Intermediate Chaos

Project Information

IChaos

Grant agreement ID: 647133

DOI

10.3030/647133

Project closed

EC signature date 17 July 2015

Start date 1 January 2016

End date 30 June 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 696 936,74

EU contribution

€ 1 696 936,74

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 4 - IChaos (Intermediate Chaos)

Reporting period: 2020-07-01 to 2021-06-30

Summary of the context and overall objectives of the project

The project ICHAOS has developed a mathematical theory of intermediate chaos, the regime of transition between order and strong chaos. The key achievements of the project include a detailed study of the analytic and probabilistic properties of point processes
connected to the theory of random matrices that goes back to the work in of John Wishart in statistics in the 1920’s as well as the work of Freeman Dyson in theoretical physics in the 1950’s. The theory of random matrices combines mathematical precision with physical relevance: all its predictions have been independently experimentally confirmed.
Originating from problems in demography and having received new impetus from the development of telecommunication and resulting problems of the servicing of queues, the theory of point processes has since found applications in areas as diverse as nuclear physics and machine learning.

Another group of key achievements of the project concerns self-similar dynamical systems that arise, on the one hand, in geometry as partial isometries in one dimension, and, on the other hand, in the theory of formal languages as dynamical systems generated by substitutions. Systems with self-similarity are in particular used to model quasi-crystals.
In broad terms, the achievement of the project is precisely to have built connections between the symbolic and the geometric description of self-similar dynamical systems. Specifically, a key new object introduced by Boris Solomyak and the PI is the new symbolic spectral cocycle for self-similar systems.

The spectral cocycle is a precise way of measuring the fact that if the self-similar system is considered over carefully chosen long spans of time, then the level of complexity is the same as when the system is considered over a short interval of time.

The theory of dynamical systems is a mathematical study of evolution in time. Dynamical chaos is a broad paradigm for the study of systems whose long-term behaviour exhibits a highly sensitive dependence on small changes in the system's initial conditions. The trajectories of a chaotic dynamical system are in this case most effectively described in probabilistic terms, for example, by comparison with Brownian motion, in stark contrast with orderly quasi-periodic motion of celestial bodies. The project ICHAOS has given a quantitative description of dynamical systems exhibiting intermediate chaotic behaviour.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The key results obtained within the framework of the project ICHAOS are:
1) A precise quantitative description on the deviation from minimality for the classical sine-process. This result can be viewed as a version of the classical Kotelnikov Theorem in the context of point processes. The broad problem is that of recovering a continuous signal from samples performed at discrete moments of time, as is the case in applications. The main result achieves the desired recovery in the case of the sine-process. The PI considers this result his best in Mathematics so far.

2) A detailed quantitative description, achieved in collaboration with Boris Solomyak and using the formalism of symbolic dynamics, of the spectrum of translation flows, a class of self-similar systems arising in geometry. This description, published in 2021, is the culmination of five years´ work on the corresponding tasks of the project.

3) A theorem, obtained in collaboration with Y.Qiu and completed in 2021, on interpolation of complex-differentiable functions from their values at random samples. The approach relies on a construction due to Patterson and Sullivan and stemming from the study of discrete groups of isometries of the Lobachevsky plane.

4) A detailed geometric and probabilistic description, obtained in collaboration with Y. Qiu, A. Shamov and published in the Journal of the European Mathematical Society in 2021, of reproducing kernels of determinantal point processes. The important special case of processes in complex domains had been studied in joint work with Y.Qiu and S. Fan.

5) A detailed description of quasi-symmetries of determinantal point processes, the relationship between Palm measures of point processes. Palm measures are a mathematical tool for describing the probability of occurrence of a new random event provided one has just occurred. Conrad Palm, an engineer for Ericsson in Stockholm, introduced and studied these probabilities in order to minimize telephone communication waiting time.

6) A theorem, proved in collaboration with Andrey Dymov, on approximation of trajectories of the sine-process, a point process playing a key rôle in the study of heavy nuclei, by a generalized stochastic process related to the Gaussian Free Field.

The results of the project have received world-wide dissemination due to conferences and workshops organized within the framework of the project, as well as reports made by PI and the project team at major international conferences and workshops throughout the duration of the project.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The project ICHAOS is a quantitative study of intermediate chaos for dynamical systems.

For the purpose of this study the project has developed new tools such as the use of the action of the group of diffeomorphisms in the study of the sine-process of Freeman Dyson, the use of the Patterson-Sullivan construction of group theory in the study of complex interpolation, a general formalism of multiplicative functionals in the study of determinantal point processes, a development of the Kolmogorov 0-1 law for point processes as well as a new local property for Palm kernels of determinantal point processes (in collaboration with Y. Qiu and A. Shamov), a development of the Erdos-Kahane method fand a new spectral cocycle for the study of the intermediate chaos for self-similar dynamical systems of geometric origin (in collaboration with B. Solomyak).
Substantial progress has been achieved in the quantitative study of the spectrum of symbolic dynamical systems with intermediate chaos.

Important progress has been achieved in the problem of sampling in function spaces using realizations of point processes. In the simplest situation, sampling corresponds to determining, with the best possible precision, a continuous signal from its values at given moments of time. Sampling plays a key rôle in signal transmission, and a new methodology has been developed for the sampling of determinantal point processes.

Dynamical systems and point processes are used to model a wide variety of phenomena in the world that surrounds us. The novel methodology of the project lies at the crossroads of probability, analysis and geometry. The research of the project team has given a quantitative description of intermediate chaos in line with the description of action for the project ICHAOS.

Symbolic coding of combinatorial structures: in this case, integer partitions

The Young Graph: partitions of integers united into a graded graph