What was least satisfying to me in our high school mathematics books was the absence of any serious definition of the notion of length of a curve, of area of a surface, of volume of a solid. I promised myself I would fill this gap when I had the chance.

It was in Montpellier, during his undergraduate days, that he underwent his first real mathematical experience. He was very dissatisfied with the teaching he was receiving. He had been told how to compute the volume of a sphere or a pyramid, but no one had explained the definition of volume. It is an unmistakable sign of a mathematical spirit to want to replace the "how" with a "why". A professor of Grothendieck's assured him that a certain Lebesgue had resolved the last outstanding problems in mathematics, but that his work would be too difficult to teach. Alone, with almost no hints, Grothendieck rediscovered a very general version of the Lebesgue integral. The genesis of this first mathematical piece of work [was] accomplished in total isolation ...

Grothendieck had very few books; rather than learning things by reading, he would try to reconstruct them on his own. And he worked very hard.

A general theory of duality for locally convex spaces had to be worked out: Schwartz and I had started its study for Fréchet spaces and their direct limits, but we had met a series of problems we could not solve. We therefore proposed them to Grothendieck, and the result turned out to exceed our most sanguine expectations. In less than a year, he had solved all our problems by very ingenious new constructions; then, with the techniques he had developed, he started to work on many other questions in functional analysis.

... recalled that after Grothendieck wrote his thesis he asserted that he was no longer interested in topological vector spaces. "He told me, 'There is nothing more to do, the subject is dead'," Malgrange recalled. At that time, students were required to prepare a "second thesis", which did not contain original work but which was intended to demonstrate depth of understanding of another area of mathematics far removed from the thesis topic. Grothendieck's second thesis was on sheaf theory, and this work may have planted the seeds for his interest in algebraic geometry, where he was to do his greatest work. After Grothendieck's thesis defense, which took place in Paris, Malgrange recalled that he, Grothendieck, and Henri Cartan piled into a taxicab to go to lunch at the home of Laurent Schwartz. They took a cab because Malgrange had broken his leg skiing. "In the taxi Cartan explained to Grothendieck some wrong things Grothendieck had said about sheaf theory," Malgrange recalled.

It is no exaggeration to speak of Grothendieck's years 1959-70 at the IHES as a 'Golden Age', during which a whole new school of mathematics flourished under Grothendieck's charismatic leadership. Grothendieck's Séminaire de Géométrie Algébrique Ⓣ established the IHES as a world centre of algebraic geometry, and him as its driving force.

The mere enumeration of Grothendieck's best known contributions is overwhelming: topological tensor products and nuclear spaces, sheaf cohomology as derived functors, schemes, K-theory and Grothendieck-Riemann-Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the 'six operations'), étale cohomology and the cohomological interpretation of L-functions, crystalline cohomology, 'standard conjectures', motives and the 'yoga of weights', tensor categories and motivic Galois groups. It is difficult to imagine that they all sprang from a single mind.

He spoke with great energy at the board but taking care to recall all the necessary material. He was very precise. The presentation was so neat that even I, who knew nothing of the topic, could understand the formal structure. It was going fast but so clearly that I could take notes.

At the time he lived at Bures-sur-Yvette, rue de Moulon, in a little white pavilion, with a ground floor and one storey. His office there was austere, and cold in the winter. He had a portrait of his father in pencil, and also on the table there was the mortuary mask of his mother. Behind his desk he had filing cabinets. When he wanted some document, he would just turn back, and find it in no time. He was well organized.

The Grothendieck I knew at this time was a very impressive person, and when I say this I am not thinking only of mathematics. Shourik, as I called him, was one of the strongest and most charismatic people I have ever met. I think of him as a character straight out of Dostoyevsky. He was also a person of great kindness and generosity. He seemed always to be in good spirits, with great mental equilibrium and also, in his own way, a certain joie de vivre. At the time, he had the capacity to be able to sleep when he wanted to, and for the number of hours he wanted to, in order to take up his work all the better afterward. In fact, his capacity for work was to me something miraculous.

He received the Fields Medal in 1966. In looking back at this period, one marvels at the generosity with which Grothendieck shared his ideas with colleagues and students, the energy he and his collaborators devoted to meticulous redaction, the excitement with which they set out to explore a new land.

Grothendieck built on work of Weil and Zariski and effected fundamental advances in algebraic topology. He introduced the idea of K-theory (the Grothendieck groups and rings). He revolutionised homological algebra in his celebrated "Tohoko paper".

Grothendieck's first lectures - which he describes as "general orientation talks" - were given in Hanoi. But because of intensified bombing of the capital, a high-level decision was made to move everyone to the secret location of the Faculty of Mathematics of Hanoi University. Grothendieck writes: "I then spent a week and a half at Hanoi University in evacuation outside the city (about 100 km from the capital); this time was largely devoted to a more specialized seminar on categories and homological algebra, with thirty to forty listeners, most of whom had followed me from Hanoi after attending the general orientation lectures." It was a remarkable event in the history of mathematics: one of the giants of 20th century mathematics delivering a short course on homological algebra in a remote forest hideout in a desperately poor country that was being "bombed back into the stone age" (U.S. Air Force General Curtis Le May's phrase) by the most powerful military force the world had ever known.

He could be totally absorbed in mathematical ideas. One letter remarks that he is thinking of a new theory of form, and has to remind himself to eat and to sleep. He had a strong interest in detail and small things. He was against 'le snobisme', and so was delighted with a comment of Henry Whitehead: 'It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial.' He wrote that mathematics is not only about doing difficult things, but also providing the framework to make difficult things easy (thus giving new opportunities for difficult tasks!).

The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps ...

The question you raise "how can such a formulation lead to computations" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.

[Often the route to solving problems is] to bring new concepts out of the dark.