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Contract number
Time span of the project

As of 01.11.2022

Number of staff members
scientific publications
General information
Name of the project: Probabilistic methods in analysis: point processes, operators and holomorphic function spaces

Goals and objectives

The objective of the project is the development of modern directions in mathematical analysis at the Saint Petersburg State University. We expect to review new interrelations between possibility theory and analysis that arise in the study of determinantal processes and areas associated with them and their connection with conformal field theory. The main research topics and tasks are:

  1. Determinantal processes arising from physical models. We expect to study determinantal processes on a plane and analyse corresponding correlation kernels. We will study the behaviour of a model of a Coulomb gas near the spectral boundary and the behaviour of models with higher Landau levels (polyanalytic Ginibre ensembles).
  2. The inverse problem of potential theory and Schwartz functions. In the classical normal random matrix model, we expect to study the equilibrium measures of the ensemble of the corresponding Coulomb gas. We are planning to use complex dynamics methods to answer some fundamental questions related to the shapes of drops (carriers of equilibrium measures) as well as their change when the potential changes (for example, Laplacian growth).
  3. The research in the field of the uncertainty principle in harmonic analysis. This area includes problems of the completeness of exponentials and polynomials formulated by Wiener and Kolmogorov over 70 years ago, inverse spectral problems of differential operators and Krein canonical systems, the theory of de Branges spaces of entire functions, classical problems of the theory of stationary Gaussian processes, problems of signal processing etc., as well as their modern generalisations and applications.
  4. The development of a perturbation theory for linear operators. The goal of this part of the project is the research of the question of the extent to which perturbed operator functions can differ from the initial operator depending on the properties of perturbation and the function. Similar problems arise in the study of functions of several (switching and not necessarily switching) operators.

The practical value of the study

Scientific results:

The leading scientist and the academic staff of the Laboratory have conducted work to study  determinantal point processes on a line at spectral boundary points, to analyze one-parameter Schwartz function and dynamic systems related to them, researched  perturbed unbounded non-self-adjoint operators, problems of sampling in spaces of analytic functions. We researched the problems of uniqueness (as well as related problems of interpolation and sampling) for a class of reproducing kernel Hilbert spaces of analytical functions. It is for a finite subset of the Ω region that we studied the conditions of decrease (to zero) of some (infinite) set of derivatives of this function at this points, accordingly, the problem is to determine the non-degeneracy of the solution. For the classical Bargmann–Fock space we have studied a model in the two-point case — at zero all the even derivatives of functions of this class disappear and all the odd derivatives  of some Fock shift. It turned out that the only solution of such a problem is trivial, moreover,   for such a partition of integers there is neither sampling, nor interpolation, in other words, in some sense we produced a bound state. Our employees have researched the one-dimensional problem of the description of points of bounded radial variation for the boundary values of the Cantor set type in terms of the generating sequence. In 1993, Bourgain demonstrated that the set of points of finite vertical variation of a positive harmonic function  of the upper semi-plane has a complete dimensionality, therefore answering the known question posed by Rudin (1955). These results were generalized by the members of the academic team  in the Rn region (they are used in the works of Jones and Mueller and Mueller and Rigler, devoted to the Anderson conjecture). We reviewed a specific, rather complexly organized, boundary function f (a  harmonic continuation of which is u) and indicated explicit conditions for the finiteness of vertical variation. In the course of our work we solved the nonlinear Riemann-Hilbert problem using the generalization of the ∂ approach of Its-Takhtajan. We have conducted a research of diagonal contractions of the Bergman kernel corresponding to the correlation kernels for random matrices. We have derived an asymptotic decomposition of orthogonal polynomials for the Bergman spaces with a weight of e-2mQ, where Q  is the . Confining two-dimensional potential. We have found the condition of neurtality on background charges with the use of a generalization of  the Gauss-Bonnet theorem. Onto flat metrics with conical features in the marked points. We have researched the representing families of reproducing kernels in spaces of analytic functions in a circle. Our researchers have solved the one-dimensional problem on the description of points  of bounded radial variation for boundary values of the type of Cantor set in the terms of the generating sequence.

Implemented results of research: 

We have obtained results of the research of the uncertainty principle in problems color correction conducted on the grounds of the Saint Petersburg State University-Huawei laboratory: we researched transformations between the RGB and XYZ color spaces, proposed several types of such transformation as well as variations of the cost function and an algorithm for finding its global minimum and tested these developments over an array of real-life data. This can be used in finding algorithms of linear transformation with set limitations, recalculating color tables with differentiation of various color parameters.

Education and career development:

  • We have organized and staged the following academic events: the conferences «Complex and Harmonic Analysis and Its Applications» (Saint Petersburg, 23-26 November 2021), «Probabilistic Techniques in Analysis: Spaces of Holomorphic functions» (Sochi, 06-10 December 2021), «Probabilistic Techniques in Analysis: Reproducing kernel Hilbert spaces» (Sochi, 20-25 October 2022).


  • «Sirius» Mathematics center (Russia): collaboration in staging conferences.
  • Euler International Mathematical Institute in Saint Petersburg (Russia): participation of employees of the Institute in conferences and a seminar organized by the Laboratory. 

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yu. belov, yu. lyubarskii, a. kulikov
Irregular Gabor frames of Cauchy kernels. Applied and Computational Harmonic Analysis, Vol. 57, 2022
p. barkhayev, yu. belov, yu. lyubarskii
Summation Method In Optimal Control Problem With Delay. Algebra and Analysis, Vol. 34, № 3, December 2022
anton baranov, yurii belov, karlheinz gröchenig
Complete interpolating sequences for the Gaussian shift-invariant space. Applied and Computational Harmonic Analysis, Vol. 61, July 2022
a. g. abanov and p. b. wiegmann
Axial-Current Anomaly in Euler Fluids. Physical Review Letters, Vol. 128, February 2022
а.б. александров, в.в. пеллер
Функции от возмущенных пар некоммутирующих диссипативных операторов. Алгебра и анализ, Т. 34, № 3, 2022.
а. д. баранов, и. р. каюмов
Оценки интегралов от производных рациональных функций в многосвязных областях на плоскости. Известия Российской академии наук. Серия математическая, 2022, 86
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