A text of Grothendieck

Starting in the autumn of 1994 I spent a year at the Institut des Hautes Etudes Scientifiques (IHES) in Bures sur Yvette near Paris, which for me at that time was an institution which seemed almost mythical. While there I discovered a bound volume of more than 1000 typewritten pages in the library. I do not know why I opened it – perhaps in those days I was tempted to open any thick book in a good library. In any case, once I had opened it I was soon captured by the text. I did not read it all, but I read most of it. The text was ‘Récoltes et Semailles’ by Alexandre Grothendieck. (This is not exactly the text occurring in the title of this post.) In English the title is ‘Reaping and Sowing’. From time to time in what follows I give English versions of text where the original is in French. These are my spontaneous translations, and I have tried to convey the essential meaning as I understand it rather than to be absolutely literal.

Recently, during a conversation with David Klawonn, we got talking about Grothendieck and memories of my time at IHES naturally came back to me. David asked me if I knew why Grothendieck changed subject from functional analysis to algebraic geometry. I did not know and I was surprised that I had never asked myself this before. I was led to do a little research in the Internet and found some things contributing to an answer although I still feel that there is a lot I do not know. While I was doing this I came across another text with the title ‘Promenade à travers une oeuvre ou L’enfant et la Mère’ (Stroll through a work or The child and the Mother). I started reading and was rapidly absorbed by the prose, just as I had been by ‘Récoltes et Semailles’ so many years before. I am not sure of the relation
between the two texts – the one I am talking about here may even be a part of (a version of) the other. A large selection of writings by Grothendieck can be found at http://www.grothendieckcircle.org/. In this text Grothendieck describes his career as a mathematician, but with a wealth of digressions. Among these I found many fascinating, both due to the content and due to the wonderful use of language. I will only discuss a couple of them here.

At one point Grothendieck describes his passage from analysis to geometry in 1955. The reader should be warned that Grothendieck’s definitions of ‘analysis’ and ‘geometry’ may not be those commonly used. Among other things he says that it was as if he was leaving arid and harsh steppes to suddenly find himself in a kind of promised land. He wants to describe the experience by the German word ‘überwältigend’ and has difficulty finding a satisfactory French translation. I personally cannot imagine experiencing a transition in that direction in this way. Nevertheless at this point, and in many other cases, I found it difficult to avoid being carried along by Grothendieck’s language even when not naturally inclined to share his point of view on the subject in question. And in a way I wanted to be carried along. Grothendieck’s judgement of himself and his own work is a curious mixture of modesty and pride. On the one hand he emphasizes his lack of talent while at the same time suggesting that he finds his own work tremendously important. His idea of his own greatest strength seems to be that he has looked carefully at things which others regarded as too unimportant to merit their attention.

I am not qualified to give an assessment of the importance of Grothendieck’s work – my own interests in mathematics are too far away from his. I do, however, have the impression that he overrates the importance of his own discoveries for mathematics as a whole. At least I can appreciate his idea of ‘ordinary’ cohomology as a derived functor. This is a case where simplicity is impressive. Grothendieck’s ideas on the central importance of his work, in particular for geometry and topology, are now vulnerable to comparisons with the work of Hamilton and Perelman leading to the proof of the Poincaré conjecture, which are of a very different kind. In this context I could not help thinking of certain parallels between Grothendieck and Perelman. I now list some. Both of them
1. are exceptionally gifted mathematicians.
2. have had difficult relations with the mathematics community and have questioned the morality of that community.
3. have refused prestigious prizes (Grothendieck the Crafoord prize, Perelman the Fields medal).
4. isolated themselves from the mathematical community at some point.
5. are uncompromising about standing up for what they believe in.
6. have a considerable charisma.
In Grothendieck’s case I could only experience the charisma through his writings; in Perelman’s case I was fortunate enough to hear the seminar in Berlin where he first talked about ideas related to his now famous work on the Poincaré conjecture. Of course the points on the above list are not all independent. Perelman has not been isolated so long and we may yet be fortunate enough to see him present us with a new breakthrough.

It is important to mention that in his writings Grothendieck sometimes seems to leave the region which most of us would call sanity. In saying this I am thinking less of the text I am discussing here than of ‘Récoltes et Semailles’ and others. When reading ‘Récoltes et Semailles’ I could not help thinking of Rousseau’s ‘Confessions’. It has a similar attractiveness for me in its earlier parts and seems to go off the rails (as seen from a frame of reference of ordinary everyday life) towards the end.

The fact that I am writing about this subject here is certainly not just due to the mathematical content of the text in question. It affected me on an emotional level and shook the limits within which my thinking usually takes place. This is not an accident. Grothendieck writes: If in ‘Récoltes et Semailles’ I am addressing someone other than myself it is not a ‘public’. I am addressing you who are reading me as a person and as a person who is alone.

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