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Invited researcher Hedenmalm Haakan Per
Time span of the project
General information
Name of the project: Probability in analysis: point processes, operators, and spaces of holomorphic functions.

Strategy for Scientific and Technological Development Priority Level: A
Goals and objectives

The goal of the project is to develop and greatly extend the ongoing activity in harmonic analysis and related fields in the St. Petersburg State University. We plan to expand the existing research in classical analysis, emphasizing modern trends and the emerging applications to other areas, such as probability, geometry and mathematical physics. Among the new subjects we plan to bring in, there are determinantal processes and related topics as well as connections with conformal field theory. Below we stress several directions of research:

  1. Determinantal processes arising from physical models. It is planned to study determinantal processes on the plane and to analyze the associated correlation kernels. We will study the behavior of the Coulomb gas model near the spectral boundary and the behavior of the higher Landau level models (polyanalytic Ginibre ensembles).

  2. Inverse problem of potential theory and Schwarz functions. In the classical random normal matrix model it is planned to study the equilibrium measures of the corresponding Coulomb gas ensembles. It is planned to use methods of complex dynamics to answer some fundamental questions concerning the shapes of the droplets (the supports of the equilibrium measures) as well as their changes under the variation of the potential (e.g. Laplacian growth).

  3. Research in the area of Uncertainty Principle in Harmonic Analysis. This area includes problems on completeness of exponentials and polynomials, inverse spectral problems for differential operators and Krein's canonical systems, the theory of de Branges spaces of entire functions, classical problems in the theory of stationary Gaussian Processes, problems in signal processing, etc., together with their modern extensions and applications.

  4. The development of perturbation theory of linear operators. The goal of this part of the project is to investigate how much the functions of the perturbed operator differ from the functions of the original operator depending on the properties of the perturbation and the function.

The practical value of the study

The main expected results include:

  1. Regarding polyanalytic Ginibre ensembles we intend to analyze the blow up kernels at spectral boundary points. We expect to obtain interesting asymptotics as the polyanalytic degree tends to infinity slower than the analytic degree of the polynomials.

  2. We expect to establish interesting symmetry properties of the mathematical material we call arithmetic jellium.

  3. In the random normal matrix model, it is planned to obtain a complete asymptotic expansion of the free energy.

  4. It is supposed to investigate the iterations of Schwarz functions for a number of droplets that are carriers of the equilibrium measure of Coulomb gas ensembles. It is planned to describe the topological properties of the corresponding Julia sets.

  5. We plan to apply the inverse problem method to find soliton-type solutions of the Hele--Shaw equation, and using Schwarz function, we will try to identify the integrable structures corresponding to various type of singularities on the moving boundary.

  6. We expect to achieve new understanding and develop new algorithms for solving inverse spectral problems for canonical Hamiltonian systems and, in particular, to generalize the classical Gelfand-Levitan theory in the area of spectral analysis of differential operators to broader classes of operators including those with singular potentials.

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