Hardcover ISBN:  9781470443832 
Product Code:  CAR/34 
List Price:  $63.00 
MAA Member Price:  $47.25 
AMS Member Price:  $47.25 
Electronic ISBN:  9781470448813 
Product Code:  CAR/34.E 
List Price:  $63.00 
MAA Member Price:  $47.25 
AMS Member Price:  $47.25 

Book DetailsThe Carus Mathematical MonographsVolume: 34; 2018; 268 ppMSC: 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 delightfilled romp across a threeway unexpected connection between complex analysis, linear algebra, and projective geometry.
The book begins with Blaschke products, complexanalytic 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.ReadershipUndergraduate and graduate students interested in geometry, complex analysis, and linear algebra.

Table of Contents

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


Additional Material

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


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 Book Details
 Table of Contents
 Additional Material
 Reviews

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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 delightfilled romp across a threeway unexpected connection between complex analysis, linear algebra, and projective geometry.
The book begins with Blaschke products, complexanalytic 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