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This Week's Finds in Mathematical Physics (Week 255)

  1. Aug 13, 2007 #1
    Also available as http://math.ucr.edu/home/baez/week255.html

    August 11, 2007
    This Week's Finds in Mathematical Physics (Week 255)
    John Baez

    I've been roaming around Europe this summer - first Paris, then
    Delphi and Olympia, then Greenwich, then Oslo, and now back to
    Greenwich. I'm dying to tell you about the Abel Symposium in
    Oslo. There were lots of cool talks about topological quantum
    field theory, homotopy theory, and motivic cohomology.

    I especially want to describe Jacob Lurie and Ulrike Tillman's
    talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen's
    talks on string topology, Stephan Stolz's talk on cohomology and
    quantum field theory, and Fabien Morel's talk on A^1-homotopy
    theory. But this stuff is sort of technical, and I usually try
    to start each issue of This Week's Finds with something you don't
    need a PhD to enjoy.

    So, here's a tour of the Paris Observatory:

    1) John Baez, Astronomical Paris,
    http://golem.ph.utexas.edu/category/2007/07/astronomical_paris.html

    Back when England and France were battling to rule the world,
    each had a team of astronomers, physicists and mathematicians
    devoted to precise measurement of latitudes, longitudes, and times.
    The British team was centered at the Royal Observatory here in
    Greenwich. The French team was centered at the Paris Observatory,
    and it featured luminaries such as Cassini, Le Verrier and Laplace.

    In "week175", written during an earlier visit to Greenwich, I
    mentioned a book on this battle:

    2) Dava Sobel, Longitude, Fourth Estate Ltd., London, 1996.

    It's a lot of fun, and I recommend it highly.

    There's a lot more to say, though. The speed of light was first
    measured by Ole Romer at the Paris Observatory in 1676. Later,
    Henri Poincare worked for the French Bureau of Longitude. Among
    other things, he was the scientific secretary for its mission to
    Ecuador.

    To keep track of time precisely all over the world, you need to
    think about the finite speed of light. This may have spurred
    Poincare's work on relativity! Here's a good book that argues
    this case:

    3) Peter Galison, Einstein's Clocks, Poincare's Maps: Empires
    of Time, W. W. Norton, New York, 2003. Reviewed by Robert Wald
    in Physics Today at http://www.physicstoday.org/vol-57/iss-9/p57.html

    I met Galison in Delphi, and it's clear he like to think about
    the impact of practical stuff on math and physics.

    I was in Delphi for a meeting of "Thales and Friends":

    4) Thales and Friends, http://www.thalesandfriends.org

    This is an organization that's trying to bridge the gap between
    mathematics and the humanities. It's led by Apostolos Doxiadis,
    who is famous for this novel:

    5) Apostolos Doxiadis, Uncle Petros and Goldbach's Conjecture,
    Bloomsbury, New York, 2000. Review by Keith Devlin at
    http://www.maa.org/reviews/petros.html

    There's a lot I could say about this meeting, but I just want
    to advertise a forthcoming book by Doxiadis and a computer
    scientist friend of his. It's a comic book - sorry, I mean
    "graphic novel"! - about the history of mathematical logic
    from Russell to Goedel:

    6) Apostolos Doxiadis and Christos Papadimitriou, Logicomix,
    to appear.

    I saw a partially finished draft. I think it does a good job
    of explaining to nonmathematicians what the big deal was with
    mathematical logic around the turn of the last century... and
    how these ideas eventually led to computers. It's also a fun
    story.

    If you're eager for summer reading and can't wait for Logicomix,
    you might try this other novel by Papadimitrou:

    7) Christos Papadimitriou, Turing (a Novel about Computation),
    MIT Press, Boston, 2003.

    It's a history of mathematics from the viewpoint of computer
    science, as told by a computer program named Turing to a
    lovelorn archaeologist. I haven't seen it yet.

    Okay - enough fun stuff. On to the Abel Symposium!

    8) Abel Symposium 2007, at http://abelsymposium.no/2007

    Actually this was a lot of fun too. A bunch of bigshots were
    there, including a bunch who didn't even give talks, like Eric
    Friedlander, Ib Madsen, Jack Morava, and Graeme Segal.

