Explore the intricacies of Non-Commutative Function Theory in this 54-minute lecture, part of a mini-course series presented by Michael Jury from the University of Florida. Delve into advanced mathematical concepts such as rational functions, linear pencils, and the Hardy space. Examine the properties of left shift and left multiplier operators, and investigate the Burling theorem. Gain insights into inner-outer factorization, factorization theorems, and the characteristics of inner and outer functions. Analyze the relationship between polynomials and rational functions, and study zero sets, including detailed zeros and Blaschke inner factors. This lecture, part of the Focus Program on Analytic Function Spaces and their Applications, offers a deep dive into complex mathematical theory for advanced students and researchers in the field.