Analysis Seminar of the Rényi Institute, Spring of 2025


March 27, Thursday, 12:15--13:15, Tondós lecture room

Speaker: Mihály Weiner, BME & Rényi Institute

Title: Hypothesis testing and a new characterization of matrix geometric means

Abstract: TBA.


March 13, Thursday, 12:15--13:15, Tondós lecture room

Speaker: András Gilyén, Rényi Institute

Title: Quantum generalizations of Glauber and Metropolis dynamics

Abstract: TBA.


February 27, Thursday, 12:15--13:15, Tondós lecture room

Speaker: Balázs Maga, Rényi Institute

Title: Permutation limits and Shannon entropy

Abstract: Studying analytic limit objects of discrete structures has become one of the main directions of combinatorics recently, which owes its success to the fact that quite often statements in the discrete world can be converted to analytic ones and vice versa, vastly enriching the available tools in dealing with important questions. In my talk, I am going to discuss permutons, i.e., permutation limits, which are simply probability measures on the unit square with uniform marginals. Such a measure naturally gives rise to a distribution over the length n permutations for any n>0. We are interested in the asymptotic properties of the Shannon entropy of these distributions, which we call the pattern entropy sequence. After discussing some simple examples, I am going to speak about an exciting connection to Kolmogorov-Sinai entropy for permutons supported by graphs of measure-preserving functions, demonstrating that given certain regularity conditions, the pattern entropy sequence has a linear growth rate. I will also touch upon how one can prove via Baire category arguments the existence of irregular permutons, i.e., ones for which no natural normalization yields a convergent sequence. A simple construction for random permutons comes from the random automorphism of the rooted $d$-ary tree. If time permits, I will also speak about some surprising probabilistic phenomena I encountered upon studying its pattern entropy sequence.


February 20, Thursday, 12:15--13:15, Tondós lecture room

Speaker: Haojie Ren, Technion, Israel

Title: Hausdorff Dimension of Solenoidal Attractors

Abstract: Solenoidal attractors are fractal sets arising in skew product systems. In this talk, I will introduce their definition, provide a simple geometric description, and give a brief overview of their research history. I will then focus on a special class of skew product maps, discussing my works on the SRB measures and the solenoidal attractors induced by these maps. The proofs draw on Hochman's entropy growth tools for convolutions and techniques from my recent collaboration with Shen. I will also introduce some open questions in the field.


February 13, Thursday, 12:15--13:15, Tondós lecture room

Speaker: Geuntaek Seo, POSTECH, South Korea

Title: Nonconvex Wasserstein Gradient Flows: A Variational Problem and Asymptotic Convergence

Abstract: This talk consists of two parts: The first part deals with the geometry of minimizers for certain free energy functionals related to aggregation-diffusion equations. As shown by Lim and McCann [ARMA, 2021], the minimizers of the interaction energy with certain mildly repulsive potentials are equi-distributed on $\Delta^n$, the vertices of the $n$-simplex. Using $\Gamma$-convergence, we show that the minimizers of the free energy functional are compactly supported and equi-distributed near $\Delta^n$ when the diffusion effect corresponds to the porous medium diffusion. This is a joint work with Tongseok Lim (Purdue). In the second part, we discuss the asymptotic convergence of nonlinear PDEs described by gradient flows for nonconvex functionals. Our goal in this research is to establish the generalization of the Lojasiewicz-Simon theory to nonconvex Wasserstein gradient flows. This is a joint work in progress with Beomjun Choi and Seunghoon Jeong (POSTECH).


February 6, Thursday, 12:15--13:15, Tondós lecture room

Speaker: Sascha Troscheit, Uppsala University, Sweden

Title: Continuum trees of real functions and their graphs

Abstract: The Brownian continuum tree (CRT) is an important random metric space that was extensively investigated in the 1990s. It can be constructed by a change of metric from a Brownian excursion function on [0,1]. This change of metric can be applied to all continuous circle mappings to give a continuum tree associated with the function. In 2008, Picard proved that analytic properties of the function are connected to the dimension theory of its tree: the upper box dimension of the continuum tree coincides with the variation index of the contour function. We will provide a short and direct proof of Picard's theorem through the study of packings. The methods used will inspire different notions of variations and variation indices, and we will link the dimension theory of the tree with the dimension theory of the graph of its contour function. The title and abstract may be familiar to some of you who came to the one-day fractal meeting at BME about a year ago. This talk includes some updates and strengthened results. (Joint work with Maik Gröger.)


January 30, Thursday, 12:15--13:15, Tondós lecture room

Speaker: Zoltán Balogh, University of Bern, Switzerland

Title: Logarithmic Sobolev Inequality on Surfaces via OMT

Abstract: In the first part of this talk I present a proof of the Logarithmic Sobolov Inequality (LSI) in the Euclidean space, that is based on the method of optimal mass transport (OMT). In the second part of the talk I show, that this method is strong enough to obtain a version of the LSI in curved spaces such as Euclidean surfaces. In this case the statement of the LSI involves the mean curvature of the surface. The talk is based on a joint work with Alexandru Kristaly.

Last modified: 14.02.2025