Introduction to Isospectrality

The book discusses the theory behind Mark Kac's question, which involves a mathematical problem that led to sophisticated mathematics. It presents Sunada's method and the construction of non-isometric yet isospectral drum membranes, as first discovered by Gordon–Webb–Wolpert. The book is suitable for lecture courses and offers researchers an elegant, self-contained reference on the topic of isospectrality.

Format: Paperback / softback
Length: 238 pages
Publication date: 14 September 2022
Publisher: Springer International Publishing AG

This captivating question, famously coined by Mark Kac, unveils a profound mathematical conundrum that has spurred the development of sophisticated mathematics. This textbook presents the theoretical framework underlying this problem, making it accessible to students for the first time. Specifically, this book offers a comprehensive exploration of Sunada's method and the construction of non-isometric yet isospectral drum membranes, as initially discovered by Gordon–Webb–Wolpert. The book commences with an introductory chapter on Spectral Geometry, emphasizing isospectrality and providing a sweeping overview (without proofs) of the Sunada–Bérard–Buser strategy. The remainder of the book comprises three chapters. Chapter 2 delves into the realm of flat surfaces, offering an elementary treatment and outlining Buser's combinatorial method for constructing a flat surface with a specified set of isometries (a Buser surface). Chapter 3 establishes the fundamental isospectrality theorems and elucidates the transplantation technique on Buser surfaces. Chapter 4 constructs Gordon–Webb–Wolpert domains from Buser surfaces and establishes their isospectrality. Lavishly illustrated and supplemented by four substantial appendices, this book is well-suited for lecture courses aimed at students who have completed introductory graduate courses in algebra, analysis, differential geometry, and topology. It also serves as a refined and self-contained reference for researchers interested in the field of isospectrality.

Weight: 479g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031171222
Edition number: 1st ed. 2022