AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 34 Finding Ellipses What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other Ulrich Daepp Pamela Gorkin Andrew Shaffer Karl Voss 10.1090/car/034 Finding Ellipses What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other AMS/MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 34 Finding Ellipses What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other Ulrich Daepp Pamela Gorkin Andrew Shaffer Karl Voss Committee on Books Jennifer J. Quinn, Chair 2017–2018 Editorial Committee Fernando Gouvea, Editor Francis Bonahon Alex Iosevich Gizem Karaali Kristin Estella Lauter Steven J. Miller David P. Roberts 2010 Mathematics Subject Classification. Primary 47A05, 47A12, 30J10, 15-02, 15A60, 51-02, 51M04, 51N35. For additional information and updates on this book, visit www.ams.org/bookpages/car-34 Library of Congress Cataloging-in-Publication Data Names: Daepp, Ulrich, author. Title: Finding ellipses: What Blaschke products, Poncelet’s theorem, and the numerical range know about each other / Ulrich Daepp [and three others]. Description: Providence, Rhode Island: MAA Press, An imprint of the American Math- ematical Society, [2018] | Series: The Carus mathematical monographs; volume 34 | Includes bibliographical references and index. Identifiers: LCCN 2018021655 | ISBN 9781470443832 (alk. paper) Subjects: LCSH: Ellipse. | Conic sections. | Geometry, Projective. | AMS: Operator theory – General theory of linear operators – General (adjoints, conjugates, products, inverses, domains, ranges, etc.). msc | Operator theory – General theory of linear operators – Numerical range, numerical radius. msc | Functions of a complex variable – Function theory on the disc – Blaschke products. msc | Linear and multilinear algebra; matrix theory – Research exposition (monographs, survey articles). msc | Linear and multi- linear algebra; matrix theory – Basic linear algebra – Norms of matrices, numerical range, applications of functional analysis to matrix theory. msc | Geometry – Research exposition (monographs, survey articles). msc | Geometry – Real and complex geom- etry – Elementary problems in Euclidean geometries. msc | Geometry – Analytic and descriptive geometry – Questions of classical algebraic geometry. msc Classification: LCC QA559 .F56 2018 | DDC 516/.152–dc23 LC record available at https://lccn.loc.gov/2018021655 Copying and reprinting. 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Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 23 22 21 20 19 18 Contents Preface vii Part 1 1 Chapter 1 The Surprising Ellipse 3 Chapter 2 The Ellipse Three Ways 13 Chapter 3 Blaschke Products 23 Chapter 4 Blaschke Products and Ellipses 35 Chapter 5 Poncelet’s Theorem for Triangles 47 Chapter 6 The Numerical Range 61 Chapter 7 The Connection Revealed 75 Intermezzo 85 Chapter 8 And Now for Something Completely Different... Benford’s Law 87 Part 2 101 Chapter 9 Compressions of the Shift Operator: The Basics 103 Chapter 10 Higher Dimensions: Not Your Poncelet Ellipse 121 Chapter 11 Interpolation with Blaschke Products 133 Chapter 12 Poncelet’s Theorem for 푛-Gons 147 v vi Contents Chapter 13 Kippenhahn’s Curve and Blaschke’s Products 159 Chapter 14 Iteration, Ellipses, and Blaschke Products 177 On Surprising Connections 195 Part 3 199 Chapter 15 Fourteen Projects for Fourteen Chapters 201 15.1 Constructing Great Ellipses 201 15.2 What’s in the Envelope? 201 15.3 Sendov’s Conjecture 206 15.4 Generalizing Steiner Inellipses 210 15.5 Steiner’s Porism and Inversion 213 15.6 The Numerical Range and Radius 222 15.7 Pedal Curves and Foci 224 15.8 The Power of Positivity 228 15.9 Similarity and the Numerical Range 231 15.10 The Importance of Being Zero 234 15.11 Building a Better Interpolant 237 15.12 Foci of Algebraic Curves 241 15.13 Companion Matrices and Kippenhahn 245 15.14 Denjoy–Wolff Points and Blaschke Products 251 Bibliography 255 Index 263 Preface [Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing—one great, glo- rious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing.... They are intimately interconnected, and they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That’s what mathematics is to me. –Paul Halmos, Celebrating 50 Years of Mathematics Discovering surprising connections between one area of mathematics and another area is not only exciting, it is useful. Laplace