It’s hard to overstate the importance of the work of Oscar Zariski. He learned algebraic geometry in early twentieth century Italy, picking up the passion for geometric intuition that characterized the Italian school but realizing soon that the subject needed surer foundations. He found these foundations in abstract algebra and, together with André Weil, remodeled the subject using those tools, thus preparing the way for the revolution that came with the work of Serre and (most of all) Grothendieck. He went well beyond foundations, of course, making important contributions to the geometry of algebraic surfaces, to the study of singularities and their resolution, and much else. Just the list of his PhD students, which includes such names as David Mumford, Heisuke Hironaka, Michael Artin, Shreeram Abhyankar, Robin Hartshorne, Steven Kleiman, and Pierre Samuel, would suffice to establish Zariski’s historical importance to the subject.

That was Zariski’s real life, in his own mind: “Geometry is the real life,” he is quoted as saying. This book, then, is about his “unreal” life, about the man who was born Ascher Zaritsky in Russia and renamed Oscar Zariski in Italy, the man who eventually became a towering mathematical figure in the United States.

*The Unreal Life* was originally published in 1991, and is here reprinted unchanged. Because the Zariski archives at Harvard were closed until 2001, Carol Parikh based her work mostly on interviews. Interviews with Zariski were conducted by Ann Kostant with the help of Mumford and Hironaka, who seem to have encouraged Parikh to turn them into this book. Parikh also talked to Zariski’s family and to various mathematicians who interacted with him. Zariski’s wife Yole provided many letters and photographs.

One may get a sense of the style and focus of this biography by comparing the section about Zariski’s time in São Paulo to the corresponding section in André Weil’s *Apprenticeship of a Mathematician*. Both men were there at the same time, so that we have parallel accounts of the period. Both are short, but the account in Parikh’s book is mainly about Zariski’s conversations with Weil. We get no feeling at all about what the city was like, what the students were like, or what he thought about the mathematicians he met there. (Presumably, he was not impressed!) Weil’s account, by including the occasional bit of detail, gives a richer picture of the place. The difference, I suspect, is partly due to lack of sources: since Zariski and Yole were there together, there are no letters to her about what it was like for him, and there seem also to have been few non-mathematical memories. Perhaps also the timing was wrong: it was while he was in São Paulo that Zariski found out what had happened to his family during the Holocaust.

While the year in São Paulo is the clearest example, one comes to realize that it’s the same way throughout. Except in the chapters dealing with Zariski’s childhood, we hear mostly about Zariski and other mathematicians, about teaching and interacting with PhD students, and about Zariski’s mathematical ideas, dreams, and ambitions. There is a little bit about the social dynamics of some of the mathematics departments in which Zariski worked, but very little about family and friends, cultural pursuits, joys and pains. One doesn’t get any idea of what the old Harvard mathematics department at 2 Divinity Ave looked like, and the same is true in general: there is very little sense of place, even when such a sense would have been important. It is definitely Zariski the mathematician who is in focus here.

Zariski is described as an inspiring lecturer, but we never get a good sense of what his lectures were like. There is a long quote from Mumford attempting to describe them, but I’m afraid it didn’t make much sense to me. Clearly he did something right, however, since several of his students seem to have arrived at Harvard intending to study a different subject. Taking a course with Zariski changed their plans.

He seems to have been an exceptionally good mentor and talent scout, having noticed very early, for example, that there was something special about Abhyankar and Hironaka. But he also could be formidable and terrifying: when he tried to have “informal” conversations with students, it often felt to them like an examination.

Those who like to tell “stories about mathematicians” will find many new ones here. I enjoyed the account of how Zariski set out to write a book on commutative algebra and had to be “rescured” by Pierre Samuel after becoming “overwhelmed by the proliferation of manuscript pages.” There is also a fantastic Emmy Noether story, told by Zariski in an interview. Noether was lecturing on ideal theory and algebraic number theory at the Institute for Advanced Studies in Princeton. Zariski decided to attend despite the fact that he didn’t really understand everything, because of Noether’s enthusiasm for her subject: “Just watching her was fun, and of course, I felt that here is a person who gets enthusiastic about algebra, so there is probably a good deal to get enthusiastic about.” And then there’s this: “Once, for example, when she was lecturing, her slip came down. She bent down, pulled off the slip, threw it into the corridor, and kept on lecturing.” Collectors of mathematical apocrypha will enjoy this book.

Parikh doesn’t include my favorite Zariski story, however, which is told by James Milne in his own Mathematical Apocrypha web page:

In the early sixties, Grothendieck visited Harvard while Zariski was still a faculty member. Once, while Zariski was lecturing in a seminar, Grothendieck kept asking him why he didn’t prove his result for all schemes, not just varieties, but Zariski simply responded that it didn’t work. Eventually, Grothendieck could stand it no longer and went to the blackboard and began writing down a proof for schemes. While he did so, Zariski wrote down a counter-example. When Grothendieck realized he was wrong, Zariski said (in his heavily accented Russo-Italian English) “In my time, I have had to learn many languages.”

This certainly fits the picture of Zariski that Parikh draws. His interaction with the new methods introduced by Serre and Grothendieck was uneasy: he understood the ideas and their power, but preferred not to use them himself. He did not, however, seem to resent it if his students used the new methods.

Short and easy to read, *The Unreal Life* is both pleasant and useful. There are many photographs, especially of Zariski himself, printed adequately if not well. There is an index and a list of Zariski’s publications.

The book includes a preface by David Mumford, directed to non-mathematicians, which tries to explain how algebra and geometry are connected in order to give some hint of what Zariski’s work was about. In Appendix A one finds several accounts of Zariski’s work by Mumford and others. Neither the preface nor the appendix work very well, alas. The preface attempts the impossible and fails gracefully. The appendix is well intentioned, but getting these ideas across to anyone but a (well-trained) algebraic geometer requires more detail and more expository sugar.

One must regret, as well, that this is a reprint and not a new edition. Eight years after 2001, one would hope that consulting the archives would yield further insights.

Overall, however, Parikh has given us a fascinating glimpse of an incredibly talented and interesting man. Her book will be a welcome addition to any library’s shelf of mathematical biographies.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.