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This Week's Finds in Mathematical Physics (Week 258)

  1. Nov 27, 2007 #1
    Also available as http://math.ucr.edu/home/baez/week258

    November 25, 2007
    This Week's Finds in Mathematical Physics (Week 258)
    John Baez

    Happy Thanksgiving! Today I'll talk about a conjecture by Deligne
    on Hochschild cohomology and the little 2-cubes operad.

    But first I'll talk about... dust!

    I began "week257" with some chat about about dust in a binary star
    system called the Red Rectangle. So, it was a happy coincidence
    when shortly thereafter, I met an expert on interstellar dust.

    I was giving some talks at James Madison University in Harrisonburg,
    Virginia. They have a lively undergraduate physics and astronomy
    program, and I got a nice tour of some labs - like Brian Utter's
    granular physics lab.

    It turns out nobody knows the equations that describe the flow of
    grainy materials, like sand flowing through an hourglass. It's a
    poorly understood state of matter! Luckily, this is a subject where
    experiments don't cost a million bucks.

    Brian Utter has a nice apparatus consisting of two clear plastic
    sheets with a bunch of clear plastic disks between them - big
    "grains". And, he can make these grains "flow". Since they're
    made of a material that changes its optical properties under stress,
    you can see "force chains" flicker in and out of existence as lines
    of grains get momentarily stuck and then come unstuck!

    These force chains look like bolts of lightning:

    1) Brian Utter and R. P. Behringer, Self-diffusion in dense
    granular shear flows, Physical Review E 69, 031308 (2004).
    Also available as arXiv:cond-mat/0402669.

    I wonder if conformal field theory could help us understand these
    simplified 2-dimensional models of granular flow, at least near some
    critical point between "stuck" and "unstuck" flow. Conformal field
    theory tends to be good at studying critical points in 2d physics.

    Anyway, I'm digressing. After looking at a chaotic double pendulum
    in another lab, I talked to Harold Butner about his work using radio
    astronomy to study interstellar dust.

    He told me that the dust in the Red Rectangle contains a lot of PAHs -
    "polycyclic aromatic hydrocarbons". These are compounds made of
    hexagonal rings of carbon atoms, with some hydrogens along the edges.
    On Earth you can find PAHs in soot, or the tarry stuff that forms
    in a barbecue grill. Wherever carbon-containing materials suffer
    incomplete combustion, you'll find PAHs.

    Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens -
    a wonder of quantum resonance. You've probably heard about
    naphthalene, which is used for mothballs. This consists of two
    hexagonal rings stuck together. True PAHs have more. "Anthracene"
    and "phenanthrene" consist of three rings. "Napthacene", "pyrene",
    "triphenylene" and "chrysene" consist of four, and so on:

    2) Wikipedia, Polycyclic aromatic hydrocarbon,
    http://en.wikipedia.org/wiki/Polycyclic_aromatic_hydrocarbon

    In 2004, a team of scientists discovered anthracene and pyrene in the
    Red Rectangle! This was first time such complex molecules had been
    found in space:

    3) Uma P. Vijh, Adolf N. Witt, and Karl D. Gordon, Small polycyclic
    aromatic hydrocarbons in the Red Rectangle, The Astrophysical
    Journal, 619 (2005) 368-378.

    By now, lots of organic molecules have been found in interstellar
    or circumstellar space. There's a whole "ecology" of organic
    chemicals out there, engaged in complex reactions. Life on planets
    might someday be seen as just an aspect of this larger ecology.

    I've read that about 10% of the interstellar carbon is in the form
    of PAHs - big ones, with about 50 carbons per molecule. They're
    common because they're incredibly stable. They've even been found
    riding the shock wave of a supernova explosion!

    PAHs are also found in meteorites called "carbonaceous chondrites".
    These space rocks contain just a little carbon - about 3% by weight.
    But, 80% of this carbon is in the form of PAHs.

