Most people who are interested in mathematics probably know the story. The Poincaré Conjecture was one of the most famous open problems in mathematics, a question about geometry which conjectured that the analogue of a familiar fact about ordinary spheres was also true for their higher dimensional analogues. Though the claim seemed reasonable, for a long time nobody was able to prove it was true and hence we had to consider the possibility that it was not. It was therefore a big deal when the Russian mathematician Perelman found a proof that the conjecture was indeed true. For his contributions, Perelman was to be awarded the prestigious Fields Medal...but for his own personal reasons Perelman refused to accept the coveted prize. Even after IMU president John Ball visited him in Russia to persuade him to attend the International Congress of Mathematicians in Madrid in 2006 where the prize would be awarded, Perelman declined the honor.

This work consists primarily of imagined dialogues between Perelman and Ball (including some which, mysteriously and almost magically, take place in Ball's mind while he sits alone in his hotel) during Ball's brief visit to Saint Petersburg in advance of the ICU. Although the author met with Ball to discuss his idea for writing this book, he claims that Ball refused to discuss the details of his conversations with Perelman. To the extent that those dialogues are merely a product of Zaouati's imagination, this is a work of fiction.

And fiction may be an appropriate forum for raising some of the essentially philosophical questions about mathematics that the book either addresses directly or merely brings up for the reader's consideration. To what extent are mathematicians noble seekers of truth as opposed to pompous glory-hounds? Are medals and cash prizes a sensible way to attract attention and talent to the field or mathematics or are the sources of unacceptable corruption? Should the credit for advances in the field of mathematics be shared among the many researchers who made small contributions along the way or only to the final one? Of course, the book does not answer these questions, but unlike the question of whether the Poincaré Conjecture was correct, they may not actually have objective answers.

Readers who enjoy (or who want to learn about) the history of mathematics will get something from the discussions and anecdotes about mathematicians that are sprinkled throughout this work. Those mentioned include: Alexander Grothendieck, Richard Hamilton, Gang Tian, John Morgan, Shing-Tung Yau, and Olga Ladyzhenskaya. Furthermore, one learns a bit about the history of the Soviet Union and what it was like to grow up as a Jew there.

Since I knew from the start both that Ball would fail in his "mission" and that the conversations were just the author's creations, I did not expect to enjoy the game of cat-and-mouse between the two mathematicians. Yet, I did! It was a bit like watching an episode of Columbo (an observation that the book itself makes when Ball pesters Perelman with one last question). A good example of that is their discussion of Grothendieck in which each one of them uses the lesson of that other famous reclusive mathematician to make a very convincing point supporting their "side" in the argument, only to see the other use him to make an equally convincing counter-argument. It was like watching a volley between tennis pros.

Now that I've said some positive things about the book, let me return to my initial concerns. One of the things that I have learned from my hobby of reviewing "mathematical fiction" is that people seem to love the idea that mathematicians have mental disorders and neuroses. Mathematician characters are so frequently portrayed as paranoid, schizophrenic, demented, absent-minded, anti-social, evil, etc. I am quite skeptical that this stereotype is justified. (The use of a small number of examples like John Nash Jr., Gödel and now Perelman as evidence suggests to me that there is not actually any statistical evidence to back it up.) Moreover, regardless of whether there is any correlation between such mental disorders and mathematical interest/ability, I believe the focus on that aspect in both mathematical fiction and popular mathematics is harmful to the discipline. In particular, I think it discourages young people from studying or pursuing careers in mathematics. That is why it bothers me that Perelman's personality and his "torment" are the focus of so much attention rather than the mathematics.

I am therefore quite pleased that "Perelman's Refusal" explicitly contradicts many of the stereotypes:

Let me be clear: I do not know whether this is a more accurate portrayal of Grigori Perelman than those stereotypes, but I am glad that the author made this choice.

The Author's Note contains a standard disclaimer saying that he is no expert in mathematics. ("My knowledge of mathematics is too sketchy and too far back.") In fact, he gets most of the mathematical discussion in the book right. Strangely, the thing that he gets wrong (repeatedly) is his description of the Poincaré Conjecture itself. Towards the beginning of the book one reads:

That description is wrong because it makes no mention of the requirement that loops in the manifold must be contractible to a point. (An 3-dimensional torus is a compact space without boundaries which is not topologically a sphere.) Ironically, at the end of the book the opposite mistake is made (the contractibility but not the compactness is mentioned). The book keeps claiming that by proving the conjecture Perelman has "discovered the shape of the universe", which I consider to be ridiculous hype. (I suppose the universe could have the topology of a higher dimensional sphere, but at this point in time we have absolutely no way of knowing that.) Finally, the book gives the impression that mathematicians all knew the conjecture was true even before it was proved, but I don't think that is accurate. Individual mathematicians may have their own opinions about open conjectures, but I think we all honestly admit that in the absence of a proof the truth is unknown (and could surprise us). Even if Zaouati's disclaimer leaves him blameless in all these instances since he could not be expected to know better, I can blame the editors at the AMS who should have.

A few more random remarks:

• Zaouati is also the CEO of an asset management company and author of non-fiction books such as Positive Finance: A Toolkit for Responsible Transformation.
• The book was first published in 2017 in French under the title Les Refus de Grigori Perelman and was translated by Rachel Zerner for the 2021 American publication which I read.
• Although the book has "A Novel" as its subtitle, the AMS publication does not feel like a novel. It is the size and shape of their mathematical monographs and printed in a large font on only 130 pages.