For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

This study traces the creation—in 1896—and evolution—through the outbreak of World War II—of the Colloquium lecture series of the American Mathematical Society (AMS). It documents how this innovation fostered a new sort of mathematical exchange and, in so doing, allowed the AMS to serve more effectively both as an agent of research-level mathematical communication and as a more truly national mathematical organization.

This is the diary of an observant mathematician who documented the drama of the resolution of Fermat's Last Theorem as it unfolded around him from 1993 to 1995. Pierre Fermat claimed around 1637, in the most famous marginalia in the history of mathematics, to have a proof of the theorem that xn + yn = zn has no whole number solutions for n greater than 2. The other principal figure is the British mathematician Andrew Wiles, who emigrated to Princeton University (...) in the 1980s and who in 1993, at what for mathematicians is the relatively advanced age of forty, announced a proof of FLT. In the course of Wiles's initial exposition to mathematicians, before a full‐fledged proof could be published, a gap was discovered. This was too late to forestall announcements in the world press of an imminent proof. What had been an almost secretive devotion to the task by Wiles now became a rather more public and pressurized effort over the course of nearly a year. Wiles wondered at times if success could be achieved, but with the help of other mathematicians the gap was filled in and the full formal proof published in 1995.The statement of the theorem is likely to be the only part of the 1990s story that a nonspecialist can immediately appreciate, and the author has not especially addressed his book to the nonspecialist. Rather, this account is analogous to a well‐illustrated exhibition guidebook that serves as a memento and record to those who visited in person but that is not intended fully to inform those who did not experience it. For a broader account of the historical and mathematical background a reader can consult the three principal books published previously on the same topic by Amir Aczel, Simon Singh, and Alfred J. van der Poorten. The author knew many of the main players, attended most of the key talks, took notes, made recordings, and photographed. A major part of the book is a selection of sixty‐two photographs from the total of three thousand made by the author. He asked questions of Wiles and others and recorded both formal and informal reactions that he observed around him during the many public and nonpublic presentations that were made. Also, in keeping with the journal nature of his work, he has recorded his own reactions. Appendixes include an excerpt from Wiles's 1995 Annals of Mathematics publication and a synopsis by Ram Murty of the key results by Gerhard Frey, Jean‐Pierre Serre, and Kenneth Ribet that formed a basis for Wiles's work.It may interest historians and archivists to learn of the extent to which the primary sources for this story include e‐mail, web pages, photographs, and newspaper and television interviews. Though we recognize this reality of contemporary science, a diary account probably brings the issue home more directly than would studies by professional historians, who might feel obligated to do more of their own interviews, for example, rather than rely on the media. Mathematics has rarely had such a public spotlight turned on it—the proof of the four‐color theorem in 1976 is the only comparable event that comes to mind. Consequently, it has opened an unusual window into how mathematicians do mathematics. (shrink)

Charles W. Curtis is a prominent mathematician who has made important contributions to the field of representation theory. His textbooks in this field have been classics for a long time. In Pioneers of Representation Theory he has set out to present the historical development of the main ideas of the discipline, from the work of Georg Ferdinand Frobenius in the 1890s up to 1960. In addition to Frobenius, the book focuses mainly on three other “pioneers”: William Burnside, Issai Schur, and (...) Richard Brauer.Curtis states that his interest in the history of representation theory arose gradually, and jointly with that of Reiner, in the process of writing their textbooks. Thus, a main aim of the book is to show how the pioneers reached their important results while using only the mathematics then available to them. In order to do so, Curtis presents successive and thorough accounts of the works in which the new and seminal ideas of the pioneers were introduced and then developed. At the same time, however, he simplifies some of the original arguments by translating them into modern terminology.In an attempt to make the book appealing and accessible to a wider circle of potential readers, some of the necessary mathematics is explained. Thus Chapter 1 presents an introduction to the basic ideas and problems of nineteenth‐century algebra and number theory; later on, as the need arises, additional mathematical sections addressed to nonexperts are included. Of course, even this very clear presentation will be accessible only to readers with at least a graduate‐level knowledge of algebra.Curtis also attempts to provide some historical context. He is particularly concerned to clarify the main problems that the work of each of the mathematicians discussed here was originally meant to resolve. In a book of this kind this is surely one of the major benefits that the reader may expect to obtain.There are biographical sketches of the major figures involved, as well as additional information about other participants. Some illuminating documents are quoted, and Curtis supplies brief accounts of the mathematical traditions within which each of the pioneers was educated and worked. He acknowledges, however, that Pioneers of Representation Theory has only a limited claim as a work of historical research. He thus provides references to more historically oriented work that interested readers can consult. In particular, he mentions several articles by Thomas Hawkins; it should be stressed that much of Hawkins's work has recently been published in book form: Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926 . Curtis's and Hawkins's books complement each other in many important ways, and each should appeal to readers of the other.Charles Curtis has written an impressive, authoritative, and well‐informed book on a difficult subject. Mathematicians and historians of twentieth‐century mathematics with the relevant background will find it difficult but rewarding reading. Along the way, historiographical questions may arise. To what extent is it possible to translate, as Curtis does, the original proofs of the pioneers into more modern terminology and yet remain faithful to the sources? In most cases, however, it will require the technical expertise of the author himself to answer questions like this one properly. We can only hope that the firm starting point this book provides will lead Curtis to further historical research in this and related fields. (shrink)