    Here's an interview with a scientist who thinks PAHs were important
    precursors of life on Earth:

    5) Aromatic world, interview with Pascale Ehrenfreund,
    Astrobiology Magazine, available at
    http://www.astrobio.net/news/modules.php?op=modload&name=News&file=article&sid=1992

    And here's a book she wrote, with a chapter on organic molecules
    in space:

    6) Pascale Ehrenfreud, editor, Astrobiology: Future Perspectives,
    Springer Verlag, 2004.

    Harold Butner also told me about dust disks that have been seen around
    the nearby stars Vega and Epsilon Eridani. By examining these disks,
    we may learn about planets and comets orbiting these stars. Comets
    emit a lot of dust, and planets affect its motion.

    Mathematicians will be happy to know that *symplectic geometry*
    is required to simulate the motion of this dust:

    7) A. T. Deller and S. T. Maddison, Numerical modelling of
    dusty debris disks, Astrophys. J. 625 (2005), 398-413.
    Also available as arXiv:astro-ph/0502135

    Okay... now for a bit about Hochschild cohomology. I want to
    outline a conceptual proof of Deligne's conjecture that the
    Hochschild cochain complex is an algebra for the little 2-cubes
    operad. There are a bunch of proofs of this by now. Here's a
    great introduction to the story:

    8) Maxim Kontsevich, Operads and motives in deformation
    quantization, available as arXiv:math/9904055.

    I was inspired to seek a more conceptual proof by some conversations
    I had with Simon Willerton in Sheffield this summer, and this paper
    of his:

    9) Andrei Calderou and Simon Willerton, The Mukai pairing, I:
    a categorical approach, available as arXiv:0707.2052

    But, while trying to write up a sketch of this more conceptual
    proof, I discovered that it had already been worked out:

    10) P. Hu, H. Kriz and A. A. Voronov, On Kontsevich's Hochschild
    cohomology conjecture, available at arXiv:math.AT/0309369.

    This was a bit of a disappointment - but also a relief. It
    means I don't need to worry about the technical details: you
    can just look them up! Instead, I can focus on sketching the
    picture I had in mind.

    If you don't know anything about Hochschild cohomology, don't worry!
    It only comes in at the very end. In fact, the conjecture
    follows from something simpler and more general. So, what you
    really need is a high tolerance for category theory, homological
    algebra and operads.

    First, suppose we have any monoidal category. Such a category
    has a tensor product and a unit object, which we'll call I. Let
    end(I) be the set of all endomorphisms of this unit object.

    Given two such endomorphisms, say

    f: I -> I

    and

    g: I -> I

    we can compose them, getting

    f o g: I -> I

    This makes end(I) into a monoid. But we can also tensor f and
    g, and since I tensor I is isomorphic to I in a specified way,
    we can write the result simply as

    f tensor g: I -> I

    This makes end(I) into a monoid in another, seemingly different
    way.

    Luckily, there's a thing called the Eckmann-Hilton argument which
    says these two ways are equal. It also says that end(I) is a
    *commutative* monoid! It's easiest to understand this argument
    if we write f o g vertically, like this:

    f

    g

    and f tensor g horizontally, like this:

    f g

    Then the Eckmann-Hilton argument goes as follows:

    f 1 f g 1 g
    = = g f = =
    g g 1 1 f f

    Here 1 means the identity morphism 1: I -> I. Each step in the
    argument follows from standard stuff about monoidal categories.
    In particular, an expression like

    f g

    h k

    is well-defined, thanks to the interchange law

    (f tensor g) o (h tensor k) = (f o h) tensor (g o k)

    If we want to show off, we can say the interchange law says we've got
    a "monoid in the category of monoids" - and the Eckmann-Hilton
    argument shows this is just a monoid. See "week100" for more.

    But the cool part about the Eckmann-Hilton argument is that we're
    just moving f and g around each other. So, this argument has a
    topological flavor! Indeed, it was first presented as an argument
    for why the second homotopy group is commutative. It's all about
    sliding around little rectangles... or as we'll soon call
    them, "little 2-cubes".