transforms al- low us to switch between differential equations and algebraic equations, the fundamental theorem of algebra can be proved using complex analy- sis, and group theory plays a fundamental role in the study of public-key cryptography. If we can think of an object from different angles, these different perspectives aid our understanding. It is in that vein thatwe have written this book: We look at three seemingly different stories and a hidden connection between them. The first of our stories is complex—that is, it is about complex func- tion theory. In complex analysis, linear fractional transformations are among the first functions that we study. We understand their geometric and function-theoretic properties, and we know that they play an im- portant role in the study of analytic functions. In the class of linear frac- tional transformations, the automorphisms of the unit disk stand out for the geometric and analytic insight they provide. The functions that we focus on are a natural generalization of the disk automorphisms; they are vii viii Preface finite products of these automorphisms and are called Blaschke prod- ucts, in honor of the German geometer Wilhelm Blaschke. While we know, for example, that automorphisms of the disk map circles to cir- cles (if we agree that lines are circles), much less is known about the geometry of Blaschke products. The second story comes from projective geometry. Here we focus on Poncelet’s closure theorem, a beautiful geometric result about conics in- scribed in polygons that are themselves inscribed in conics. We devote some attention to precursors of Poncelet’s theorem, including Chapple’s formula and Fuss’s theorem. In the final part of this book we will also consider variations on this theme, including Steiner’s porism. The final story is about the numerical range of a matrix. The numer- ical range is the range of a quadratic form associated with a matrix but restricted to the unit sphere of ℂ푛. The numerical range always con- tains the eigenvalues of the matrix, and it often can tell you more about a matrix than the eigenvalues can. While questions about the numerical ranges of 2 × 2 matrices are (relatively) easily analyzed, consideration of 푛 × 푛 matrices for 푛 > 2 is deeper. Even 3 × 3 matrices lead to deep and interesting questions! What is the hidden connection between these three stories? Curiously, the connection is an object we have not discussed much yet: It is the ellipse. In fact, this book is really a story about the ellipse, an object studied by the Greek mathematicians Menaechmus, Euclid, Ap- polonius of Perga, and Pappus, among others. Later, Kepler developed a description of planetary motion, leading to his first law: The orbit ofa planet is elliptical with one focus at the Sun. In spite of this long history, ellipses never cease to surprise us. It turns out that ellipses provide a remarkable amount of information about Blaschke products. Ellipses can tell us when a Blaschke product has a nontrivial factorization with respect to composition. And when we study the dynamics of Blaschke products, ellipses can tell us what to expect. The ellipse even establishes a connection between Blaschke products and a useful tool for detecting tax fraud, known as Benford’s law. But why does an ellipse know anything about these particular ra- tional functions? That is the story we wish to tell, and it is a story that Preface ix relies heavily on properties of the numerical range of a class of operators and that story, in turn, relies on projective geometry. The first part of this book (Chapter 1 through Chapter 7) focuses on Blaschke products that are products of three disk automorphisms, el- lipses that are inscribed in triangles, and 2 × 2 matrices. In this case, ellipses suggest directions to study. Each Blaschke product of degree 3 is naturally associated with an ellipse, as is each 2×2 matrix. The relation- ship between Blaschke products, 2 × 2 matrices, and Poncelet’s theorem will be revealed in Chapter 7. This part of the book also introduces the basic ideas of two-dimensional projective geometry and provides a proof of Poncelet’s theorem in the event that the circumscribing polygons are triangles. After concluding Part 1, the reader will be treated to an inter- mezzo (Chapter 8), as we focus on the surprising connection between Poncelet’s theorem and Benford’s law.