Point processes are well studied objects in probability theory, with applications in many different disciplines such as nuclear physics, materials science, telecommunications, astronomy, artificial intelligence (machine learning) and economics, among others. Some interesting point processes can be obtained as eigenvalues of random matrices or as zeros of series expansions with random coefficients.
These random configurations of points, unlike Poisson processes, show no clumping and are related with "optimal" distributions of points (e.g., Fekete points). The study of these point processes provides a strong interaction between probability theory (stochastic processes) and different branches in classical analysis: complex analysis, numerical analysis, potential theory, reproducing kernel function spaces, orthogonal polynomials, Riemann-Hilbert problems, etc.
The purpose of this CIMPA school is to focus on two particularly active areas which are representative of this interaction. The study of determinantal point processes and of zeros of random gaussian functions. Determinantal point processes appear in the study of eigenvalues of random matrices with complex entries and also of log-gas systems. Since their introduction into the physics community in the 50s, they have influenced many different branches of mathematics: potential theory, Riemann-Hilbert problems, orthogonal polynomials, differential equations and operator theory.
The study of zeros of random gaussian analytic functions, or zeros of series expansions with random coefficients, is a classical subject in complex analysis. The relation of this processes with reproducing kernel function spaces has given a new impulse to their study using methods from complex analysis, complex geometry and numerical analysis.
The scientific activities planned for this school pursue two main goals. On the one hand, to strengthen the research activities in probability theory by promoting interactions with groups working on classical analysis, not only in Argentina, but also in the neighboring countries. On the other hand, we look forward to attract PhD as well as postdoctoral students, to these research areas, that nowadays are not very active in the Argentinian mathematical community.