An orthogonal polynomials approach to Poncelet’s theorem, numerical range, Blaschke products, and beyond…

For several decades, the relationship between Blaschke products, Poncelet’s Theorem, and numerical range of completely non-unitary contractions has been the focus of extensive research. Our goal is to discuss a connection between these topics and the theory of orthogonal polynomials on the unit circle. This recently discovered approach allows to prove several new results, to interpret the existing theory in a new context, and also to understand further connections with other areas of geometry and analysis. This is a joint work with Brian Simanek and Barry Simon.