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Invited researcher Hedenmalm Haakan Per
Time span of the project
2021-2023
General information
Name of the project: Probabilistic methods in analysis: point processes, operators and holomorphic function spaces

Strategy for Scientific and Technological Development Priority Level: а
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
  1. In Ginibre ensembles theory, we expect to analyse the asymptotic behaviour of kernels at spectral boundary points. We will obtain a new precise asymptotic behaviour for the case when the degree of polyanalyticity tends to infinity slower than the analytic degree of the polynomials
  2. To determine the properties of the mathematical material known as arithmetic jellium.
  3. In the model of normal random matrices, we expect to obtain a complete asymptotic decomposition of free energy.
  4. Our researchers will investigate iterations of the Schwartz functions for a number of droplets – carriers of the equilibrium measure of Coulomb gas ensembles. It is planned that we describe the topological properties of the corresponding Julia sets.
  5. We are planning to apply the inverse problem method to find soliton-type solutions to the Hele–Shaw equation and, using the Schwartz function, determine the integrated structures corresponding to various types of features on a moving boundary.
  6. Our team expect to develop new algorithms for the solution of inverse spectral problems for canonical Hamiltonian systems, in particular, to generalise the classical Gelfand–Levitan theory in the domain of spectral analysis of differential operators to wider classes of operators, including those with singular potential.

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