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

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© 2018 by the author. All rights reserved. Printed in the United States of America. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 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. In the second part of the book (Chapter 9 through Chapter 14) we consider the connection between ellipses and Blaschke products that are products of more than three analytic disk automorphisms. We will see that though the boundaries of the numerical ranges of the matrices we study need not be elliptical, the boundaries always satisfy a Poncelet-like property. In addition, when the boundary is elliptical, it provides insight into the function-theoretic behavior of the corresponding Blaschke prod- uct. Because Poncelet’s ideas are intimately connected to our presenta- tion, we provide a recent and beautiful proof of the general theorem. This part of the book is also a bit more challenging than the first part, mathematically speaking, but we strive for a self-contained treatment, providing appropriate proofs and references. Finally, the third part of the book provides a range of exercises and projects, from straightforward exercises for someone first entering the field to potential research problems. Each chapter has a corresponding project that is usually introduced with new material. To be prepared to work through a project, active reading is required. Projects include things we find particularly interesting; for example, we discuss Sendov’s conjecture, interpolation, inversion, and the spectral radius. In addi- tion, we provide suggestions for the creation of several algorithms that will enhance one’s understanding. We hope that the reader will pick one x Preface

(or more) of these projects and use it (or them) to develop a deeper ap- preciation of the subject, to write an honors thesis, or to begin a research project. The necessary background and a starter bibliography for each project is provided, but we encourage readers to do a thorough search of the literature.

Using this book. The first two parts of the book are meant to tellthe story we have just described; a story that is meant to be read. However, this book can be used in various ways. It can serve as a reference for independent study, as the text for a capstone course in mathematics, or as a reference for a researcher. We have written this book for an active reader—paper and pencil in hand—and though there are a few exercises embedded in this reading, it is Part 3 of the text that includes a significant number of exercises as well as projects. These projects provide a range of both material and depth by including exercises well within the reach of every reader, suggested research papers, and research problems that have been open for many years.

Applets. There are several interactive applets designed to go along with the book, and we encourage using the applets while reading. If we feel that the time is ripe for experimentation, we will direct the reader to the applet using the symbol ,.1 Our applets illustrate various results in the text, such as the main result on the connection of Blaschke prod- ucts to ellipses. The proof of Poncelet’s theorem that we present depends on Pascal’s and Brianchon’s theorem, and we provide applets associated with each of these results. In addition, we have applets that illustrate the mapping behavior of Blaschke products (including locating fixed points in the closed unit disk), composition with Blaschke products, and these applets also produce the numerical range of particular matrices. It has been our experience that these tools are not only fun to work with, they also enhance one’s understanding and are great for testing conjectures.

Acknowledgments. The authors wish to thank the MAA Editorial Board of the Carus Series for many suggestions that have improved the manuscript. We also thank the editors at the AMS for their invaluable help preparing this book. We are grateful to Bucknell University for its

1http://pubapps.bucknell.edu/static/aeshaffer/v1/ Preface xi support and to the many students we have had who have played an im- portant role in the creation of this book. In particular, we wish to thank Robert C. Rhoades, Benjamin Sokolowsky, Nathan Wagner, and Beth Skubak Wolf. We are also grateful to Kelly Bickel, David Farmer, Gunter Semmler, Elias Wegert, and Jonathan Partington for valuable discus- sions and to Bucknell L&IT for technical support. Pamela Gorkin’s work was supported by Simons Foundation Collaboration Grant 243653.

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(푢퐻2)⟂, 112 ⟨⟨푥⟩⟩, 90 퐴, 117 ⟨푥, 푦⟩, 13 퐴⋆, 62 ⌊푥⌋, 98 퐵퐻2, 109 ℂ, 14 퐶(퐴), 163 ℂ∗, 134 푛 퐶휑, 236 ℂ , 13 2 퐶푀푎,휙 ∶ ℬ → ℬ, 192 ℙ (ℂ), 163 퐻2, 106 ℙ2(ℝ), 48 퐻2(픻), 105 ℝ∗, 134

퐻퐴 + 푖퐾퐴, 161 핋, 17 + 퐼ℓ, 148 퐇 , 134 − 퐼ℓ,푚, 148 퐇 , 134 퐽3, 164 ℬ, 192 퐽푛, 103 풮푛, 103 퐾ᵆ, 112 ‖퐴‖, 63, 76 퐿2(핋), 104 ‖푥‖, 13

