A beginner's view of noncommutative realizations

Noncommutative analysis is a topic of interest for lots of us in the operator theory community, and realizations are an important and useful tool. However, many of the references are quite abstract and not geared towards beginners. This talk gives an introduction to noncommutative rational functions and realizations — not in full generality, but with detailed computations and examples.

Further reading: Helton–Mai–Speicher (arXiv:1511.05330), from which most of the talk is adapted; Helton–McCullough–Vinnikov on convex rational functions (§2 and Appendix A); Berstel–Reutenauer, Noncommutative Rational Series with Applications (Ch. 1, §5 and §7).