Grothendieck and Motives

The latest issue of the Notices of the AMS contains several things very much worth reading. There’s the second part of a wonderful biographical article about Grothendieck written by Allyn Jackson (for some comments about the first part, see an earlier posting).

There’s also an excellent short expository piece by Barry Mazur that explains a bit about one of Grothendieck’s influential and still only partially understood ideas, that of a “motive”. In algebraic geometry the standard ways of defining topological invariants of topological spaces are of limited use, and one wants a much more algebraic notion of such an invariant. This is what a motive is supposed to somehow provide, but to even show that such conjectural motives have the properties one would like requires solving perhaps the biggest open problem in algebraic geometry, the Hodge Conjecture.

Finally there’s a thought-provoking piece called The Elephant in the Internet by Daniel Biss about the effect of the internet on the mathematics literature. It contains some comments about the difference between standards in physics and mathematics, including an analogy of mathematics as classical and physics as popular music. His conclusion that “our current relationship to the Internet has the undeniable effect of degrading the sacrosanct status of the mathematical text” seems to me excessive and it’s a shame that he feels “hesitant to post my papers online; it always feels a little like leaving my infant in a dumpster.” I have some sympathy for his worry that preprint archives and contact with the more journalistic physics literature may make the mathematics literature much less authoritative than it used to be (this was also the concern of a similar article by Jaffe and Quinn published in the AMS Bulletin in 1993). But the lost golden age that Biss yearns for was not so golden. Much of the math literature was written to very high standards of rigor, but often in ways that made such uncompromising demands on the reader that virtually no one who was not already an expert could hope to understand what was being said. The fact that the internet has provided venues for much sloppier, unpolished, but more expository articles also has its very positive aspects.

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5 Responses to Grothendieck and Motives

  1. sol says:

    Just trying to keep pace Thomas Larsson:)

    As physicists know, “accidents: do not appear without a reason. When performing a long and difficult calculation, and then suddenly having thousands of unwanted terms miraculously add up to zero, physicists know that this does not happen without a deeper, underlying reason. Today, physicists know that these “accidents” are an indication that a symmetry is at work. For strings, the symmetry is called conformal symmetry, the symmetry of stretching and deforming the string’s world sheet

    Hyperspace, by Michio Kaku, Page 173

  2. Chris W. says:

    John Moffat posted a preprint in August written along somewhat similar lines, although the tone of his introductory discussion was unmistakably negative — along the lines of “I’m not sure it’s worth it, but let’s take this idea [the multiverse] seriously and see what we can do with it.” It’s worth reading as a review.

  3. Thomas Larsson says:

    Here we go again. This time it’s Dine.

  4. sol says:

    Since demystification is the desire, then what do we realize when math generates these principles that espouse new theories?

    Why 10 Dimensions?

    When strings vibrate in space-time, they are described by a mathematical function called the Ramanujan modular function.26 This term appears in the equation:27

    [1-(D – 2)/24]
    where D is the dimensionality of the space in which the strings vibrate. In order to obey special relativity 9and manifest co-variance), this term must equal 0, which forces D to be 26. This is the origin of the 26 dimensions in the original string theory.

    In the more general Ramanujan modular function, which is used in current superstring theories, the twenty-four is replaced by the number eight, making D equal to 10.28

    In other words, the mathematics require space-time to have 10 dimensions in order for the string theory to be self-consistent, but physicists still don’t know why these particular numbers have been selected.

  5. sol says:

    In La Clef des Songes he ( alexander-grothendieck)explains how the reality of dreams convinced him of God’s existence.

    Do we see some pattern here when we consider Ramanujan’s Modular Functions and the source from which these numbers were pulled? As if from some fifth dimensional realization(abstract realizations)?

    A recent author has suggested that math ability derives from the brain abilities used in social understanding. Think of living in a tribe or small town where “everybody knows everybody”. By growing up in such an environment you know not only everyone else’s name, but their preferences and personal characteristics. You are freely able to think what so-and-so and such-and-such would talk about if they had a conversation. And it is proposed that mathematicians have this same ability, only with the abstract things they think about and discuss, rather than people.

    Maybe Peter, I could then have convoluted the math realm with further ideas here as expressed in the Elephant link.

    Sometimes it appears as mysterious? But really, when you look at how a “Cab number” could have been calculated, one wonder ‘s what nonsense mathematicians could raise, without recognizing the consistancy of this math that comes into existance? 🙂

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