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Publisher:

American Mathematical Society

Publication Date:

2000

Number of Pages:

196

Format:

Hardcover

Price:

30.00

ISBN:

978-0-8218-2670-6

Category:

General

[Reviewed by , on ]

Robert McGuigan

01/18/2001

Probably we are all familiar with the story of Andrew Wiles's proof of Fermat's Last Theorem. Wiles, working alone and in secret for years, thought he had proved Fermat's Last Theorem. He announced his result at a conference in Cambridge, England, in June of 1993. During the refereeing process a gap was found in the proof. Six months of effort by Wiles led, with some help from Richard Taylor, to the successful completion of the proof. How many of us would like to have been "present at the creation", hanging around Princeton, attending teas and gossiping with inside figures, listening to explanatory lectures and attending celebratory conferences? The book under review is a diary-style recounting of the author's experience as a Fermat "groupie", following the story from its beginning in exciting e-mails and attending and reporting on most of the many conferences and lectures devoted to celebrating and explaining this marvelous work. Mozzochi is a mathematician who published research papers in the area of arithmetic algebraic geometry and number theory, was located in Princeton at the time, and was well positioned to talk to important figures in the Fermat story and be a recipient of grapevine information.

Popularizations of the story focus on the drama of Wiles's devoted pursuit of this problem, the heartbreak of the gap in the proof, and the triumph of the final solution. Of necessity they give only the sketchiest description of the mathematics involved. Mozzochi's *Diary*, written by a mathematician for mathematicians, gives much more detail about the mathematics and the research process. There is also a lot of detail about the people involved, not just Wiles. Through this diary the reader gets to meet almost every major player in the story. A wonderful addition to the text is the inclusion of 62 photographs, almost all taken by the author, of just about everyone who gave a lecture on the proof or otherwise made an important contribution. We are given a rare description of the functioning and collegiality of an extensive community of research mathematicians.

The book begins with a recounting of how the author, at the Institute for Advanced Studies, learned of Wiles's announcement at the conference in Cambridge, England. We then learn of the email circulation that brought the news to much of the number theory research community, the details of the finding of the famous gap, the anguish-filled six months of effort ending with the successful completion of the proof, and of the many lectures given by Wiles and others explaining the proof. Instead of Mozzochi's only discussing Wiles's feelings when the gap was discovered we also find out how many other mathematicians reacted to the situation. In addition to being an interesting account of the activity surrounding Wiles's work on the Fermat problem, Mozzochi's book provides historical documentation, as well. Conferences and lectures are described very precisely, down to the date, time, and even the name and room number of the hall in which lectures took place. For example, "On April 21st Wiles delivered a lecture on his proof in room 220 of the Durham Laboratory of Engineering at Yale University between 3:30 and 4:45 PM." Copies of e-mail messages circulated to the number theory research community are included.

An attempt is made to make a careful tracing of mathematical influences leading up to the final proof. For example, Mozzochi takes care to give due credit to Gerhard Frey, who is somewhat neglected in other accounts, for his early contributions to the idea, ultimately proved correct by Kenneth Ribet, that the Shimura-Taniyama conjecture implies Fermat's Last Theorem. The reactions of many mathematicians to the discovery of the gap in Wiles's original attempted proof are described. We learn of the frustrated competitive feelings of some other researchers who felt that Wiles, trying to fill the gap in the proof, waited too long to release the correct part of his original manuscript. Other mathematicians, having copies of the manuscript through serving as referees when it was first submitted to *Inventiones Mathematicae*, were careful not to use it in their lectures before Wiles officially released it. An inquiry is made, citing primary sources, into the proper attribution of the Shimura-Taniyama conjecture. Everything is properly footnoted and documented. An extensive bibliography of 131 items supports the text. The book concludes with a brief account of the proof of the full Shimura-Taniyama conjecture by Conrad, Diamond, Taylor, and Breuil. Appendices include a substantial quote from the introduction to Wiles's *Annals of Mathematics* paper containing the correct proof, and a reprinting of Ram Murty's review of the published proceedings of the instructional conference held at Boston University in August of 1995.

This book will undoubtedly be extremely useful to future historians of mathematics. It is very enjoyable to read as well. Though there is a lot of mathematical terminology used without explanation, it is possible to read and enjoy the book just for the stories about the people involved and to get a taste of the excitement in the number theory community at the time. Wiles's work makes all of us proud to be mathematicians. This book can make us feel closer to the way in which Wiles's work was received and is being assimilated by the mathematical world. I recommend it as an addition to both personal and institutional libraries.

Robert McGuigan (mcguigan@cs.wsc.ma.edu) chairs the mathematics department at Westfield State College, Westfield, Massachusetts. His broad mathematical interests include number theory, geometry, analysis functional and otherwise and elementary and secondary pedagogy. He is a Japanese linguist, teaching the language at Westfield. A devotee of the game of go, he has published many translations of writings on go. He is an avid amateur musician, playing violin and viola in local chamber groups and orchestras.

- February 10, 1994
- September 19, 1994
- August 9, 1995
- Cast of characters
- List of photographs
- Acknowledgements and notes
- Appendix A: An excerpt from the Introduction to Wiles's
*Annals of Mathematics*paper - Appendix B: List of speakers, Cambridge University, June 1993
- Appendix C: List of speakers, Boston University, August 1995
- Murty's review
- Bibliography
- Index

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