Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other

This unusual book is a recent entry in the MAA’s Carus Mathematical Monograph series. Its theme revolves around some surprising connections among complex analysis, geometry, and linear algebra. It is not so much a textbook as a resource for capstone courses, independent study or research. It is best suited for those with at least an advanced undergraduate background in mathematics.

The book is divided into three parts. The first part develops and reveals the surprising connection in its most satisfying form. In the second part ,we go deeper to see what we get and what we lose in generalizing the basic results. A third part provides a collection of exercises and projects.

The elements of the surprising connection consist of: finite Blaschke products (products of linear fractional transformations of a special form that are automorphisms of the unit disk in the complex plane); Poncelet’s theorem for triangles (a result from projective geometry); and the numerical range of a matrix (the set of values of the quadratic form \( x^{*}Ax \) where \( x \) is a complex number of unit magnitude).

What’s the connection? If a Blaschke product \( B(z) \) of three automorphisms of the unit disk sends \( 0 \) to \( 0 \), then the intersection over all \( \lambda \) in the unit circle of the triangular regions formed using the three distinct zeros of \( B(z) – λ = 0 \) as vertices is an ellipse with foci at the zeros of \( B(z)/z \). This ellipse is the boundary of the numerical range of a matrix \( 2 \times 2 \) matrix \( A \) and its foci are at the eigenvalues of \( A \). Furthermore, that ellipse is inscribed in infinitely many triangles that are themselves inscribed in the unit circle, one triangle for each point of the circle. That ellipse is a 3-Poncelet ellipse and it is a central fixture of the book.

The surprise isn’t revealed until well into the book. The first part of the book is written as a story of discovery. It begins with constructions of the ellipse itself and continues by introducing the three apparently unrelated subjects that are assembled to create the surprise. The reader is introduced to the basics of Blaschke products, Poncelet’s projective geometry and his theorem for triangles, and the linear algebra underlying the idea of a matrix’s numerical range. All the material in this part of a book is within the range of a good undergraduate.

The second part of the book is more advanced. Its goal is to explore connections between the ellipse and Blaschke products that are products of more than three automorphisms of the unit disk. It turns out that the boundaries of the numerical range of the relevant matrices need not be elliptical, but that the boundaries satisfy a Poncelet-like property. The operators that provide matrices are compressions of a shift operator associated with finite Blaschke products. When the boundary is elliptical, the Blaschke product has a special form. The mathematical reach of this part is significantly greater. The chapters include elements of Lebesgue theory, Hardy spaces, functional analysis, operator theory and more.

The third part of the book is an introduction to further research. The authors have devised projects and provided some exercises for each of the chapters in the book. Some of these projects, particularly the latter ones, might be very challenging.

Not much of the story here is told in a strictly linear form. The first part is the most direct, but it has several digressions. The second part is more complex and more digressive. The text has something of a tree structure with three main trunks and many branches. Even strong students might find themselves far out on a branch trying to reconnect to main features of the argument.

This is an intriguing way to provide attractive and accessible subjects for undergraduates in a research setting. Each of the three main topics offers many opportunities for deeper exploration, and their collision in the context of this book makes them even more appealing. The challenges are not insignificant. Even the best students might find themselves paging backward and forward in the book, feeling frustrated while trying to make connections.

An extensive bibliography provides support for a considerable variety of investigations. The authors have also created several interactive applets on their website that the reader is directed to explore at certain points during the book, such as when the Blaschke product’s connection to ellipses is described.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.