**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

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

Share this page
*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.