Mathematics students and researchers often react to Alexandre Grothendieck’s legendary fame in the world of mathematics by asking just what the man did to earn him so brilliant a reputation. But as legitimate as it is, the question is difficult to answer, because of the particularly abstruse nature of his mathematics and the wealth of notions he introduced, some of which have become so natural and so familiar to mathematicians that it is easy to forget that there was ever an actual individual who first brought them out into the light, whereas others are so abstract that even experts may take years to grasp them. It is not merely a matter of stating some powerful, striking result that he proved, although there are many of these. But more deeply, Grothendieck’s work is based on a whole system of recasting old ideas in new ways, and it transformed the entire area of algebraic geometry essentially beyond recognition.

This book attempts to provide a reasonable explanation of what made Grothendieck the mathematician that he was. Thirteen articles written by people who knew him personally—some who even studied or collaborated with him over a period of many years—portray Grothendieck at work, explaining the nature of his thought through descriptions of his discoveries and contributions to various subjects, and with impressions, memories, anecdotes, and some biographical elements. Seeing him through the eyes of those who knew him well, the reader will come away with a better understanding of what made Grothendieck unique.