    Next, let's consider a version of this argument that holds only
    "up to homotopy". This will apply when we have not a *set*
    of morphisms from any object X to any object Y, but a *chain
    complex* of morphisms.

    Instead of getting a set end(I) that's a commutative monoid, we'll
    get a cochain complex END(I) that's a commutative monoid "up to
    coherent homotopy". This means that the associative and commutative
    laws hold up to chain homotopies, which satisfy their own laws up to
    homotopy, ad infinitum.

    More precisely, END(I) will be an "algebra of the little 2-cubes
    operad". This implies that for every configuration of n little
    rectangles in a square:

    ---------------------
    | |
    | ----- |
    | ----- | | |
    || | | | |
    || | | | |
    | ----- | | |
    | ----- |
    | ---------------- |
    | | | |
    | ---------------- |
    | |
    ---------------------

    we get an n-ary operation on END(I). For every homotopy between
    such configurations:

    --------------------- ---------------------
    | | | ----- |
    | ----- | || | ---- |
    | ----- | | | || | | | |
    || | | | | || | | | |
    || | | | | || | | | |
    | ----- | | | ---> | ----- | | |
    | ----- | | ----- |
    | ---------------- | | ------- |
    | | | | | | | |
    | ---------------- | | ------- |
    | | | |
    --------------------- ---------------------

    we get a chain homotopy between n-ary operations on END(I). And
    so on, ad infinitum.

    For more on the little 2-cubes operad, see "week220". In fact,
    what I'm trying to do now is understand some mysteries I described
    in that article: weird relationships between the little 2-cubes
    operad and Poisson algebras.

    But never mind that stuff now. For now, let's see how easy it is to
    find situations where there's a chain complex of morphisms between
    objects. It happens throughout homological algebra!

    If that sounds scary, you should refer to a book like this as you
    read on:

    10) Charles Weibel, An Introduction to Homological Algebra,
    Cambridge U. Press, Cambridge, 1994.

    Okay. First, suppose we have an abelian category. This provides a
    context in which we can reason about chain complexes and cochain
    complexes of objects. A great example is the category of R-modules
    for some ring R.

    Next, suppose every object X in our abelian category has an
    "projective resolution" - that is, a chain complex

    d_0 d_1 d_2
    X_0 <--- X_1 <--- X_2 <--- ...

    where each guy X_i is projective, and the homology groups

    ker (d_i)
    H_i = -------------
    im (d_{i-1})

    are zero except for H_0, which equals X. You should think of
    a projective resolution as a "puffed-up" version of X that's
    better for mapping out of than X itself.

    Given this, besides the usual set hom(X,Y) of morphisms from the
    object X to the object Y, we also get a cochain complex which I'll
    call the "puffed-up hom":

    HOM(X,Y)

    How does this work? Simple: replace X by a chosen projective
    resolution

    X^0 <--- X^1 <--- X^2 <--- ...

    and then map this whole thing to Y, getting a cochain complex

    hom(X^0,Y) ---> hom(X^1,Y) ---> hom(X^2,Y) ---> ...

    This chain complex is the puffed-up hom, HOM(X,Y).

    Now, you might hope that the puffed-up hom gives us a new category
    where the hom-sets are actually cochain complexes. This is morally
    true, but the composition

    o: HOM(X,Y) x HOM(Y,Z) -> HOM(X,Z)

    probably isn't associative "on the nose". However, I think it should
    be associative up to homotopy! This homotopy probably won't satisfy
    the law you'd hope for - the pentagon identity. But, it should
    satisfy the pentagon identity up to homotopy! In fact, this should
    go on forever, which is what we mean by "up to coherent homotopy".
    This kind of situation is described by an infinite sequence of shapes
    called "associahedra" discovered by Stasheff (see "week144").

    If this is the case, instead of a category we get an "A-infinity
    category": a gadget where the hom-sets are cochain complexes and the
    associative law holds up to coherent homotopy. I'm not sure the
    puffed-up hom gives an A-infinity category, but let's assume so and
    march on.

