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Homogenous dynamics, arithmetic and equidistribution

Periodic Reporting for period 3 - HomDyn (Homogenous dynamics, arithmetic and equidistribution)

Okres sprawozdawczy: 2022-06-01 do 2023-11-30

A major motivation behind everything we do in the HomDyn project is the remarkable fact that tools from the mathematical theory of dynamical systesm, a mathematical field whose origins can be traced to an attempt to understand planatary motion where the equations are too complicated to be solved explicitly but for which one can still build a rubust and detaield theory can be useful to study obejcts that are very non-dynamical in nature, for instance arithmetic objects like integer points.

The dynamics we study happen on a special kind of space that have a lot of symmetry called homogenous spaces, and the study of actions on these spaces that respect the symmetry is called homogenous dynamics. We are using a very wide toolbox that includes dynamical systems theory (and in particular, the probabilistic part of the theory known as ergodic theory), the theory of algebraic groups, number theory, arithmetic combinatorics and spectral theory to study homogenous dynamics, and build two-way bridges between homogenous dynamics and problems in arithmetic, quantum dynamics, graphs theory and other topic.
Our main aim is to progress tha state-of-the-art in mathematical topics related to the research project, and demonstrate the remarkable interconnections between different mathematical areas that make mathematics such a fascinating topic; but the topics involved are very related to more practical questions, in particular to quasi-randomness: how a deterministic process can generate output that can be used as a usefull substitute for a purely random input.

The project addresses some long term and well known mathematical conjectures, as well as central issues in homogenous dynamics such as understanding rates of equidistribution (which is closely related to the degree of precision of quasi randomness).
Towards the goal of the project our team coducted research especially in three direction: rigidity of multiparameter flows, random walks related to arithmetic, and quantitative behaviour of unipotent flows.

The COVID epidemic forced us to cancel an international workshop/school planned on project related topics, including on outputs of our project; we compensted by giving many Zoom seminar talks, some with wide audience (in one Zoom talk, we had about 300 listeners, which is a lot for a technical mathematical seminar)
Highlights of our work that already appear as preprints include in particular significant advances in understanding dynamics on "complicate" -geoemtrically inifnite homogenous spaces, on random walks, and on diophantine approximations.

The reseach is progressing well, and we anticipate meeting the objectives of the project.