Periodic Reporting for period 4 - HomDyn (Homogenous dynamics, arithmetic and equidistribution)
Période du rapport: 2023-12-01 au 2024-11-30
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.
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.
Towards the goal of the project our team conducted research especially in four direction:
* quantitative behavior of unipotent flows
* rigidity of multiparameter flows
* random walks related to arithmetic
* hyperbolic geometry in low dimensions.
The project was held in a challenging environment, where we had to content with both the COVID epidemic and worsening security situation in Israel. Despite that we were able to complete our plan successfully, and have achieved major advances in the research directions covered by the DoW.
The COVID epidemic forced us to postpone an international workshop/school planned on project related topics, including on outputs of our project; but once having a workshop was again feasible, we organized a high profile international one week program that in particular served as a venue for us to disseminate research related to the proposal and to learn first-hand about relevant developments in Europe and beyond.
During the COVID epidemic we compensated by having project members giving many Zoom seminar talks, some with wide audience. After the epidemic subsided team members gave workshop and conference talks, in Europe and beyond, about our achievements. Four graduate students graduated during the project, and in addition one is expected to submit his thesis a couple of months after the end of the project. We were also fortunate to have recruited a strong group of postdoctoral researchers.
* quantitative behavior of unipotent flows
* rigidity of multiparameter flows
* random walks related to arithmetic
* hyperbolic geometry in low dimensions.
The project was held in a challenging environment, where we had to content with both the COVID epidemic and worsening security situation in Israel. Despite that we were able to complete our plan successfully, and have achieved major advances in the research directions covered by the DoW.
The COVID epidemic forced us to postpone an international workshop/school planned on project related topics, including on outputs of our project; but once having a workshop was again feasible, we organized a high profile international one week program that in particular served as a venue for us to disseminate research related to the proposal and to learn first-hand about relevant developments in Europe and beyond.
During the COVID epidemic we compensated by having project members giving many Zoom seminar talks, some with wide audience. After the epidemic subsided team members gave workshop and conference talks, in Europe and beyond, about our achievements. Four graduate students graduated during the project, and in addition one is expected to submit his thesis a couple of months after the end of the project. We were also fortunate to have recruited a strong group of postdoctoral researchers.
A major achievement in our project has been to obtain effective versions of landmark results of Ratner and Margulis that were known only in qualitative form previously. This progress has unexpectedly been interconnected with progress by other researchers on seemingly unrelated topics (decoupling and projection theorems). Our progress has had high impact and high visibility in both areas, and was a major impetus for a "hot-topic" workshop bringing together researchers from both of these (previously unrelated) areas in Berkeley at the well known SLMath institute.
Through this, and through other ways, we obtained long sought after results about both rational and irrational quadratic forms.
We have found new ways of connecting homogeneous dynamics and hyperbolic geometry, including in the theory of infinite volume hyperbolic manifolds with complicated geometry, and discovered a new "critical exponent" gap principle both in hyperbolic 3 dimensional manifolds and in certain important moduli spaces related to rational billiards.
Through this, and through other ways, we obtained long sought after results about both rational and irrational quadratic forms.
We have found new ways of connecting homogeneous dynamics and hyperbolic geometry, including in the theory of infinite volume hyperbolic manifolds with complicated geometry, and discovered a new "critical exponent" gap principle both in hyperbolic 3 dimensional manifolds and in certain important moduli spaces related to rational billiards.