    Suppose we take any object X in our abelian category. Then we get
    a cochain complex

    END(X) = HOM(X,X)

    equipped with a product that's associative up to coherent homotopy.
    Such a thing is known as an "A-infinity algebra". It's just an
    A-infinity category with a single object, namely X.

    Next suppose our abelian category is monoidal. (To get the tensor
    product to play nice with the hom, assume tensoring with any object
    is right exact.) Let's see what happens to the Eckmann-Hilton
    argument. We should get a version that holds "up to coherent
    homotopy".

    Let I be the unit object, as before. In addition to composition:

    o: END(I) x END(I) -> END(I)

    tensoring should give us another product:

    tensor: END(I) x END(I) -> END(I)

    which is also associative up to coherent homotopy. So, END(I) should
    be an A-infinity algebra in two ways. But, since composition
    and tensoring in our original category get along nicely:

    (f tensor g) o (h tensor k) = (f o h) tensor (g o k)

    END(I) should really be an A-infinity algebra in the category of
    A-infinity algebras!

    Given this, we're almost done. A monoid in the category of monoids
    is a commutative monoid - that's another way of stating what the
    Eckmann-Hilton argument proves. Similarly, an A-infinity algebra in
    the category of A-infinity algebras is an algebra of the little
    2-cubes operad. So, END(I) is an algebra of the little 2-cubes
    operad.

    Now look at an example. Fix some algebra A, and take our
    monoidal abelian category to have:

    A-A bimodules as objects
    A-A bimodule homomorphisms as morphisms

    Here the tensor product is the usual tensor product of bimodules,
    and the unit object I is A itself. And, as Simon Willerton pointed
    out to me, END(I) is a chain complex whose homology is familiar:
    it's the "Hochschild homology" of A.

    So, the cochain complex for Hochschild cohomology is an algebra of
    the little 2-cubes operad! But, we've seen this as a consequence
    of a much more general fact.

    To wrap up, here are a few of the many technical details I glossed
    over above.

    First, I said a projective resolution of X is a puffed-up version of
    X that's better for mapping out of. This idea is made precise
    in the theory of model categories. But, instead of calling it a
    "puffed-up version" of X, they call it a "cofibrant replacement" for
    X. Similarly, a puffed-up version of X that's better for mapping
    into is called a "fibrant replacement".

    For a good introduction to this, try:

    11) Mark Hovey, Model Categories, American Mathematical Society,
    Providence, Rhode Island, 1999.

    Second, I guessed that for any abelian category where every object has
    a projective resolution, we can create an A-infinity category using
    the puffed-up hom, HOM(X,Y). Alas, I'm not really sure this is true.

    Hu, Kriz and Voronov consider a more general situation, but what I'm
    calling the "puffed-up hom" should be a special case of their "derived
    function complex". However, they don't seem to say what weakened sort
    of category you get using this derived function complex - maybe an
    A-infinity category, or something equivalent like a quasicategory or
    Segal category? They somehow sidestep this issue, but to me it's
    interesting in its own right.

    At this point I should mention something well-known that's similar
    to what I've been talking about. I've been talking about the
    "puffed-up hom" for an abelian category with enough projectives.
    But most people talk about "Ext", which is the cohomology of the
    puffed-up hom:

    Ext^i(X,Y) = H^i(HOM(X,Y))

    And, while I want

    END(X) = HOM(X,X)

    to be an A-infinity algebra, most people seem happy to have

    Ext(X) = H(HOM(X,X))

    be an A-infinity algebra. Here's a reference:

    12) D.-M. Lu, J. H. Palmieri, Q.-S. Wu and J. J. Zhang,
    A-infinity structure on Ext-algebras, available as
    arXiv:math.KT/0606144.

    I hope they're secretly getting this A-infinity structure on
    H(HOM(X,X)) from an A-infinity structure on HOM(X,X). They don't
    come out and say this is what they're doing, but one promising
    sign is that they use a theorem of Kadeishvili, which says that
    the cohomology of an A-infinity algebra is an A-infinity algebra.

