https://sites.google.com/view/dcrwgt/homepage
The Dioscuri Center for Random Walks in Geometry and Topology is funded under the Dioscuri program, co-implemented by the Polish Ministry of Education and Science and the German Federal Ministry of Education and Research. The project spans five years starting from September 1, 2023, and is led by Mikołaj Frączyk (link to his page). The research focuses on the geometry and topology of locally symmetric spaces, employing probabilistic methods, particularly random walks. This area is at the forefront of global mathematics, with revolutionary contributions from mathematicians such as William Thurston, Gregory Margulis, and Mikhail Gromov. Despite numerous advancements, the study of locally symmetric spaces remains an active field in mathematics with many unresolved problems.
The research aims to address the following questions:
How does the volume of a compact locally symmetric space control its topological complexity?
How rapidly do the dimensions of spaces of automorphic forms with given parameters grow in natural arithmetic sequences of locally symmetric spaces?
How can we classify locally symmetric spaces with infinite volume?
How do random walks behave on locally symmetric spaces?
Read More o
https://tanja-schindler.github.io/
The purpose of the project is to study limit laws and mixing properties for weakly chaotic systems. The theory of strongly chaotic systems has been extensively developed in the last decades and there are now methods that can be used in studying their statistical properties: e.g. a functional analytic approach (transfer operators, anisotropic Banach spaces) or a geometric approach (distribution of stable and unstable foliations).
However much less is known for systems which do not have such a strong chaotic behavior and in particular are of zero entropy. The project aims to develop general methods to study many natural classes of systems which exhibit slower (or weaker) notions of randomness (or chaos), e.g. suspension flows. The research focuses on mixing and statistical properties of such systems with the goal of characterizing their chaotic behavior and their comparison with strongly chaotic systems. In particular, the research entails the infinite measure case where the theory has not yet been developed and where possibly other notions of mixing have to be developed.
Read More o
https://cenmatre.id.uj.edu.pl/start
The Flagship Project Central European Mathematical Research Lab is funded by the Excellence Initiative program at Jagiellonian University, and it is led by Adam Kanigowski (link to his page). The research focuses on dynamical systems, ergodic theory, and their connections with number theory, geometry, and probability theory. More precisely, the research focuses on the randomness and chaoticity of smooth systems on manifolds, the classification problems of measure preserving systems, and homogenous and non-homogenous dynamics in interactions with analytic number theory, including sparse equidistribution problems. The center conducts research both on problems with zero any positive entropy. In the case of positive entropy, it aims to understand the interactions between ergodic and statistical properties, such as the relationships between Bernoullicity or the K-property and the rate of mixing or limit theorems. For systems with zero entropy, which often defines non-chaotic behavior, the study includes the appearance of mixing properties and higher-order mixing, the distribution of orbits, and connections. The recent research topics also entail so-called ergodic theorems, examining the behavior of orbits sampled along a small but arithmetically rich subset of natural numbers (e.g., prime numbers). This part of the research naturally connects with the study of interactions between smooth dynamical systems and number theory.
By introducing new tools and techniques, the research constitutes a significant contribution to dynamical systems and ergodic theory (in particular, to sparse equidistribution problems and understanding chaotic properties of smooth systems). Successful tackling of several longstanging open problems locates the research in the world class in ergodic theory.