Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

This Week's Finds in Mathematical Physics (Week 220)

  1. Oct 12, 2006 #1
    Also available as http://math.ucr.edu/home/baez/week220.html

    August 31, 2005
    This Week's Finds in Mathematical Physics - Week 220
    John Baez

    Work on quantum gravity has seemed stagnant and stuck for the
    last couple of years, which is why I've been turning more
    towards pure math.

    Over in string theory they're contemplating a vast "landscape" of
    possible universes, each with their own laws of physics - one or
    more of which might be ours. Each one is supposed to correspond
    to a different "vacuum" or "background" for the marvelous unifying
    M-theory that we don't completely understand yet. They can't
    choose the right vacuum except by the good old method of fitting
    the experimental data. But these days, this time-honored method
    gets a lot less airplay than the "anthropic principle":

    1) Leonard Susskind, The anthropic landscape of string theory,
    available as hep-th/0302219.

    Perhaps this is because it's more grandiose to imagine choosing
    one theory out of a multitude by discovering that it's among the
    few that supports intelligent life, than by noticing that it
    correctly predicts experimental results. Or, perhaps it's because
    nobody really knows how to get string theory to predict experimental
    results! Even after you chose a vacuum, you'd need to see how
    supersymmetry gets broken, and this remain quite obscure.

    There's still tons of beautiful math coming out of string theory,
    mind you: right now I'm just talking about physics.

    What about loop quantum gravity? This line of research has always
    been less ambitious than string theory. Instead of finding the
    correct theory of everything, its goal has merely been to find
    *any* theory that combines gravity and quantum mechanics in a
    background-free way. But, it has major problems of its own:
    nobody knows how it can successfully mimic general relativity
    at large length scales, as it must to be realistic! Old-fashioned
    perturbative quantum gravity failed on this score because it
    wasn't renormalizable. Loop quantum gravity may get around this
    somehow... but it's about time to see exactly how.

    Loop quantum gravity follows two main approaches: the so-called
    "Hamiltonian" or "spin network" approach, which focuses on the
    geometry of space at a given time, and the so-called "Lagrangian"
    or "spin foam" approach, which focuses on the geometry of
    spacetime.

    In the last couple of years, the most interesting new work in the
    Hamiltonian approach has focussed on problems with extra symmetry,
    like black holes and the big bang. Here's a nontechnical
    introduction:

    2) Abhay Ashtekar, Gravity and the quantum, available as
    gr-qc/0410054.

    and here's some new work that treats the information loss
    puzzle:

    3) Abhay Ashtekar and Martin Bojowald, Black hole evaporation:
    a paradigm, Class. Quant. Grav. 22 (2005) 3349-3362. Also
    available as gr-qc/0504029.

    However, by focusing on solutions with extra symmetry, one puts
    off facing the hardest aspects of renormalization, or whatever
    its equivalent might be in loop quantum gravity.

    The other approach - the spin foam approach - got stalled when
    the most popular model seemed to give spacetimes made mostly of
    squashed-flat "degenerate 4-simplexes". Various papers have
    found an effect like this: see "week198" for more details. So,
    there's definitely a real phenomenon going on here. However,
    its physical significance remains a bit obscure. The devil is
    in the details.

    In particular, even though the *amplitude* for a single large
    4-simplex in the Barrett-Crane model is dominated by degenerate
    geometries, certain *second derivatives* of the amplitude might not -
    and this may be what really matters. Carlo Rovelli has recently
    come out with a paper on this:

    4) Carlo Rovelli, Graviton propagator from background-independent
    quantum gravity, available as gr-qc/0508124.

    If the idea holds up, I'll be pretty excited. If not, I'll be
    bummed. But luckily, I've already gone through the withdrawal
    pains of switching my focus away from quantum gravity. When you
    do theoretical physics, sometimes you feel the high of discovering
    hidden truths about the physical universe. Sometimes you feel the
    agony of suspecting that those "hidden truths" were probably just
    a bunch of baloney... or, realizing that you may never know.
    Ultimately nature has the last word.

    Math is (at least for me) a less nerve-racking pursuit, since
    the truths we find can be confirmed simply by discussing them:
    we don't need to wait for experiment. Math is just as grand as
    physics, or more so. But it's more wispy and ethereal, since it's
    about pure pattern in general - not the particular magic patterns
    that became the world we see. So, the stakes are lower, but the
    odds are higher.

