Mathematicians: An Outer View of the Inner World (Princeton University Press, $35) presents 92 photographic portraits by Mariana Cook, with an introduction by Robert Clifford Gunning (who also sits for one of the portraits) and an afterword by Brandon Fradd (who instigated the project). The book’s concept is simple and elegant: On each right-hand page is a portrait of a mathematician, printed in rich black and white, and on the facing left-hand page is an autobiographical essay by the subject of the photograph. The portraits are carefully composed and lighted, with a certain formality to them. A few of the subjects are captured in situ, with equation-covered blackboards behind them; most others are posed before a plain black dropcloth; quite a few (for reasons unexplained) sit or stand in shrubbery. With just two exceptions (Henri Cartan and Noam Elkies) all of the subjects look straight into the lens.
From this outer view of mathematicians, can you really learn anything about their inner world? The faces are deeply engaging on a human level, but it would be a stretch to suppose that body language reveals much about the capacity for abstract reasoning or theorem-proving. The essays carry that burden. Most of them tell the how-I-became-a-mathematician story; a few talk about what it feels like to do mathematics at the highest level; other authors undertake to explain some particular aspect of their work.
This is a family album for the mathematical community. In some cases the family relations are biological: There are two Fefferman brothers (Charles and Robert), three Browder brothers (William, Felix and Andrew), and two father-and-daughter pairings (Heisuke and Eriko Hironaka and George and Kate Adebolo Okikiolu). Perhaps more important are the academic lines of descent; for example, Michael Atiyah is the “academic father” (the thesis advisor) of Frances Kirwan and Simon Donaldson.
A family album, but not quite the entire family. By and large this is the world of mathematics as seen from Princeton, New Jersey. Fully half the subjects have some connection to Princeton University or to the neighboring Institute for Advanced Study. There are also disciplinary leanings, with an emphasis on number theory and the various modern branches of geometry (algebraic, differential, symplectic). Some more statistics: Among the 92 subjects, 73 work in North America (though only 41 were born here), 22 have won a Fields Medal (the premier prize in mathematics), 15 are women, only three are under age 40 (Terence Tao, Manjul Bhargava, Maryam Mirzakhani).
The photograph above shows Kenneth Ribet of the University of California, Berkeley, who seems to be enacting one of Zeno’s paradoxes: Achilles is overtaken by a speeding tortoise.—Brian Hayes