Daepp U., Gorkin P., Shaffer A., Voss K. Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other

American Mathematical Society (AMS), 2019. — 282 p. — (The Carus Mathematical Monographs 34). — ISBN10: 147044383X.Mathematicians delight in finding surprising connections between seemingly disparate areas of mathematics. Whole domains of modern mathematics have arisen from exploration of such connections consider analytic number theory or algebraic topology. Finding Ellipses is a delight-filled romp across a three-way unexpected connection between complex analysis, linear algebra, and projective geometry. The book begins with Blaschke products, complex-analytic functions that are generalizations of disk automorphisms. In the analysis of Blaschke products, we encounter, in a quite natural way, an ellipse inside the unit disk. The story continues by introducing the reader to Poncelet's theorem a beautiful result in projective geometry that ties together two conics and, in particular, two ellipses, one circumscribed by a polygon that is inscribed in the second. The Blaschke ellipse and the Poncelet ellipse turn out to be the same ellipse, and the connection is illuminated by considering the numerical range of a matrix. The numerical range is a convex subset of the complex plane that contains information about the geometry of the transformation represented by a matrix. Through the numerical range of matrices, we learn more about the interplay between Poncelet's theorem and Blaschke products. The story ranges widely over analysis, algebra, and geometry, and the exposition of the deep and surprising connections is lucid and compelling. Written for advanced undergraduates or beginning graduate students, this book would be the perfect vehicle for an invigorating and enlightening capstone exploration. The exercises and collection of extensive projects could be used as an embarkation point for a satisfying and rich research project. You are invited to read actively using the accompanying interactive website, which allows you to visualize the concepts in the book, experiment, and develop original conjectures.The Surprising Ellipse
The Ellipse Three Ways
Blaschke Products
Blaschke Products and Ellipses
Poncelet’s Theorem for Triangles
The Numerical Range
The Connection Revealed Intermezzo
And Now for Something Completely Different...Benford’s Law
Compressions of the Shift Operator: The Basics
Higher Dimensions: Not Your Poncelet Ellipse
Interpolation with Blaschke Products
Poncelet’s Theorem for 𝑛-Gons
Kippenhahn’s Curve and Blaschke’s Products
Iteration, Ellipses, and Blaschke Products
On Surprising Connections
Fourteen Projects for Fourteen Chapters