푀푒, 162 휔+, 242 푀 , 177 휔−, 242 푎,휙 ←→ 푃+, 113 푢푣, 54, 179 푃−, 113 휓푐, 169, 177 푃ᵆ, 113 휎(퐸), 123, 232 ̃ 푆, 108 푘푎, 107 푆⟂, 71 tr(퐴), 16, 234 ⋆ 푆 , 111 휑푎, 114 2 푆ᵆ, 113 푒푗, standard basis 퐿 (핋), 104 푛 푈휆, 119 푒푗, standard basis ℂ , 63 푊(퐴), 14 푓(푛), 92

푋푝,푞, 148 푓푟, 105 푓(푛)̂ , 105 푘푎, 107 푤(퐴), 232

263 264 Index

푤(푇), 223 Braikenridge–MacLaurin theorem, 57 adjoint, 16 Brianchon’s theorem, 55 affine part, 59 Brianchon, Charles, 57 algebraic curve, 51 applets ,, x Cauchy kernel, 107 Blaschke product tools, 29–33, 121,Cauchy–Bunyakovsky–Schwarz in- 159, 170, 177, 209, 251 equality, 66, 79, 108

experimental tools, 138 chain of circles between 풞1 and 풞2, geometry tools, 8, 53, 56, 217 213 astroid, 204 Chapple, William, 45 Chapple–Euler formula, 43–46, 175, Bedford, Frank, 88 197 Bell, Eric Temple, 1 circular point, 242 Benford’s Law, 88 circumscribing a conic, 55 Beurling’s theorem, 111 class 풮푛, 103, 162 Blaschke Colosseum, 3 curve, 129, 174 companion matrix, 245 ellipse complete orthonormal basis, 105 (푛, 푝)-ellipse, 184 composition operator, 192, 236 3-ellipse, 33, 43, 60, 82 compressed shift operator, 113 product, 19, 23, 28 compression, 71 characterization, 19, 26 matrix, 117 composition, 159, 169, 171 conic, 51 decomposable, 169 associated matrix, 51 degree, 19 degenerate, 52 degree-2, 35 constructing an ellipse degree-3, 37–44, 82 envelope method, 201 degree-4, 169–174 folding, 7–10 finite, 19 pin and string method, 201 infinite, 111 contraction, 76 normalized, 23 curve properties, 28, 37–38 algebraic, 51 unicritical, 253 class, 163, 241 Blaschke, Wilhelm, 19 degree, 163, 241 Borcea, J., 209 dual, 51, 163, 241 boundary generating curve, 163 Index 265

real part, 163 perimeter, 5–7 tangent line, 51 Ellipse, President’s Park South, 3 cyclic quadrilateral, 221 elliptical range theorem, 16, 62, 68– 71, 164, 173, 222 Dégot, J., 207 envelope, 202, 204 degenerate, 52 Euclidean norm, 13 Denjoy–Wolff point, 251 Euler, Leonhard, 45 Denjoy–Wolff theorem, 251 Descartes’s circle theorem, 221 Fatou’s radial limit theorem, 106 Descartes, René, 85 field of values, 61 diagonal, 55 first digit, 88 direct sum focus, 242 of matrices, 166 ellipse, 3 disk automorphism, 23, 134, 177 real, 242 elliptic, 178 singular, 243 canonical, 182 Fourier coefficient, 105 characterization, 178 Frantz’s theorem, 179 convex order, 182 frequency, 90 order, 182 Fuss’s theorem, 175, 197 hyperbolic, 178 Fuss, Nicolaus, 45 parabolic, 178 dual, 51, 241–242 Gauss–Lucas theorem, 206 dual curve, 163 Gelfand’s questions, 90, 98 dual equation, 241 general position duality, 47, 57 of lines, 55 of points, 53 eigenvalue, 14, 232 Gergonne, Joseph, 57 Hermitian matrix, 161 glissette, 224 eigenvector, 14 ellipse, 3 half-plane area enclosed, 5 lower, 134 construction, 8–10 upper, 134 diameter, 3 Halmos, Paul, vii, 125 equation, 4 Hardy space, 105 focus, 3 Hermitian matrix major axis, 3 eigenvalue, 161 minor axis, 4 numerical range, 161 266 Index hexagon Kippenhahn, Rudolf, 123, 159–160 diagonal, 55 inscribed in conic, 53 Lagrange form, 140 opposite sides, 53 Lebesgue space, 104 opposite vertices, 55 Lebesgue, Henri, 104 principal diagonal, 55 line at infinity, 48 side, 53 linear fractional transformation, 134 simple, 53 Möbius transformations, 134 hexagrammum mysticum theorem, Maclaurin expansion, 7 57 Maclaurin, C., 7 Hilbert space, 76 major axis, 3 homogeneous polynomial, 51 Mandart inellipse, 212 hyperbolic Marden, M., 210 convex hull, 209 matrix convex set, 209 adjoint, 16 hyperbolic geometry, 208 associated to conic, 51 Illiev, L., 206 Cartesian decomposition, 161 inner function, 109 companion, 245 inner product direct sum, 166 퐿2(핋), 104 Hermitian, 160, 222 standard, 13 imaginary part, 161 inscribed in a conic, 53 nilpotent, 222 interpolation, 134, 237–241 nonnegative, 228 invariant subspace, 109, 111 norm, 63, 76 inverse of a point, 216 normal, 62 inversion, 215–221 normal, numerical range, 68, 75 iterate, 92, 182, 251 positive, 228 positive definite, 222 Jensen’s theorem, 206 real part, 161 Jordan block, 78, 103, 131, 164, 168, reducible, 124, 246 222 self-adjoint, 62, 160 similar, 63 Kepler, Johannes, 6 trace, 16, 69, 234 King, Jonathan, 91 unitarily equivalent, 63 Kippenhahn curve, 163 unitary, 62, 75 Kippenhahn’s theorem, 163 unitary decomposition, 123 Index 267 maximum modulus theorem, 26 opposite vertices, 55 measure, 95–97 orthogonal complement, 71 Mersenne, Marin, 57 orthogonal projection, 71 minor axis, 4 oval, 165 model space, 112 parallel postulate, 208 negative pedal curve, 227 Parseval’s identity, 105 Nelson, Wayne James, 87 Pascal line, 57 nephroid, 205 Pascal’s theorem, 53 nested form, 240 Pascal, Blaise, 57 Newcomb, Simon, 88 pedal curve, 227 Newton form, 238 pedal point, 227 nilpotent, 222 Perron–Frobenius theorem, 228 nonnegative matrix, 228 phase portrait, 30–32 norm Poincaré model, 208 Euclidean, 13 points at infinity, 48 matrix, 63 polygonal chain vector, 13 inscribed in conic, 53 normalized Blaschke product, 23 polynomial numerical radius homogenous, 51 matrix, 224, 232 Poncelet operator, 223 curve, 121, 130 numerical range, 14, 61 ellipse circular disk, 174 3-ellipse, 60, 82 elliptical disk, 159, 168, 171 4-ellipse, 175 Hermitian matrix, 161, 222 convex, 175 normal matrix, 75 property, 121 properties, 63, 68 Poncelet’s theorem, 92 alternate version, 157 operator false statements, 158 quasi-nilpotent, 223 for triangles, 18, 57 bounded, 76 general, 156 contraction, 77 Poncelet, Jean-Victor, 47 nilpotent, 222 porism, 213 norm, 77 positive matrix, 228 shift, 108 principal diagonal, 55 opposite sides, 53 268 Index projection, 71, 113 Steiner inellipse, 39, 211 projective geometry, 47 Steiner’s porism, 213 projective coordinates, 48 Steiner’s theorem, 210 projective space, 48 Stone–Weierstrass theorem, 26 Ptolemy’s theorem, 221 strongly real of positive type, 135, 239 quasi-nilpotent, 223 support line, 124, 162–164 radial limit, 106 Takenaka–Malmquist basis, 114 radical axis, 219 tangent line, 51 rational function, 135 tangential equation, 241 degree, 135 Toeplitz, Otto, 61 reference circle, 216 Toeplitz–Hausdorff theorem, 61–62, reproducing kernel, 107 71–73 normalized, 107, 114 trace, 16, 234 Traité des propriété projectives des Saratov notebook, 47 figures, 48 Schmeisser, G., 207 Schur’s theorem, 64 unit circle, 17 Sendov’s conjecture, 206 unitary 1-dilation, 76–82, 119 shift unitary dilation, 76 backward, 111 forward, 109 vector, 49 shift operator, 108 cross product, 49 adjoint, 111 dot product, 49 compressed, 113 stochastic, 229 side of a polygonal chain, 53 Siebeck’s theorem Walsh’s two-circle theorem, 206 general, 243 Weyl’s theorem, 98 triangles, 211 zero inclusion question, 236 Soddy’s formula, 221 spectral radius, 231 spectral theorem for normal matrices, 67 spectrum, 123, 232 standard form, 117 standard inner product, 13 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS

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 circum- scribed 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 2 × 2 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 n × n 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 exposi- tion 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 exer- cises 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.

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