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Publisher:

Birkhäuser

Publication Date:

2012

Number of Pages:

539

Format:

Hardcover

Price:

129.00

ISBN:

9783034802147

Category:

Monograph

[Reviewed by , on ]

Felipe Zaldivar

06/25/2012

Shafarevich is a well-known and accomplished mathematician. His contributions to number theory and algebraic geometry set him apart. A quick look at his *Collected Mathematical Papers* (Springer, 1988) gives a glimpse to the many topics where he made important and lasting contributions: class field towers, the inverse problem of Galois theory for solvable groups, moduli of K3 surfaces, finiteness results and conjectures on the arithmetic and geometry of elliptic curves, abelian varieties, algebraic curves and number fields. In addition to that, he is also remembered for the scientific school he founded and nourished in Moscow: Manin, Piatetski-Shapiro, Tyurina, Arakelov, Moisheson, Dolgachev, Kostrikin, Golod, Kolyvagin and many others were at some time students of Shafarevich and contributed to his many seminars. It is not an exaggeration to say that Algebraic Geometry and Arithmetic Algebraic Geometry flourished in the Soviet Union thanks, in great part, to the work and dedication of I. R. Shafarevich.

Shafarevich has also written several well-received textbooks and monographs, from *Basic Algebraic Geometry* (Springer, 1974) and *Number Theory,* coauthored with Z. I. Borevich, (Academic, 1966) to *Algebraic Surfaces*, with contributions by Manin, Moisheson, Tyurin and others (AMS, 1967). The mathematical community recognized Shafarevich’s contributions early and he holds many awards and memberships to various learned societies.

Those contributions to science would have made a biography of Shafarevich more than desirable. But there is another aspect of Shafarevich’s thinking, perhaps not as well known as his mathematical one, which took Shafarevich into the limelight and made him into a public figure beginning in the late 1980s.

Perhaps a quick review of what was known in the West of Shafarevich’s non-mathematical activities would help to put the book under review in context. It was known that in the 1960s or 1970s Shafarevich had joined a Human Rights Committee and became involved in human right activism in the Soviet Union. Thus, Shafarevich was now a “dissident” and was punished in his own country in many ways.

There were not many details, however, about Shafarevich’s thought, with the exception, perhaps, of a lecture that he delivered when presented with the Heinemann Award of the Göttingen Academy of Sciences. The lecture was published in German in 1973 and translated into English in 1981: “On Certain Tendencies in the Development of Mathematics” (*The Mathematical Intelligencer*, Vol. **3**, Number 4, 1981, 182-184). A first reading of this article would quickly classify it as the usual Platonism in mathematics. But there were some clues on Shafarevich’s ideas, for example in the concluding statement that “mathematics can … serve as model for the solution of the fundamental problem of our time: To reveal a supreme religious aim and purpose for mankind’s cultural activity.”

There were no more than the usual acknowledgements or dismissals of this form of Neo-Platonism, perhaps recognizing its exceptional nature since it came from a scientist in the Soviet Union. The next incident was the infamous “*Russophobia* affair”. In 1988, Shafarevich’s manuscript *Russophobia* was published and became known in Russia and in the West. Many statements in this book would change the public image of Shafarevich from a human rights activist and dissident to that of an extremist on the right and an anti-Semite.

This brings us to the book under review. To begin with, as it is clear from the title of the book, it contains almost nothing about the mathematical accomplishments of Shafarevich. The book is almost completely devoted to the political and religious writings and public activities of I. R. Shafarevich, from his early years (Shafarevich was born in 1923) and traumatic experiences before the Second World War with a few pages with some reminiscences on his formation and then next to nothing about his mathematical activity. Then comes a short chapter devoted to his human rights activism in the 1960s, including his association with Sakharov and Solzhenitsyn.

The rest of the book, basically from chapter five to the end, is completely devoted to: 1) Documenting, discussing and explaining Shafarevich’s political, religious and philosophical writings, and 2) Reviewing some of the public activities of Shafarevich in the rapidly changing scene in Russia from the 1990s to this century. There are, of course, many controversial issues involved.

I don’t know most of the original sources quoted by the author, hence am not able to pass judgment on the author’s scholarship. Assuming it is solid, the book should enable the interested reader to form her or his own opinion on the non-mathematical thoughts and actions of I. R. Shafarevich.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.

See the table of contents in pdf format.

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