeneous needle polynomials play a crucial role in the study of the asymptotics of the Christoffel functions and $L^p$ Markov type estimates, which in turn lead to Marcinkiewicz-Zygmund type discretization results.
December 5: Cai-Yun Ma (BME)
Title: One-sided multifractal analysis of Gibbs measures on the lineAbstract
In this talk, I will discuss the one-sided multifractal analysis of Gibbs measures supported on a self-conformal set on the real line. More precisely, let $K$ be the attractor of a $C^{1+\delta}$ iterated function system on the real line satisfying the strong separation condition. Let $\mu_{\psi}$ be a Gibbs measure on $K$ associated with a continuous potential $\psi$. As a main result, we obtain the Hausdorff dimension and packing dimension of the one-sided local dimensions of $\mu_{\psi}$. The talk is based on joint work with De-Jun Feng.