    Finally, the really interesting part: how do we make an A-infinity
    algebra in the category of A-infinity algebras into an algebra of
    the little 2-cubes operad? This is the heart of the "homotopy
    Eckmann-Hilton argument".

    I explained operads, and especially the little k-cubes operad,
    back in "week220". The little k-cubes operad is an operad in
    the world of topological spaces. It has one abstract n-ary operation
    for each way of sticking n little k-dimensional cubes in a big
    one, like this:

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

    A space is called an "algebra" of this operad if these abstract
    n-ary operations are realized as actual n-ary operations on the
    space in a consistent way. But, when we study the homology
    of topological spaces, we learn that any space gives a chain complex.
    This lets us convert any operad in the world of topological spaces
    into an operad in the world of chain complexes. Using this, it also
    makes sense to speak of a *chain complex* being an algebra of the
    little k-cubes operad. For that matter, a cochain complex.

    Let's use "E(k)" to mean the chain complex version of the little
    k-cubes operad.

    An "A-infinity algebra" is an algebra of a certain operad called
    A-infinity. This isn't quite the same as the operad E(1), but it's
    so close that we can safely ignore the difference here: it's
    "weakly equivalent".

    Say we have an A-infinity algebra in the category of A-infinity
    algebras. How do we get an algebra of the little 2-cubes operad,
    E(2)?

    Well, there's a way to tensor operads, such that an algebra of
    P tensor Q is the same as a P-algebra in the category of Q-algebras.
    So, an A-infinity algebra in the category of A-infinity algebras is
    the same as an algebra of

    A-infinity tensor A-infinity

    Since A-infinity and E(1) are weakly equivalent, we can turn this
    algebra into an algebra of

    E(1) tensor E(1)

    But there's also an obvious operad map

    E(1) tensor E(1) -> E(2)

    since the product of two little 1-cubes is a little 2-cube.
    This too is a weak equivalence, so we can turn our algebra of
    E(1) tensor E(1) into an algebra of E(2).

    The hard part in all this is showing that the operad map

    E(1) tensor E(1) -> E(2)

    is a weak equivalence. In fact, quite generally, the map

    E(k) tensor E(k') -> E(k+k')

    is a weak equivalence. This is Proposition 2 in the paper by
    Hu, Kriz and Voronov, based on an argument by Gerald Dunn:

    13) Gerald Dunn, Tensor products of operads and iterated loop
    spaces, Jour. Pure Appl. Alg 50 (1988), 237-258.

    Using this, they do much more than what I've sketched: they
    prove a conjecture of Kontsevich which says that the Hochschild
    complex of an algebra of the little k-cubes operad is an algebra
    of the little (k+1)-cubes operad!

    That's all for now. Sometime I should tell you how this is related
    to Poisson algebras, 2d TQFTs, and much more. But for now, you'll
    have to read that in Kontsevich's very nice paper.

    -----------------------------------------------------------------------
    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/twfcontents.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
  3. Nov 27, 2007 #2
    John Baez wrote:
    >
    > Also available as http://math.ucr.edu/home/baez/week258
    >
    > November 25, 2007
    > This Week's Finds in Mathematical Physics (Week 258)
    > John Baez

    [snip]

    > Anyway, I'm digressing. After looking at a chaotic double pendulum
    > in another lab, I talked to Harold Butner about his work using radio
    > astronomy to study interstellar dust.
    >
    > He told me that the dust in the Red Rectangle contains a lot of PAHs -
    > "polycyclic aromatic hydrocarbons". These are compounds made of
    > hexagonal rings of carbon atoms, with some hydrogens along the edges.
    > On Earth you can find PAHs in soot, or the tarry stuff that forms
    > in a barbecue grill. Wherever carbon-containing materials suffer
    > incomplete combustion, you'll find PAHs.
    >
    > Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens -
    > a wonder of quantum resonance. You've probably heard about
    > naphthalene, which is used for mothballs. This consists of two
    > hexagonal rings stuck together. True PAHs have more. "Anthracene"
    > and "phenanthrene" consist of three rings. "Napthacene", "pyrene",
    > "triphenylene" and "chrysene" consist of four, and so on:

    [snip]

    > By now, lots of organic molecules have been found in interstellar
    > or circumstellar space. There's a whole "ecology" of organic
    > chemicals out there, engaged in complex reactions. Life on planets
    > might someday be seen as just an aspect of this larger ecology.
    >
    > I've read that about 10% of the interstellar carbon is in the form
    > of PAHs - big ones, with about 50 carbons per molecule. They're
    > common because they're incredibly stable. They've even been found
    > riding the shock wave of a supernova explosion!

    [snip]

    PAH chemistry is 2-D. Until you diddle graphene, how much fun can
    chickenwire be? LOTS! - in 3-D. Fullerenes, nanotubes... and
    Ambassador Kosh's three-edged sword (complete with Jedi fluorescence),
    following. Possibly superconductive for physicists' enjoyment.

    Anthracene plus benzyne (Diels-Alder reaction) giving trypticene is a
    common undergrad lab prep. 9,9-Bianthracene is easy. Turn the crank
    and get locked rotor bitrypticene more than likely,

    http://www.mazepath.com/uncleal/bitrypt.png
    Stereogram, like a Magic Eye.

    HyperChem assigns +34.899 kcal/mole overall. Cute molecule.
    FeCl3/MeNO2 oxidation to ring couple giving 25.407 kcal/mole.

    http://www.mazepath.com/uncleal/bitrypta.png

    It's *more* stable. We *like* long rigid strings of forced
    conjugation - fluorescence and metallic conductivity. Alkyl,
    polyoxypropylene or polyoxyethylene, etc. chains on the anthracene for
    (liquid crystal) solubility. Flank a central benzene with two fused
    thiophenes (from commercial 1,2,4,5 benzenetetracarboxylic acid,
    pyromellitic acid) to replace anthracene. That can lead to real fun.

    10,10-Dibromo-9,9-bianthracene gives bridgehead
    alpha,omega-dibromoditrypticene polymerizable by molten sodium or
    whatnot. Monobromo monomer for endcapping oligomers. Consider
    constrained conjugated pi pretties like

    http://www.mazepath.com/uncleal/bitrypt2a.png
    then
    http://www.mazepath.com/uncleal/bitrypt2b.png

    If the wings are thiophenes with sulfurs pointing out you've got a
    lusty (super)conducting rigidly conjugated three-bladed pi system that
    loves to ordered-array upon gold.

    The polybarrelene core is also interesting,
    WIRED 15(12) 214 (2007)
    Feel the power of the C3-symmetry synthon. Easier than staffanes,
    more rigid than ladderanes.

    o-Benzyne (1,2-didehydrobenzene) is adequately stable in vacuum or
    isolated in bulk in hemicarcerand hosts, formed by UV photolysis.
    Astronomic assignment of optical absorptions to organic molecules has
    not gone all that well. Perhaps the folks are missing a dimension.

    --
    Uncle Al
    http://www.mazepath.com/uncleal/
    (Toxic URL! Unsafe for children and most mammals)
    http://www.mazepath.com/uncleal/lajos.htm#a2
     
  4. Nov 27, 2007 #3
    John Baez wrote:
    >
    > Also available as http://math.ucr.edu/home/baez/week258
    >
    > November 25, 2007
    > This Week's Finds in Mathematical Physics (Week 258)
    > John Baez

    [snip]

    > Anyway, I'm digressing. After looking at a chaotic double pendulum
    > in another lab, I talked to Harold Butner about his work using radio
    > astronomy to study interstellar dust.
    >
    > He told me that the dust in the Red Rectangle contains a lot of PAHs -
    > "polycyclic aromatic hydrocarbons". These are compounds made of
    > hexagonal rings of carbon atoms, with some hydrogens along the edges.
    > On Earth you can find PAHs in soot, or the tarry stuff that forms
    > in a barbecue grill. Wherever carbon-containing materials suffer
    > incomplete combustion, you'll find PAHs.
    >
    > Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens -
    > a wonder of quantum resonance. You've probably heard about
    > naphthalene, which is used for mothballs. This consists of two
    > hexagonal rings stuck together. True PAHs have more. "Anthracene"
    > and "phenanthrene" consist of three rings. "Napthacene", "pyrene",
    > "triphenylene" and "chrysene" consist of four, and so on:

    [snip]

    > By now, lots of organic molecules have been found in interstellar
    > or circumstellar space. There's a whole "ecology" of organic
    > chemicals out there, engaged in complex reactions. Life on planets
    > might someday be seen as just an aspect of this larger ecology.
    >
    > I've read that about 10% of the interstellar carbon is in the form
    > of PAHs - big ones, with about 50 carbons per molecule. They're
    > common because they're incredibly stable. They've even been found
    > riding the shock wave of a supernova explosion!

    [snip]

    PAH chemistry is 2-D. Until you diddle graphene, how much fun can
    chickenwire be? LOTS! - in 3-D. Fullerenes, nanotubes... and
    Ambassador Kosh's three-edged sword (complete with Jedi fluorescence),
    following. Possibly superconductive for physicists' enjoyment.

    Anthracene plus benzyne (Diels-Alder reaction) giving trypticene is a
    common undergrad lab prep. 9,9-Bianthracene is easy. Turn the crank
    and get locked rotor bitrypticene more than likely,

    http://www.mazepath.com/uncleal/bitrypt.png
    Stereogram, like a Magic Eye.

    HyperChem assigns +34.899 kcal/mole overall. Cute molecule.
    FeCl3/MeNO2 oxidation to ring couple giving 25.407 kcal/mole.

    http://www.mazepath.com/uncleal/bitrypta.png

    It's *more* stable. We *like* long rigid strings of forced
    conjugation - fluorescence and metallic conductivity. Alkyl,
    polyoxypropylene or polyoxyethylene, etc. chains on the anthracene for
    (liquid crystal) solubility. Flank a central benzene with two fused
    thiophenes (from commercial 1,2,4,5 benzenetetracarboxylic acid,
    pyromellitic acid) to replace anthracene. That can lead to real fun.

    10,10-Dibromo-9,9-bianthracene gives bridgehead
    alpha,omega-dibromoditrypticene polymerizable by molten sodium or
    whatnot. Monobromo monomer for endcapping oligomers. Consider
    constrained conjugated pi pretties like

    http://www.mazepath.com/uncleal/bitrypt2a.png
    then
    http://www.mazepath.com/uncleal/bitrypt2b.png

    If the wings are thiophenes with sulfurs pointing out you've got a
    lusty (super)conducting rigidly conjugated three-bladed pi system that
    loves to ordered-array upon gold.

    The polybarrelene core is also interesting,
    WIRED 15(12) 214 (2007)
    Feel the power of the C3-symmetry synthon. Easier than staffanes,
    more rigid than ladderanes.

    o-Benzyne (1,2-didehydrobenzene) is adequately stable in vacuum or
    isolated in bulk in hemicarcerand hosts, formed by UV photolysis.
    Astronomic assignment of optical absorptions to organic molecules has
    not gone all that well. Perhaps the folks are missing a dimension.

    --
    Uncle Al
    http://www.mazepath.com/uncleal/
    (Toxic URL! Unsafe for children and most mammals)
    http://www.mazepath.com/uncleal/lajos.htm#a2
     
  5. Nov 27, 2007 #4
    John Baez wrote:
    >
    > Also available as http://math.ucr.edu/home/baez/week258
    >
    > November 25, 2007
    > This Week's Finds in Mathematical Physics (Week 258)
    > John Baez

    [snip]

