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This Week's Finds in Mathematical Physics (Week 267)

  1. Jul 30, 2008 #1
    Also available at http://math.ucr.edu/home/baez/week267.html

    July 23, 2008
    This Week's Finds in Mathematical Physics (Week 267)
    John Baez

    After the workshop on categorical groups in Barcelona, I went to
    Granada - the world capital of categorical groups! Pilar Carrasco,
    an expert on this subject, had kindly invited me to spend a week
    there and give some talks. Even more kindly, she put me in a hotel
    right next to the Alhambra. So, this week I'll tell you about some
    categorical groups I saw in the Alhambra.

    I've long been fascinated by that melting-pot of cultures in southern
    Spain called Andalusia. I wrote about it back in "week221". It was
    invaded by Muslims in 711 AD, and became a center for mathematics and
    astronomy from around 930 AD to around the 1200s, when the city of
    Toledo, recaptured by Catholic Spaniards, became the center of a big
    translation industry - translating Arabic and Hebrew texts into Latin.
    This was important for the transmission of ancient Greek writings into
    Western Europe.

    The Alhambra was built after the true heyday of Andalusia, in the
    era when Muslims had almost been pushed out by the Catholics. Its
    construction was begun by Muhammed ibn Nasr, founder of the Nasrid
    Dynasty - the last Muslim dynasty in Spain.

    In 1236, Ferdinand III of Castile captured the marvelous city of
    Cordoba, "ornament of the world". Ibn Nasr saw which way the wind
    was blowing, and arranged to pay tribute to Ferdinand and even help
    him take the city of Seville in return for leaving his city -
    Granada - alone. He started building the Alhambra in 1238. It was
    completed in the late 1300s.

    For the mathematician, one striking thing about the Alhambra is
    the marvelous tile patterns. On my visit, I took photos of all
    the tiles I could see:

    1) John Baez, Alhambra tiles,

    Some people claim that tilings with all 17 possible "wallpaper groups"
    as symmetries can be found in the Alhambra. This article rebuts that
    claim with all the vehemence such an academic issue deserves, saying
    that only 13 wallpaper groups are visible:

    2) Branko Grünbaum, The emperor's new clothes: full regalia, G-string,
    or nothing?, with comments by Peter Hilton and Jean Pedersen, Math.
    Intelligencer 6 (1984), 47-56.

    As mentioned in "week221", this page shows 13 of the 17:

    3) Steve Edwards, Tilings from the Alhambra,
    http://www2.spsu.edu/math/tile/grammar/moor.htm [Broken]

    Of the remaining four, two seem completely absent in Islamic
    art - the groups called "pgg" and "pg". Both are fairly low on
    symmetries, so they might have been avoided for lack of visual

    Let me describe them, just for fun. You can learn the definition of
    wallpaper groups, and learn a lot more about them, from this rather
    wonderful article:

    4) Wikipedia, Wallpaper group,

    The group pgg is the symmetry group of a popular zig-zag method of
    laying bricks, shown in the article above. The only rotations in this
    group are 180-degree rotations: you can rotate any brick 180 degrees
    around its center, and the pattern comes back looking the same. There
    are no reflections in this group. But, there are "glide reflections"
    in two diagonal directions: a "glide reflection" is a combination of a
    translation along some line and a reflection across that line.

    The group pg is a subgroup of pgg. If we take our zig-zag pattern of
    bricks and break the 180-degree rotation symmetry somehow, the
    remaining symmetry group is pg. This group contains no reflections
    and no rotations. It contains translations along one axis and "glide
    reflections" along another.

    For more on tilings, try this book. Among other things, it points
    out that there's a lot more beauty and mathematical structure in
    tilings than is captured by their symmetry groups!

    5) Branko Grünbaum and G. C. Shephard, Tilings and Patterns,
    New York, Freeman, 1987.

