A Mad Day’s Work: from Grothendieck to Connes and Kontsevich the Evolution of Concepts of Space and Symmetry

To add to the chorus of praise by referring to my own experience would be of little interest, but I am in no way forgetting the facilities for work provided by the Institut des Hautes Études Scientifiques (IHES) for so many years, particularly the constantly renewed opportunities for meetings and exchanges. While there have been some difficult times, there is no point in dwelling on them. One of the great virtues of the institute was that it erected no barriers between mathematics and theoretical physics. There has always been a great deal of interpenetration of these two areas of interest, which has only increased over time. From the very beginning Louis Michel was one of the bridges due to his devotion to group theory. At present, when the scientific outlook has changed so greatly over the past forty years, the fusion seems natural and no one wonders whether Connes or Kontsevich are physicists or mathematicians. I moved between the two fields for a long time when to do so was to run counter to the current trends, and I welcome the present synthesis. Alexander Grothendieck dominated the first ten years of the institute, and I hope no one will forget that. I knew him well during the 50s and 60s, especially through Bourbaki, but we were never together at the institute—he left it in September 1970 and I arrived in July 1971. Grothendieck did not derive his inspiration from physics and its mathematical problems. Not that his mind was incapable of grasping this area—he had thought about it secretly before 1967—but the moral principles that he adhered to relegate physics to the outer darkness, especially after Hiroshima. It is surprising that some of Grothendieck’s most fertile ideas regarding the nature of space and symmetries have become naturally wed to the new directions in modern physics. It is this unexpected marriage—and its occasionally comical aspects—that I would like to talk about here. “A mad day’s work”, as you know, is the subtitle given to The Marriage of Figaro by Beaumarchais. From a certain distance there is less cause for astonishment; the concepts of space and symmetry are so fundamental