Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen Houghton Mifflin Harcourt, 242 pp., $26.00

Masha Gessen’s Perfect Rigor is a fascinating biography of Grigory (Grisha) Perelman, the fearsomely brilliant and notoriously antisocial Russian mathematician. Perelman proved the Poincaré Conjecture, one of mathematics’ most important and intractable problems, in 2002—almost a century after it was first posed, and just two years after the Clay Mathematics Institute offered a one-million-dollar prize for its solution.

Gessen herself grew up in the former Soviet Union, is roughly Perelman’s age, and has a mathematical background, which facilitated her interviews with many of his classmates, mentors, teachers, and colleagues. Not surprisingly, she did not interview the reclusive mathematician or his mother, with whom he currently lives. But the others give a convincing picture not only of him but also of the strange world of Soviet mathematics, which was divided between the official, rigid mathematical establishment and the informal mathematical counterculture. The former, because of its historical importance to engineering and military projects, was supported by the Party and the government; the latter consisted of scholars who loved mathematics for its own sake and used it as a way to escape the stultifying influence of officious apparatchiks.

Born in 1966 to Jewish parents, Perelman came of age when this distinction was breaking down during the era of glasnost and perestroika. By the time he was ten he began to show a talent for mathematics, and his mother, who had abandoned her own graduate work in the field in order to raise him, enrolled him in an after-school math club coached by Sergei Rukshin, a mathematics undergraduate at Leningrad University. Rukshin was a troubled youth who became obsessed with mathematics and gradually developed a rigorous, distinctive, and very effective method of teaching problem-solving. Over the last twenty years, approximately half of all Russian entrants to the International Mathematical Olympiad have studied with him.

Only nineteen himself when he met Perelman, Rukshin stayed in contact with him from his first after-school math club until, it seems, a relatively recent break. He found that the not yet adolescent Perelman, described by Gessen as “an ugly duckling among ugly ducklings…pudgy and awkward,” was already unusually deliberate and precise in his thinking. Alexander Golovanov, who studied math alongside Perelman, said that Rukshin’s growing commitment to and love for Perelman came to give meaning to his own life. Like many a competitive sports coach, Rukshin hated it when his charges engaged in anything other than his sport. This was an unnecessary restriction in Perelman’s case since from the beginning he seemed uninterested in girls or anything other than mathematics.

When Perelman was fourteen, Rukshin spent the summer tutoring him in English; he accomplished in a few months what generally took…