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

Ulrich Daepp Bucknell University, Lewisburg, PA
Pamela Gorkin Bucknell University, Lewisburg, PA
Andrew Shaffer Bucknell University, Lewisburg, PA
Karl Voss Bucknell University, Lewisburg, PA
MAA Press: An Imprint of the American Mathematical Society
Available Formats:
Hardcover ISBN: 978-1-4704-4383-2
Product Code: CAR/34
List Price: $63.00 MAA Member Price:$47.25
AMS Member Price: $47.25 Electronic ISBN: 978-1-4704-4881-3 Product Code: CAR/34.E List Price:$63.00
MAA Member Price: $47.25 AMS Member Price:$47.25
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AMS Member Price: $70.88 Click above image for expanded view Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other Ulrich Daepp Bucknell University, Lewisburg, PA Pamela Gorkin Bucknell University, Lewisburg, PA Andrew Shaffer Bucknell University, Lewisburg, PA Karl Voss Bucknell University, Lewisburg, PA MAA Press: An Imprint of the American Mathematical Society Available Formats:  Hardcover ISBN: 978-1-4704-4383-2 Product Code: CAR/34  List Price:$63.00 MAA Member Price: $47.25 AMS Member Price:$47.25
 Electronic ISBN: 978-1-4704-4881-3 Product Code: CAR/34.E
 List Price: $63.00 MAA Member Price:$47.25 AMS Member Price: $47.25 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$94.50 MAA Member Price: $70.88 AMS Member Price:$70.88
• Book Details

The Carus Mathematical Monographs
Volume: 342018; 268 pp
MSC: Primary 47; 30; 15; 51;

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.

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

• Chapters
• 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$\ldots$ 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

• Reviews

• 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
• Request Review Copy
Volume: 342018; 268 pp
MSC: Primary 47; 30; 15; 51;

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.

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

• Chapters
• 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$\ldots$ 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
• 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
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