Seminar in Mathematical Modeling and Analysis - Rafał Czyż

The seminars in Mathematical Modeling and Analysis are aimed at researchers, employees, and students.

This week's seminar is given by Speaker: Rafał Czyż, Jagiellonian University.

Title: Infinite-Dimensional Determinants and Pluripotential Theory

Abstract: In my talk, I will present some basic facts from pluripotential theory in Hilbert spaces. It is well known that in $\mathbb C^n$ smooth maximal plurisubharmonic functions $u$ can be characterized by the complex Monge-Ampère operator, namely by condition $(dd^c u)^n=0$. We want to obtain similar characterization in infinite dimensional Hilbert spaces. The notion of determinant is needed to generalize the Monge-Ampère operator to infinite dimensional Hilbert spaces. I will present three concepts of determinant: trace and determinant class operators defined for compact operators, Fuglede-Kadison determinant defined in some special von Neumann algebras, and Fujii-Seo normalized determinant defined for positive operators.