Monotonicity, Convexity, and Realization of Noncommutative Functions

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.

Intro
Classical motivation
Goals of the talk
Representation of Pick functions
Connection with positive matrices
Matrix inequalities
The functional calculus
Special matrix functions
Classical representations: Löwner-Nevanlinna
Classical representations: Kraus
Thematic ideas in classical proofs
The matrix universe (more usual)
Noncommutative sets
Noncommutative functions
nc matrix functions
Löwner's Theorem
free Nevanlinna representation
What is the royal road?
Royal road approach
A royal road: dominions
Royal road theorem
Proof idea
A road to realizations
Convex functions
Kraus theorem
The butterfly realization
Further considerations