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.