    Speaking of math, I really want to talk about the Streetfest - the
    conference in honor of Ross Street's 60th birthday. It was a real
    blast: over sixty talks in two weeks in two cities, Sydney and
    Canberra. However, I accidentally left my notes from those talks
    at home before zipping off to Calgary for a summer school on
    homotopy theory:

    5) Topics in Homotopy Theory, graduate summer school at the
    Pacific Institute of Mathematics run by Kristine Bauer and Laura
    Scull. Recommended reading material available at
    http://www.pims.math.ca/science/2005/05homotopy/reading.html

    So, I'll say a bit about what I learned at this school.

    Dan Dugger spoke about motivic homotopy theory, which was
    *great*, because I've been trying to understand stuff
    from number theory and algebraic geometry like the Weil
    conjectures, etale cohomology, motives, and Voevodsky's proof
    of the Milnor conjecture... and thanks to his wonderfully
    pedagogical lectures, it's all starting to make some sense!

    I hope to talk about this someday, but not now.

    Alejandro Adem spoke about orbifolds and group cohomology.
    Purely personally, the most exciting thing here was seeing
    that orbifolds can also be seen as certain kinds of topological
    groupoids, or stacks, or topoi... so that various versions of
    "categorified topology" are actually different faces of the
    same thing!

    I may talk about this someday, too, but not now.

    I spoke about higher gauge theory and its relation to Eilenberg-
    Mac Lane spaces. I may talk about that too someday, but not now.

    Dev Sinha spoke about operads, and besides explaining the basics,
    he said a couple of things that really blew me away. So, I want
    to talk about this now.

    For one, the homology of the little k-cubes operad is a graded
    version of the Poisson operad! For two, the little 2-cubes
    operad acts on the space of thickened long knots!

    But for this to thrill you like it thrills me, I'd better say a
    word about operads - and especially little k-cubes operads.

    Operads, and especially the little k-cubes operads, were
    invented by Peter May in the early 1970s to formalize the
    algebraic structures lurking in "infinite loop spaces". In
    "week149" I explained what infinite loop spaces are, and how
    they give generalized cohomology theories, but let's not get
    bogged down in this motivation now, since operads are actually
    quite simple.

    In its simplest form, an operad is a gizmo that has for each
    n = 0,1,2,... a set O(n) whose elements are thought of as n-ary
    operations - operations with n inputs. It's good to draw such
    operations as black boxes with n input wires and one output:

    \ | /
    \ | /
    \ | /
    -----
    | f |
    -----
    |
    |

    For starters these operations are purely abstract things that
    don't actually operate on anything. Only when we consider a
    "representation" or "action" of an operad do they get incarnated
    as actual n-ary operations on some set. The point of operads is
    to study their actions.

    But, for completeness, let me sketch the definition of an operad.
    An operad tells us how to compose its operations, like this:

    \ / \ | / |
    \ / \ | / |
    ----- ----- -----
    | b | | c | | d |
    ----- ----- -----
    \ | /
    \ | /
    \ | /
    \ | /
    \ | /
    \ | /
    \ | /
    -----
    | a |
    -----
    |
    |

    Here we are composing a with b,c, and d to get an operation with 6
    inputs called a o (b,c,d).

    An operad needs to have a unary operation serving as the identity
    for composition. It also needs to satisfy an "associative law"
    that makes a composite of composites like this well-defined:


    \ / | \ | / \ /
    \ / | \ | / \ /
    --- --- --- ---
    | | | | | | | |
    --- --- --- ---
    \ | / /
    \ | / /
    \ | / /
    ----- ----- -----
    | | | | | |
    ----- ----- -----
    \ | /
    \ | /
    \ | /
    \ | /
    \ | /
    \ | /
    \ | /
    -----
    | |
    -----
    |
    |

    (This picture has a 0-ary operation in it, just to emphasize
    that this is allowed.)

    That's the complete definition of a "planar operad". In a
    full-fledged operad we can do more: we can permute the inputs
    of any operation and get a new operation:

    \ / /
    / /
    / \ /
    / /
    / / \
    \ | /
    -----
    | |
    -----
    |
    |

    This gives actions of the permutation groups on the sets O(n).
    We also demand that these actions be compatible with composition,
    in a way that's supposed to be obvious from the pictures. For
    example:


    \ | / | \ / \\\ / / /
    \ | / | \ / \\/ / /
    --- --- --- /\\ / /
    | a | | b | | c | / \\/ /
    --- --- --- / / /
    \ / / / / /\\
    \ / / / | | \\\
    \ / / / | | \\\
    / / --- --- ---
    / \ / = | b | | c | | a |
    / / --- --- ---
    / / \ \ | /
    \ | / \ | /
    ----- -----
    | d | | d |
    ----- -----
    | |
    | |

    and similarly for permuting the inputs of the black boxes on
    top.

