Jurij Volcic

Symmetries in noncommuting variables: in and out

Abstract: Matrices, unlike numbers, typically do not commute. This simple fact makes the study of functions in several matrix or operator variables inherently noncommutative, a theme that has been driven by advances in multivariable operator theory, random matrices, and quantum information theory. The last decade has seen rapid developments of complex analysis, geometry and optimisation tailored to noncommuting variables. This pair of lectures explores two facets of symmetry within this framework.

The first lecture reviews the story and the progress on the joint similarity problem, which concerns distinguishing matrix tuples up to a choice of coordinate system. The second lecture turns to noncommutative polynomial and rational functions invariant under permutations of their variables, and investigates their structure.