    > Anyway, I'm digressing. After looking at a chaotic double pendulum
    > in another lab, I talked to Harold Butner about his work using radio
    > astronomy to study interstellar dust.
    >
    > He told me that the dust in the Red Rectangle contains a lot of PAHs -
    > "polycyclic aromatic hydrocarbons". These are compounds made of
    > hexagonal rings of carbon atoms, with some hydrogens along the edges.
    > On Earth you can find PAHs in soot, or the tarry stuff that forms
    > in a barbecue grill. Wherever carbon-containing materials suffer
    > incomplete combustion, you'll find PAHs.
    >
    > Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens -
    > a wonder of quantum resonance. You've probably heard about
    > naphthalene, which is used for mothballs. This consists of two
    > hexagonal rings stuck together. True PAHs have more. "Anthracene"
    > and "phenanthrene" consist of three rings. "Napthacene", "pyrene",
    > "triphenylene" and "chrysene" consist of four, and so on:

    [snip]

    > By now, lots of organic molecules have been found in interstellar
    > or circumstellar space. There's a whole "ecology" of organic
    > chemicals out there, engaged in complex reactions. Life on planets
    > might someday be seen as just an aspect of this larger ecology.
    >
    > I've read that about 10% of the interstellar carbon is in the form
    > of PAHs - big ones, with about 50 carbons per molecule. They're
    > common because they're incredibly stable. They've even been found
    > riding the shock wave of a supernova explosion!

    [snip]

    PAH chemistry is 2-D. Until you diddle graphene, how much fun can
    chickenwire be? LOTS! - in 3-D. Fullerenes, nanotubes... and
    Ambassador Kosh's three-edged sword (complete with Jedi fluorescence),
    following. Possibly superconductive for physicists' enjoyment.

    Anthracene plus benzyne (Diels-Alder reaction) giving trypticene is a
    common undergrad lab prep. 9,9-Bianthracene is easy. Turn the crank
    and get locked rotor bitrypticene more than likely,

    http://www.mazepath.com/uncleal/bitrypt.png
    Stereogram, like a Magic Eye.

    HyperChem assigns +34.899 kcal/mole overall. Cute molecule.
    FeCl3/MeNO2 oxidation to ring couple giving 25.407 kcal/mole.

    http://www.mazepath.com/uncleal/bitrypta.png

    It's *more* stable. We *like* long rigid strings of forced
    conjugation - fluorescence and metallic conductivity. Alkyl,
    polyoxypropylene or polyoxyethylene, etc. chains on the anthracene for
    (liquid crystal) solubility. Flank a central benzene with two fused
    thiophenes (from commercial 1,2,4,5 benzenetetracarboxylic acid,
    pyromellitic acid) to replace anthracene. That can lead to real fun.

    10,10-Dibromo-9,9-bianthracene gives bridgehead
    alpha,omega-dibromoditrypticene polymerizable by molten sodium or
    whatnot. Monobromo monomer for endcapping oligomers. Consider
    constrained conjugated pi pretties like

    http://www.mazepath.com/uncleal/bitrypt2a.png
    then
    http://www.mazepath.com/uncleal/bitrypt2b.png

    If the wings are thiophenes with sulfurs pointing out you've got a
    lusty (super)conducting rigidly conjugated three-bladed pi system that
    loves to ordered-array upon gold.

    The polybarrelene core is also interesting,
    WIRED 15(12) 214 (2007)
    Feel the power of the C3-symmetry synthon. Easier than staffanes,
    more rigid than ladderanes.

    o-Benzyne (1,2-didehydrobenzene) is adequately stable in vacuum or
    isolated in bulk in hemicarcerand hosts, formed by UV photolysis.
    Astronomic assignment of optical absorptions to organic molecules has
    not gone all that well. Perhaps the folks are missing a dimension.

    --
    Uncle Al
    http://www.mazepath.com/uncleal/
    (Toxic URL! Unsafe for children and most mammals)
    http://www.mazepath.com/uncleal/lajos.htm#a2
     
  6. Nov 30, 2007 #5
    John Baez wrote:

    > So, the cochain complex for Hochschild cohomology is an algebra of
    > the little 2-cubes operad! But, we've seen this as a consequence
    > of a much more general fact.


    What the precise formulation of this more general fact? If its exist.
     
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