    Voila!

    Now, operads make sense in various contexts. So far we've been
    talking about operads that have a *set* O(n) of n-ary operations
    for each n. These have actions on *sets*, where each guy in O(n)
    gets incarnated as a *function* that eats n elements of some set
    and spits out an element of that set.

    But historically, Peter May started by inventing operads that have
    a *topological space* of n-ary operations for each n. These like
    to act on *topological spaces*, with the operations getting
    incarnated as *continuous maps*.

    Most importantly, he invented an operad called the "little k-cubes
    operad". Here O(n) is the space of ways of putting n disjoint
    little k-dimensional cubes in a big one. We don't demand that
    the little cubes are actually cubes: they can be rectangular
    boxes. We do demand that their walls are nicely lined up with
    the walls of the big cube:


    ---------------------
    | |
    | ----- |
    | ----- | | |
    || | | | |
    || | | | | typical
    | ----- | | | 3-ary operation in the
    | ----- | little 2-cubes operad
    | ---------------- |
    | | | |
    | ---------------- |
    | |
    ---------------------

    This is an operation in O(3), where O is the little 2-cubes
    operad. Or, at least it would be if I labelled each of the 3
    little 2-cubes - we need that extra information.

    We compose operations by sticking pictures like this into
    each of the little k-cubes in another picture like this!
    I should draw you an example, but I'm too lazy. So, figure
    it out yourself and check the associative law.

    The reason this example is so important is that we get an action
    of the little k-cubes operad whenever we have a "k-fold loop
    space".

    Starting from a space S equipped with a chosen point *, the
    k-fold loop space Omega^k(S) is the space of all maps from
    a k-sphere into S that send the north pole to the point *. But
    this is also the space of all maps from a k-cube into S sending
    the boundary of the k-cube to the point *.

    So, given n such such maps, we can glom them together using an
    n-ary operation in the little k-cubes operad:

    ---------------------
    |*********************|
    |***********-----*****|
    |*-----****| |****|
    || |***| |****|
    || |***| |****|
    | ----- ***| |****|
    |***********-----*****|
    |***----------------**|
    |**| |*|
    |***----------------**|
    |*********************|
    ---------------------

    where we map all the shaded stuff to the point *. We get
    another map from the k-cube to S sending the boundary to *.
    So:

    ANY k-FOLD LOOP SPACE HAS AN ACTION OF
    THE LITTLE k-CUBES OPERAD!

    But the really cool part is the converse:

    ANY CONNECTED POINTED SPACE WITH AN ACTION OF
    THE LITTLE k-CUBES OPERAD IS
    HOMOTOPY EQUIVALENT TO A k-FOLD LOOP SPACE!

    This is too technical to make a good bumper sticker, so if you
    want people in your neighborhood to get interested in operads,
    I suggest combining both the above slogans into one:

    A k-FOLD LOOP SPACE IS THE SAME AS
    AN ACTION OF THE LITTLE k-CUBES OPERAD!

    Like any good slogan, this leaves out some important fine print,
    but it gets the basic idea across. Modulo some details, being a
    k-fold loop space amounts to having a bunch of operations: one
    for each way of stuffing little k-cubes in a big one!

    By the way:

    Speaking of bumper stickers, I'm in Montreal now, and there's
    a funky hangout on the Boulevard Saint-Laurent called Cafe Pi
    where people play chess - and they sell T-shirts, key rings,
    baseball caps and coffee mugs decorated with the Greek letter pi!
    The T-shirts are great if you're going for a kind of math-nerd/
    punk look; I got one to wow the students in my undergraduate
    courses. I don't usually provide links to commercial websites,
    but I made an exception for Acme Klein Bottles, and I'll make an
    exception for Cafe Pi:

    5) Cafe Pi, http://www.cafepi.ca/

    Unfortunately they don't sell bumper stickers.

    But where were we? Ah yes - the little k-cubes operad.

    The little k-cubes operad sits in the little (k+1)-cubes operad
    in an obvious way. Indeed, it's a "sub-operad". So, we can
    take the limit as k goes to infinity and form the "little
    infinity-cubes operad". Any infinite loop space gets an action
    of this... and that's why Peter May invented operads!

    You can read more about these ideas in May's book:

    6) J. Peter May, The Geometry of Iterated Loop Spaces,
    Lecture Notes in Mathematics 271, Springer, Berlin, 1972.

    or for a more gentle treatment, try this expository article:

    7) J. Peter May, Infinite loop space theory, Bull. Amer. Math.
    Soc. 83 (1977), 456-494.

