Explore a comprehensive lecture on noncommutative functions, focusing on monotonicity, convexity, and realization. Delve into classical motivations, representation of Pick functions, and connections with positive matrices. Examine matrix inequalities, functional calculus, and special matrix functions. Investigate classical representations like Löwner-Nevanlinna and Kraus, along with thematic ideas in classical proofs. Learn about the matrix universe, noncommutative sets, and noncommutative functions. Discover Löwner's Theorem, free Nevanlinna representation, and the royal road approach. Understand convex functions, Kraus theorem, and the butterfly realization. This 49-minute talk, presented by Ryan Tully-Doyle from California Polytechnic State University, is part of the Focus Program on Analytic Function Spaces and their Applications at the Fields Institute.