The goal of this project is to investigate a surprising connection between the geometry of conic sections and finite Blaschke products. Poncelet’s theorem states that if a polygon is inscribed in a conic section and circumscribes another conic section, then each point of the outer conic is a vertex of a polygon with the same property. Daepp, Gorkin and Mortini (2002) identified a surprising connection between Poncelet’s theorem and the geometry of finite Blaschke products. A finite Blaschke product is the product of finitely many automorphisms of the unit disc. Let B(z) be a Blaschke product of degree three with one zero at the origin and two other nonzero and distinct zeros. Each point on the unit circle has three preimages. The Daepp-Gorkin-Mortini result states that the (Euclidean) triangle formed by these preimages is of Poncelet type. In fact, the triangles formed by the three preimages of any point of the unit circle all circumscribe a fixed ellipse, whose foci lie at the remaining two zeros of B. In this project we’ll investigate various related questions. For instance, what happens if we replace “Euclidean triangle” by “hyperbolic triangle”? Alternatively, what is the locus spanned by all such triangles under other normalization for degree three Blaschke products? No background is needed in complex analysis; this project is accessible to students who have completed Math 241.
Completion of Calculus 3