    But Dev Sinha told us about some subsequent work by Fred
    Cohen, who computed the homology and cohomology of the little
    k-cubes operad.

    For this, we need to think about operads in the world of linear
    algebra. Here we consider operads that have a *vector space* of
    n-ary operations for each n, which get incarnated as *multilinear
    maps* when they act on some *vector space*. These are sometimes
    called "linear operads".

    An example is the operad for Lie algebras. This one is called
    "Lie". Lie(n) is the vector space of n-ary operations that one
    can do whenever one has a Lie algebra. In this example:

    Lie(0) is zero-dimensional, since there are no nullary operations
    (constants) built into the definition of Lie algebra.


    Lie(1) is one-dimensional, since the only unary operations are
    multiples of the identity operation:

    a |-> a


    Lie(2) is one-dimensional, since the only binary operations are
    multiples of the Lie bracket:

    (a,b) |-> [a,b]

    You might think we need a second guy in Lie(2), namely

    (a,b) |-> [b,a]

    but the antisymmetry of the Lie bracket says this is linearly
    dependent on the first one:

    [b,a] = -[a,b]


    Lie(3) is two-dimensional, since the only ternary operations
    are multiples of these two:

    (a,b,c) |-> [[a,b],c]
    (a,b,c) |-> [b,[a,c]]

    You might think we need a third guy in Lie(3), for example

    (a,b,c) |-> [a,[b,c]]

    but the Jacobi identity says this is linearly dependent on the
    first two:

    [a,[b,c]] = [[a,b],c] + [b,[a,c]]

    You may enjoy trying to show that the dimension of Lie(n) is
    (n-1)!, at least for n > 0. There's an incredibly beautiful
    conceptual proof, and probably lots of obnoxious brute-force
    proofs.

    There's a lot more to say about the Lie operad, but right now
    I want to talk about the Poisson operad. A "Poisson algebra"
    is a commutative associative algebra that has a bracket operation
    {a,b} making it into a Lie algebra, with the property that

    {a,bc} = {a,b}c + b{a,c}

    So, bracketing with any element is like taking a derivative: it
    satisfies the product rule.

    For this reason, Poisson algebras arise naturally as algebras of
    observables in classical mechanics - the Poisson bracket of any
    observable A with an observable H called the "Hamiltonian" tells
    you the time derivative of A:

    dA/dt = {H,A}

    This is the beginning of a nice big story.

    But, what's got me excited now is how Poisson algebras show up in
    topology!

    To understand this, we need to note that there's a linear operad
    whose algebras are Poisson algebras. That's not surprising. But,
    we can get a very similar operad in a rather shocking way, as
    follows.

    Take the little k-cubes operad. This has a space O(n) of n-ary
    operations for each n. Now take the homology of these spaces
    O(n), using coefficients in your favorite field, and get vector
    spaces H(O(n)). By functorial abstract nonsense these form a
    linear operad. And this is the operad for Poisson algebras!

    Alas, we actually have to be a bit more careful. The homology of
    O(n) with coefficients in some field is really a *graded* vector
    space over that field. So, H(O(n)) really forms an operad in the
    category of graded vector spaces. And, it's the operad whose
    algebras are graded Poisson algebras with a bracket of degree k-1.

    What's that? Well, it's like a Poisson algebra, but if a is an
    element of degree |a| and b is an element of degree |b|, then:

    ab has degree |a| + |b| (we've got a graded algebra)

    {a,b} has degree |a| + |b| + k - 1 (with a bracket of degree k-1)

    and the usual axioms for a Poisson algebra hold, but sprinkled
    with minus signs according to the usual yoga of graded vector
    spaces.

    So: whenever we have a k-fold loop space, its homology is a graded
    Poisson algebra with a bracket of degree k-1.

    To get an idea of this works, let me sketch how the product and
    the bracket work. Suppose we have an space X with an action of
    the little k-cubes operad:

    The product on homology corresponds to sticking two little cubes
    side by side. Given two points in X, this gives another point in
    X. More generally, given two homology classes a and b in X, we
    get a homology class of degree |a| + |b| in X.

    The bracket comes from taking one little cube and moving it around
    to trace out a sphere surrounding the other little cube. Given
    two points in X, this gives a (k-1)-sphere in X. More generally,
    given a homology class a in X, and a homology class b in X, we
    get a homology class {a,b} of degree |a| + |b| + k - 1.

    The equation

    {a,bc} = {a,b}c + b{a,c}

    then says "moving a around b and c is like moving a around b while
    c stands by, plus moving a around c while b stands by".

