Finding Ellipses (Carus Mathematical Monographs)

Description

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. Cover Title page Copyright Contents Preface Part 1 Chapter 1. The Surprising Ellipse Chapter 2. The Ellipse Three Ways Chapter 3. Blaschke Products Chapter 4. Blaschke Products and Ellipses Chapter 5. Poncelet’s Theorem for Triangles Chapter 6. The Numerical Range Chapter 7. The Connection Revealed Intermezzo Chapter 8. And Now for Something Completely Different… Benford’s Law Part 2 Chapter 9. Compressions of the Shift Operator: The Basics Chapter 10. Higher Dimensions: Not Your Poncelet Ellipse Chapter 11. Interpolation with Blaschke Products Chapter 12. Poncelet’s Theorem for 𝑛-Gons Chapter 13. Kippenhahn’s Curve and Blaschke’s Products Chapter 14. Iteration, Ellipses, and Blaschke Products On Surprising Connections Part 3 Chapter 15. Fourteen Projects for Fourteen Chapters 15.1. Constructing Great Ellipses 15.2. What’s in the Envelope? 15.3. Sendov’s Conjecture 15.4. Generalizing Steiner Inellipses 15.5. Steiner’s Porism and Inversion 15.6. The Numerical Range and Radius 15.7. Pedal Curves and Foci 15.8. The Power of Positivity 15.9. Similarity and the Numerical Range 15.10. The Importance of Being Zero 15.11. Building a Better Interpolant 15.12. Foci of Algebraic Curves 15.13. Companion Matrices and Kippenhahn 15.14. Denjoy–Wolff Points and Blaschke Products Bibliography Index Back Cover