    (My apologies to all the bigshots I didn't list.)

    Speaking of bigshots, Vladimir Voevodsky gave a special surprise
    lecture on symmetric powers of motives. He wowed the audience not
    only with his mathematical powers but also his ability to solve a
    technical problem that had stumped all the previous speakers! The
    blackboards in the lecture hall were controlled electronically,
    by a switch. But, the blackboards only moved a few inches before
    stalling out. So, people had to keep hitting the switch over and
    over. It was really annoying, and it became the subject of running
    jokes. People would ask the speakers: "Can't you talk and press
    buttons at the same time?"

    So, what did Voevodsky do? He lifted the blackboard by hand!
    He laughed and said "Russian solution". But, I think it's a great
    example of how he gets around problems by creative new approaches.

    It really pleased me how many talks mentioned n-categories, and
    even used them to do exciting things. This seems quite new. In
    the old days, bigshots might think about n-categories, but they'd
    be embarrassed to actually mention them, since they had a
    reputation for being "too abstract".

    In fact, Dan Freed alluded to this in his talk on topological
    quantum field theory. He said that every mathematician has
    an "n-category number". Your n-category number is the largest n
    such that you can think about n-categories for a half hour without
    getting a splitting headache.

    When Freed first invented this concept, he felt pretty
    self-satisfied, since his n-category number was 1, while for
    most mathematicians it was 0. But lately, he says, other
    people's n-category numbers have been increasing, while his has
    stayed the same.

    He said this makes him suspicious. In light of the scandals
    plaguing the Tour de France and American baseball, he suspects
    mathematicians are taking "category-enhancing substances"!

    Freed shouldn't feel bad: he was among the first to introduce
    n-categories in the subject of topological quantum field theory!
    He gave a nice talk on this, clear and unpretentious, leading
    up to a conjecture for the 3-vector space that Chern-Simons
    theory assigns to a point.

    That would make a great followup to these papers on the 2-vector
    space that Chern-Simons theory assigns to a circle:

    9) Daniel S. Freed, The Verlinde algebra is twisted equivariant
    K-theory, available as arXiv:math/0101038.

    Daniel S. Freed, Twisted K-theory and loop groups, available
    as arXiv:math/0206237.

    Daniel S. Freed, Michael J. Hopkins and Constantin Teleman,
    Loop groups and twisted K-theory II, available as
    arXiv:math/0511232.

    Daniel S. Freed, Michael J. Hopkins and Constantin Teleman,
    Twisted K-theory and loop group representations, available as
    arXiv:math/0312155.

    In a similar vein, Jacob Lurie talked about his work with Mike
    Hopkins in which they proved a version of the "Baez-Dolan cobordism
    hypothesis" in dimensions 1 and 2. I'm calling it this because
    that's what Lurie called it in his title, and it makes me feel good.

    You can read about this hypothesis here:

    10) John Baez and James Dolan, Higher-dimensional algebra and
    topological quantum field theory, J.Math.Phys. 36 (1995) 6073-6105
    Also available as arXiv:q-alg/9503002.

    It was an attempt to completely describe the algebraic structure of
    the n-category nCob, where:

    objects are 0d manifolds,
    1-morphisms are 1d manifolds with boundary,
    2-morphisms are 2d manifolds with corners,
    3-morphisms are 3d manifolds with corners,
     
  2. jcsd
  3. Aug 15, 2007 #2
    On Aug 12, 8:56 pm, b...@math.removethis.ucr.andthis.edu (John Baez)
    wrote:
    >
    > Also available ashttp://math.ucr.edu/home/baez/week255.html
    >
    > August 11, 2007
    > This Week's Finds in Mathematical Physics (Week 255)
    > John Baez
    >
    > 17) Stephan Stolz and Peter Teichner, What is an elliptic object?
    > Available at http://math.ucsd.edu/~teichner/papers.html


    That link fails, but http://math.berkeley.edu/~teichner/papers.html
    seems OK


    Cheers

    John R Ramsden
     
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