    I guess this result can be found here:

    8) Frederick Cohen, Homology of Omega^{n+1}Sigma^{n+1}X and C_{n+1}X,
    n > 0, Bull. Amer. Math. Soc. 79 (1973), 1236-1241.

    9) Frederick Cohen, Tom Lada and J. Peter May, The homology of
    iterated loop spaces, Lecture Notes in Mathematics 533, Springer,
    Berlin, 1976.

    But, I don't think these old papers talk about graded Poisson
    operads! Dev Sinha has a paper where he takes these ideas and
    distills them all into the combinatorics of graphs and trees:

    10) Dev Sinha, A pairing between graphs and trees, available
    as math.QA/0502547.

    However, what I really like is how he gets these graphs and
    trees starting from the homology and cohomology (respectively)
    of the little k-cubes operad! This seems to be lurking in here:

    11) Dev Sinha, Manifold theoretic compactifications of
    configuration spaces, available as math.GT/0306385

    But, I think (and hope) he's writing an expository article that
    will explain everything as simply as he did in his lectures!

    I have a vague feeling that this relation between the little
    k-cubes operad and the Poisson operad is part of a big picture
    involving braids and quantization. Another hint in this direction
    is Deligne's Conjecture, now proved in many ways, which says that
    the operad of singular chains coming from the little 2-disks
    operad acts on the Hochschild cochain complex of any associative
    algebra. Since Hochschild cohomology classifies the ways you
    can deform an associative algebra, this result is related to
    quantization and Poisson algebras. But, I don't get the big
    picture! This might help:

    12) Maxim Kontsevich, Operads and motives in deformation
    quantization, Lett. Math. Phys. 48 (1999) 35-72. Also available
    as math.QA/9904055.

    I'd like to ponder this now! But I'm getting tired, and I still
    need to say how the little 2-cubes operad acts on the space of
    thickened long knots.

    What's a thickened long knot? In k dimensions, it's an embedding
    of a little k-cube in a big one:

    f: [0,1]^k -> [0,1]^k

    subject to the condition that the top and bottom of the little
    cube get mapped to the top and bottom of the big one via the
    identity map. So, you should imagine a thickened long knot as
    a fat square rope going from the ceiling to the floor, all tied
    up in knots.

    There are two ways to "compose" thickened long knots.

    If you're a knot theorist, the obvious way is to stick one on top
    of the other - just like the usual composition of tangles. But if
    you just think of thickened long knots as functions, you can also
    compose them just by composing functions! This amounts to
    stuffing one knot inside another... a little hard to visualize,
    but fun.

    Anyway, it turns out that the whole little 2-cubes operad acts
    on the space of thickened long knots, with the two operations
    I just mentioned corresponding to this:


    ---------------------
    | |
    | |
    | |
    | |
    | | sticking one thickened long
    --------------------- knot on top of another
    | |
    | |
    | |
    | |
    | |
    ---------------------

    and this:

    ---------------------
    | | |
    | | |
    | | |
    | | |
    | | | sticking one thickened long
    | | | knot inside another
    | | |
    | | |
    | | |
    | | |
    | | |
    ---------------------

    This isn't supposed to make obvious sense, but the point is, there
    are lots of binary operations interpolating between these two -
    one for each binary operation in the little 2-cubes operad!

    This gives a new proof that the operation of "sticking one
    thickened long knot on top of another" is commutative up to
    homotopy.

    And, using these ideas, Ryan Budney has managed to figure out a
    lot of information about the homotopy type of the space of long
    knots. Check out these papers:

    13) Ryan Budney, Little cubes and long knots, available as
    math.GT/0309427.

    14) Ryan Budney and Frederick Cohen, On the homology of the space
    of long knots, available as math.GT/0504206.

    15) Ryan Budney, Topology of spaces of knots in dimension 3,
    available as math.GT/0506524.

    The paper by Budney and Cohen combines the two ideas I just
    described - the action of the little 2-cubes operad on thickened
    long knots and its relation to the Poisson operad. Using these,
    they show that the rational homology of the space of thickened
    long knots in 3 dimensions is a free Poisson algebra! They also
    show that the mod-p homology of this space is a free "restricted
    Poisson" algebra.

    -----------------------------------------------------------------------
    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at

    http://math.ucr.edu/home/baez/

    For a table of contents of all the issues of This Week's Finds, try

    http://math.ucr.edu/home/baez/twf.html

    A simple jumping-off point to the old issues is available at

    http://math.ucr.edu/home/baez/twfshort.html

    If you just want the latest issue, go to

    http://math.ucr.edu/home/baez/this.week.html
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted