**The Carus Mathematical Monographs**

Volume: 34;
2018;
268 pp;
Hardcover

MSC: Primary 47; 30; 15; 51;

**Print ISBN: 978-1-4704-4383-2
Product Code: CAR/34**

List Price: $63.00

AMS Member Price: $47.25

MAA Member Price: $47.25

**Electronic ISBN: 978-1-4704-4881-3
Product Code: CAR/34.E**

List Price: $63.00

AMS Member Price: $47.25

MAA Member Price: $47.25

#### Supplemental Materials

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

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*Ulrich Daepp; Pamela Gorkin; Andrew Shaffer; Karl Voss*

MAA Press: An Imprint of the American Mathematical Society

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 \(2 \times 2\) 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 \(n \times n\) 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 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.

#### Readership

Undergraduate and graduate students interested in geometry, complex analysis, and linear algebra.

#### Reviews & Endorsements

This book is part of the lovely collection 'Carus Mathematical Monographs,' which is aimed at an advanced undergraduate audience. The book is perfectly pitched to the target audience. The exposition is very clear and engaging, as the material is developed basically as a story. It addresses a broad array of topics: projective geometry (which is not very much taught nowadays), complex analysis, operator theory, interpolation, iteration, even a bit of number theory. The various results are brought up in an inquisitive manner that will convey to the reader the flavor of the research process.

-- Line Baribeau

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Copyright iv5
- Contents v6
- Preface vii8
- Part 1 114
- Chapter 1. The Surprising Ellipse 316
- Chapter 2. The Ellipse Three Ways 1326
- Chapter 3. Blaschke Products 2336
- Chapter 4. Blaschke Products and Ellipses 3548
- Chapter 5. Poncelet’s Theorem for Triangles 4760
- Chapter 6. The Numerical Range 6174
- Chapter 7. The Connection Revealed 7588
- Intermezzo 8598
- Chapter 8. And Now for Something Completely Different… Benford’s Law 87100

- Part 2 101114
- Chapter 9. Compressions of the Shift Operator: The Basics 103116
- Chapter 10. Higher Dimensions: Not Your Poncelet Ellipse 121134
- Chapter 11. Interpolation with Blaschke Products 133146
- Chapter 12. Poncelet’s Theorem for 𝑛-Gons 147160
- Chapter 13. Kippenhahn’s Curve and Blaschke’s Products 159172
- Chapter 14. Iteration, Ellipses, and Blaschke Products 177190
- On Surprising Connections 195208

- Part 3 199212
- Chapter 15. Fourteen Projects for Fourteen Chapters 201214
- 15.1. Constructing Great Ellipses 201214
- 15.2. What’s in the Envelope? 201214
- 15.3. Sendov’s Conjecture 206219
- 15.4. Generalizing Steiner Inellipses 210223
- 15.5. Steiner’s Porism and Inversion 213226
- 15.6. The Numerical Range and Radius 222235
- 15.7. Pedal Curves and Foci 224237
- 15.8. The Power of Positivity 228241
- 15.9. Similarity and the Numerical Range 231244
- 15.10. The Importance of Being Zero 234247
- 15.11. Building a Better Interpolant 237250
- 15.12. Foci of Algebraic Curves 241254
- 15.13. Companion Matrices and Kippenhahn 245258
- 15.14. Denjoy–Wolff Points and Blaschke Products 251264

- Bibliography 255268
- Index 263276
- Back Cover Back Cover1282