Serdecznie zapraszamy na cykl wykładów, które prowadzić będzie Benoit Collins (Université d\'Ottawa) na temat Discretization of non-commutative random variables. Wykłady odbywać się będą w poniedziałki i środy w godz. 16:15-18:00 w sali 711 począwszy od poniedziałku 20 kwietnia. Zapraszam, Piotr Śniady Streszczenie: In classical probability theory, any family of random variables can be approximated in a weak sense by discrete random variables. This is the essence of Lebesgue integration theory, and this process is very important both at the level of theory and at the level of applications. Non-commutative probability has been developed along similar lines as classical probability theory, with applications to various areas of mathematics and physics. Therefore the problem of approximating non-commutative random variables is quite important as well. The aim of this series of lectures is to review various aspects of the question of discretization of non-commutative random variables. The problem of whether any non-commutative random variable can be discretized was originally formulated by Alain Connes in the seventies in an operator algebraic context and language. As of today, it still stands as an open question. This question being of operator algebraic flavor, we will try to recall briefly the main operator algebraic aspects of it. However, instead of concentrating on the operator algebraic aspects of the question (which are not the main area of specialization of the lecturer, and which were the object of a less intense study over the last years), I will focus on more probability- and algebraic-minded aspects of this question that have been developed over the recent years. Although we will depict a few new strategies that could lead to proving or disproving the discretization conjecture, we will spend most of our time on a few relevant examples. In particular, we will review old and new criteria ensuring the discretizability of a given non-commutative probability space. We will also review old and new examples of discretization of random variables via random matrix theory and representation theory. And last, we will study selected positivity problems in real algebraic geometry and harmonic analysis and address the question of discretizing these problems. Here, our study will include in particular the Ho problem and the Bessis-Moussa-Villani conjecture.
