Asymptotic integration in free noncommutative function theory

Free noncommutative (nc) function theory is an analogue of classical function theory where one replaces functions between two complex vector spaces by functions between square matrices of all sizes over these vector spaces that respect matrix size, direct sums, and similarities. It has emerged in the last decade and a half as a vibrant research area with results that are sometimes similar to and sometimes strikingly different from the classical commutative setting, and with interconnections and applications ranging from free rings and free skew fields to free probability and random matrix theory to dimension independent linear matrix inequalities in systems and control. The purpose of this research in teams is to develop new relations between free probability and nc function theory by launching a systematic investigation and usage of asymptotic integration over classical compact matrix groups and their homogeneous spaces. We expect this to open the road for an asymptotic study of Hardy and Bergmann spaces of nc functions and exhibit completely new phenomena regarding Shilov boundaries and uniqueness sets in the nc setting.