Finding ellipses : what Blaschke products, Poncelet's theorem, and the numerical range know about each other / Ulrich Daepp, Pamela Gorkin, Andrew Shaffer, Karl Voss.
Publisher:
MAA Pressan imprint of the American Mathematical Society,
Copyright Date:
2018
Description:
xi, 268 pages : illustrations (some color) ; 23 cm.
Includes bibliographical references (pages 255-262) and index.
Contents:
Preface -- Part 1 -- The Surprising Ellipse -- The Ellipse Three Ways -- Blaschke Products -- Blaschke Products and Ellipses -- Poncelet's Theorem for Triangles -- The Numerical Range -- The Connection Revealed -- Intermezzo -- And Now for Something Completely Different... Benford's Law -- Part 2 -- Compressions of the Shift Operator : The Basics -- Higher Dimensions : Not Your Poncelet Ellipse -- Interpolation with Blaschke Products -- Poncelet's Theorem for n-Gons -- Kippenhahn's Curve and Blaschke's Products -- Iteration, Ellipses, and Blaschke Products -- On Surprising Connections -- Part 3 -- Fourteen Projects for Fourteen Chapters. Constructing Great Ellipses -- What's in the Envelope? -- Sendov's Conjecture -- Generalizing Steiner Inellipses -- Steiner's Porism and Inversion -- The Numerical Range and Radius -- Pedal Curves and Foci -- The Power of Positivity -- Similarity and the Numerical Range -- The Importance of Being Zero -- Building a Better Interpolant -- Foci of Algebraic Curves -- Companion Matrices and Kippenhahn -- Denjoy-Wolff Points and Blaschke Products.
Summary:
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.
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.