Probabilistic Methods in Analysis

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).

• The Laboratory conducts a seminar on probability, harmonic and complex analysis https://math-cs.spbu.ru/pta-seminar/

Collaborations:

• «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.