A major motivation behind everything we did in the HomDyn project is the remarkable fact that tools from the mathematical theory of dynamical systems, a mathematical field whose origins can be traced to an attempt to understand planetary motion where the equations are too complicated to be solved explicitly but for which one can still build a robust and detailed theory can be useful to study objects 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 homogeneous spaces, and the study of actions on these spaces that respect the symmetry is called homogeneous dynamics. We used a very wide toolbox that included 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 homogeneous dynamics, and build two-way bridges between homogeneous dynamics and problems in arithmetic, quantum dynamics, graphs theory and other topic.
We were in particular focused on how fast things happen. In homogeneous dynamics there were landmark results regarding the behavior of systems eventually. But we wanted to know how quickly this happens --- we wanted effective and quantitative version of these remarkable qualitative theorems.
Our main aim was to progress the state-of-the-art in mathematical topics related to the research project. We found new connections between different mathematical areas. The topics involved are related to more practical questions, in particular to quasi-randomness: how a deterministic process can generate output that can be used as a useful substitute for a purely random input.
