Differential and Complex Geometry: Origins, Abstractions and Embeddings by Raymond O. Wells Jr. (PDF)

Description

Differential and complex geometry are two central areas of mathematics with a long and intertwined history. This book, the first to provide a unified historical perspective of both subjects, explores their origins and developments from the sixteenth to the twentieth century.Providing a detailed examination of the seminal contributions to differential and complex geometry up to the twentieth-century embedding theorems, this monograph includes valuable excerpts from the original documents, including works of Descartes, Fermat, Newton, Euler, Huygens, Gauss, Riemann, Abel, and Nash.Suitable for beginning graduate students interested in differential, algebraic or complex geometry, this book will also appeal to more experienced readers.

User’s Reviews

Editorial Reviews: Review “The author’s writing style is generally clear and inviting … . This is a valuable book. There is sure to be something useful in it for anybody who is interested in geometry, topology, analysis or the history of mathematics.” (Mark Hunacek, MAA Reviews, February, 2018)“The book under review paves a wonderful pathway for introducing this fundamental chapter of mathematics to students: one can use this well-written volume to explain why certain results in differential and complex geometry are so crucial and important. This volume, a superb addition to the existent literature, is a work that should find its place in the library of every geometer and every topologist at work today.” (Bogdan D. Suceavă, Mathematical Reviews, 2018)“The book provides the reader with both historical and technical information regarding various sub-fields of differential and complex geometry. Undoubtedly, the author spent a lot of time researching the tackled subjects and presenting notions in such a clear and fluent way within the manuscript … . Differential and Complex Geometry: Origins, Abstractions and Embeddings is a rather valuable and extensive work bringing together research topics which are usually studied separately.” (Diana Maimut, zbMATH, 2018) From the Back Cover Differential and complex geometry are two central areas of mathematics with a long and intertwined history. This book, the first to provide a unified historical perspective of both subjects, explores their origins and developments from the sixteenth to the twentieth century.Providing a detailed examination of the seminal contributions to differential and complex geometry up to the twentieth century embedding theorems, this monograph includes valuable excerpts from the original documents, including works of Descartes, Fermat, Newton, Euler, Huygens, Gauss, Riemann, Abel, and Nash.Suitable for beginning graduate students interested in differential, algebraic or complex geometry, this book will also appeal to more experienced readers.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A little more than the first half of this book (Parts I-III) is a good historical overview that can be enjoyed by someone who’s taken a complex variables course. It covers the development of complex numbers and their geometrical representation, elliptic functions and integrals, complex analysis, and the birth of manifolds. The coverage extends from Pappus in 4th Century Alexandria to the the 1913 edition of Weyl’s Concept of a Riemann Surface.The final portion, Part IV, is a survey of 20th-Century embedding theorems by Whitney, Nash and others. Unfortunately, it requires vastly more background to be accessible: at least a year or more of study toward a graduate degree, I suspect. And whereas the first half of the book has more than a few illustrations (albeit a number of them color reproductions of pages of unillustrated text from original papers), this second half has none — even in a chapter on Riemannian manifolds, which often gets a more visual treatment.The author mentions that he spent more than 10 years writing the book. I wonder if maybe some impatience to finish it explains the major discontinuity in prerequisites needed for understanding the topics covered. A brief introduction to Part IV does provide an overview of the motivations for the subsequent chapters: but although quite accessible, it seems like an expedient. The description of how “analysis situs” became topology also seems rushed, with many decades of this interesting topic compressed into just a few paragraphs’ treatment.If you like to deepen your understanding of math from historical treatments, at least the first 11 chapters of this book (and the introduction to the 12th) will have plenty to hold your interest. But if you’re not up to speed with mid-20th Century differential geometry at a reasonably advanced level, you’ll find it hard to get much from roughly half the book. In that case, you might consider reading a library copy rather than buying your own at full price.

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