Poncelet polygons through the lenses of orthogonal polynomials on the unit circle, finite Blaschke products, and numerical ranges.

Abstract

The topics in this thesis fall in the intersection of projective geometry, complex analysis, and linear algebra. Each of these three fields gives a canonical construc-tion of families of polygons inscribed in the unit circle with interlacing vertices. In projective geometry, we consider Poncelet curves which admit a family of circum-scribed polygons. From the perspective of linear algebra, we consider the eigenvalues of a one parameter family of matrices called CMV matrices. Complex analysis gives two equivalent constructions of such polygons. One involves the preimages of points on T under a Blaschke product. The other involves the zero sets of paraorthogonal polynomials on the unit circle. In all three cases, the intersection of the polygons in this family forms part of an algebraic curve, which will be the primary focus of our study. A large part of this dissertation is dedicated to making each of these constructions precise in order to prove their equivalence. Particular attention is given to the notion of foci of an algebraic curve. In the latter chapters, we will focus on the specific case in which this inscribed curve (the intersection of the family of polygons) is an ellipse. We will give necessary and sufficient conditions in terms of the constructions of these families of n-gons for small n for which the inscribed curve is an ellipse. We will also provide algorithms for constructing such families of polygons with prescribed foci for the corresponding ellipse. We will conclude with possible directions for similar progress for larger n.