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The second Ryll-Nardzewski Day will take place on June 28, 2025. The main goal of the series is to promote Polish mathematics and to highlight Czesław Ryll-Nardzewski as an icon of the Polish school of mathematics and a distinguished citizen of Wrocław.
Lectures will be held in building C-13, room 1.27. Lunch will be served in building H-14 following the conference.
Conference programme:
11:15–11:25: opening
Rector of the Wrocław University of Science and Technology
prof. dr hab. inż. Arkadiusz Wójs
11:30–12:25: Ryll-Nardzewski Lecture
Nikos Frantzikinakis
University of Crete
Ergodic properties of bounded sequences and applications
12:30–13:10:
Piotr Biler
University of Wrocław
A nonlocal parabolic problem
13:15–13:55:
Jacek Małecki
Wrocław University of Science and Technology
Stochastic particle systems with weak repulsive forces
Abstracts of talks:
Nikos Frantzikinakis (University of Crete)
Ergodic properties of bounded sequences and applications
How can we measure and exploit randomness properties of a bounded sequence? A construction due to Furstenberg from the 70's allows us to encode the statistical properties of bounded sequences dynamically using the language of ergodic theory. This provides a different point of view, which has important advantages and has led to several breakthroughs in ergodic theory, combinatorics and, more recently, number theory. Examples include various refinements of Szemeredi’s theorem on arithmetic progressions, progress towards the Sarnak and Chowla conjecture, and the partition regularity of Pythagorean triples. I will review some related results with an emphasis on more recent developments.
Piotr Biler (University of Wrocław)
A nonlocal parabolic problem
We discuss the existence of radially symmetric solutions (evolution and self-similar cases) of the minimal Keller–Segel system in ℝd:
ut − Δu + ∇·(u∇φ) = 0,
Δφ + u = 0,
under optimal assumptions on the initial data u0 = u(0, ·). We are interested, in particular, in minimal regularity assumptions imposed on the initial data in order to a local-in-time solution exists, and size conditions for (approximate) dichotomy: global-in-time existence versus finite time blowup of solutions. The results, obtained in collaboration with Grzegorz Karch, Dominika Pilarczyk, Hiroshi Wakui and Jacek Zienkiewicz, are essentially those reported in [1] and [2].
[1] P. Biler, Singularities of solutions to chemotaxis systems, i--xxiv, 1--207; De Gruyter Series in Mathematics and Life Sciences 6, Walter de Gruyter, Berlin, 2020. ISBN 978-3-11-059789-9. DOI:10.1515/9783110599534
[2] P. Biler, G. Karch, H. Wakui, Large self-similar solutions of the parabolic-elliptic Keller--Segel model, Indiana Univ. Math. J. 72 (2023), pp. 1027–1054. DOI:10.1512/iumj.2023.72.9218
Jacek Małecki (Wrocław University of Science and Technology)
Stochastic particle systems with weak repulsive forces
The talk will be devoted to stochastic particle models with repulsive interactions, with particular emphasis on cases where the repulsive force coming from a singular drift is too weak to prevent particle collisions. We will discuss conditions ensuring the existence of such models together with an examination of their behavior, focusing on collisions, multi-collisions, and the times of collisions. The presented results will be partially drawn from joint projects with Sergio Andraus, Piotr Graczyk, and Nicole Hufnagel.
