Mathematicians: An Outer View of the Inner World

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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10 Responses to Mathematicians: An Outer View of the Inner World

  1. theoreticalminimum says:

    Nice :-)

    I actually came across Cook’s ‘Mathematician Gallery’ some time ago, when I was looking for a photo of Don Zagier. May I point out that that she published another photo book Faces of Science, which is worth having a look into. By the way, all the pictures figuring in the galleries can be viewed in higher resolution if you right-click on any one of them, and select “View Image”.

    It’s quite nice to see young brilliant mathematicians like Tao and Mirzakhani in the album :-) (following on your post about matters ICM, may I say that I carry the hope that the latter might become the first female Fields medalist). My favourite picture is that of my maître, J.-P. Serre :-)

    I would finally suggest The Unravelers as a great addition to Cook’s book (but it’s exclusively about mathematicians who have spent time at the Institut des Hautes Études Scientifiques).

  2. Anon says:

    Thanks for the post. The comments by Deligne were interesting.

  3. MathPhys says:

    Yes, that’s a very nice comment by Deligne.

    PS I didn’t know he became a Viscount.

  4. Pingback: A semana nos arXivs… « Ars Physica

  5. Deane says:

    I first noticed Cook’s photos on the front cover of a Princeton University Press mailing. It had photos of authors of books listed in the mailing, but they were the nicest photos I had ever seen of mathematicians. I even emailed PUP asking if there was any way I could get a poster version of the cover or the individual photos. I am still disappointed that none are available. But I did pre-order the book (Amazon had it at a very reasonable price). I agree that the photos and essays are both wonderful.

  6. Chris W. says:

    Loosely related: The Oded Schramm Memorial Conference was held at Microsoft Research on August 30-31.

    One of the talks given was this:

    Random triangulations as dynamical variables in quantum mechanical models
    Michael Freedman
    Abstract: I’ll discuss a simple calculation which Oded showed me how to do and its implications for taking quantum mechanical models off lattice.

  7. Chris Oakley says:

    Happy birthday, Peter!

    I am actually in NY now working for the client. As we are not far from the WTC site I was a bit worried that Al Qaeda might be back for old time’s sake. But then I realised they would not use our calendar. September 11, 2001 was 23rd Jumada all-thani 1422 (give or take a day), the anniversary of which would have been on June 16.

  8. Peter Woit says:

    Thanks Chris, both for the birthday greetings and for the obscure calendar reassurance…

  9. Chris Oakley says:

    A pleasure, and I hope that it is a better day than the one eight years ago. FWIW the dates of 23 Jumada al-thani, according to the Saudi calendar are

    1422 11-Sep-2001
    1423 01-Sep-2002
    1424 21-Aug-2003
    1425 09-Aug-2004
    1426 29-Jul-2005
    1427 18-Jul-2006
    1428 08-Jul-2007
    1429 27-Jun-2008
    1430 16-Jun-2009
    1431 06-Jun-2010
    1432 26-May-2011
    1433 14-May-2012
    1434 03-May-2013
    1435 23-Apr-2014
    1436 12-Apr-2015
    1437 01-Apr-2016

  10. YBM says:

    About mathematicians and would-be mathematicians, you could be amused to know that the Bogdanov brothers are now claiming that Alain Connes is stealing their key ideas without quoting them.