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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