A friend recently loaned me a wonderful book, the recently published Mathematicians: An Outer View of the Inner World, which consists mainly of photographs of mathematicians by Mariana Cook, paired with a page of comments from the mathematician being photographed. For more of the photos, see Mariana Cook’s web-site. The comments typically deal with the story of what led the person into mathematics, or a summary of their career, or some general thoughts on mathematics and the pleasures of studying it.
Many of these mini-essays are well-worth reading. The Viscount Deligne describes working with Grothendieck and contrasts this to some of his later experience:
When I was in Paris as a student, I would go to Grothendieck’s seminar at IHES and Jean-Pierre Serre’s seminar at the Collège de France. To understand what was being done in each seminar would fill my week. I learned a lot doing so. Grothendieck asked me to write up some of the seminars and gave me his notes. He was extremely generous with his ideas. One could not be lazy or he would reject you. But if you were really interested and doing things he liked, then he helped you a lot. I enjoyed the atmosphere around him very much. He had the main ideas and the aim was to prove theories and understand a sector of mathematics. We did not care much about priority because Grothendieck had the ideas we were working on and priority would have meant nothing. I later met other areas of mathematics where people were worried about doing something first and were hiding what they were doing form one another. I didn’t like it. There are all kinds of mathematicians, even competitive ones.
Michele Vergne has an intriguing comment about the “quantization commutes with reduction” question, which is a fundamental issue for how symmetries work in quantum physics. When you have a gauge symmetry, do you get the same thing if you first eliminate the gauge variables (go to the symplectic reduction) and then quantize, or if you quantize and then take the gauge-invariant subspace? This turns out to be a remarkably interesting mathematical question. Perhaps the best way to think about its physical significance is to take it as a criterion for any viable notion of exactly what “quantization” is, and how it is supposed to interact with the notion of symmetry.
Today I can see a dim light on a problem that has been on my mind for a long time. This is the assertion: quantization commutes with reduction. It was a beautiful conjecture of Guillemin-Sternberg, which was clearly true, but revealed itself hard to prove in general. I was able to prove an easy case. A much more difficult case was then proved by another mathematician ten years ago, using surgery. For me, this method via cuts is ugly. I would have liked to prove this conjecture with my own methods. Long after the full proof was found, I kept reorganizing my own arguments in all possible ways. If I repeated them over and over, the difficulties were bound to disappear. But they did not. These ceaseless failed attempts left a scar. I do still hope to discover where exactly the difficulty was, and today I feel I know the small hole where the difficulty was hiding. I think it can be grasped easily. Then, maybe, I will be able to formulate and prove the theorem in a much more general way. True, for thiat I need someone else’s idea, but just recently, I used a brilliant idea of one of my students to explain a very similar phenomenon. I believe it can also be used to understand this case. Anyway, I will try. Tomorrow.
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