    The mathemagician John Horton Conway has come up with a very nice
    proof that there are only 17 wallpaper groups. This is nicely
    sketched in the Wikipedia article above, but detailed here:

    6) John H. Conway, The orbifold notation for surface groups, in
    Groups, Combinatorics and Geometry, London Math. Soc. Lecture
    Notes Series 165, Cambridge U. Press, Cambridge, 1990, pp. 438-447

    Here's the basic idea. Take a wallpaper pattern and count two points
    as "the same" if they're related by a symmetry. In other words - in
    math jargon - take the plane and mod out by the wallpaper group. The
    result is a 2-dimensional "orbifold".

    In a 2d manifold, every point has a little neighborhood that looks
    like the plane. In a 2d orbifold, every point has a little
    neighborhood that looks either like the plane, or the plane mod a
    finite group of rotations and/or reflections.

    Let's see how this works for a few simple wallpaper groups.

    I'll start with the most boring wallpaper group in the world, p1.
    If you thought pg was dull, wait until you see p1. It's the symmetry
    group of this wallpaper pattern:

    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R

    This group doesn't contain any rotations, reflections or glide
    reflections - I used the letter R to rule those out. It only
    contains translations in two directions, the bare minimum allowed
    by the definition of a wallpaper group.

    If we take the plane and mod out by this group, all the points
    labelled x get counted as "the same":

    R R R R R R
    R R R R R R
    R x R x R x R x R x R
    R R R R R R
    R R R R R R
    R x R x R x R x R x R
    R R R R R R
    R R R R R R

    Similarly, all these points labelled y get counted as "the same":

    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R
    R R R R R R

    So, when we take the plane and mod out by the group p1, we get a
    rectangle with its right and left edges glued together, and with its
    top and bottom edges glued together. This is just a torus. A torus
    is a 2d manifold, which is a specially dull case of a 2d orbifold.

    Now let's do a slightly more interesting example:

    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T

    The letter T is more symmetrical than the letter R: you can reflect
    it, and it still looks the same. (If you're viewing this using some
    font where the letter T *doesn't* have this symmetry, switch fonts!)
    So, the symmetry group of this wallpaper pattern, called pm, is bigger
    than p1: it also contains reflections and glide reflections along a
    bunch of parallel lines. So now, all these points labelled x get
    counted as the same when we mod out:

    T T T T T T
    T T T T T T
    T x x T x x T x x T x x T x x T
    T T T T T T
    T T T T T T
    T x x T x x T x x T x x T x x T
    T T T T T T
    T T T T T T

    and similarly for all these points labelled y:

    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T
    T T T T T T

    but look how these points labelled z work:

    T T T T T T
    T z T z T z T z T z T
    T T T T T T
    T T T T T T
    T z T z T z T z T z T
    T T T T T T
    T T T T T T
    T z T z T z T z T z T

    There are only half as many z's per rectangle, since they lie on
    reflection lines.

    Because of this subtlety, this time when we mod out we get an orbifold
    that's not a manifold! It's the torus of the previous example, but
    now folded in half. We can draw it as *half* of one of the rectangles
    above, with the top and bottom glued together, but not the sides:

    T .
    T .
    T .

    So, it's a cylinder... but in a certain technical sense the points at
    the ends of this cylinder count as "half points": they lie on
    reflection lines, so they've been "folded in half".

    This particular orbifold looks a lot like a 2d manifold "with
    boundary". That's a generalization of a 2d manifold where some
    points - the "boundary" points - have a neighborhood that looks like
    a half-plane. But 2d orbifolds can also have "cone points" and
    "mirror reflector" points.

    What's a cone point? It's like the tip of a cone. Take a piece of
    paper, cut it like a pie into n equal wedges, take one wedge, and glue
    its edges together! This gives a cone - and the tip of this cone is a
    "cone point". We say it has "order n", because the angle around it is
    not 2pi but 2pi/n.

    Here's a more sophisticated way to say the same thing: take
    a regular n-gon and mod out by its rotational symmetries, which
    form a group with n elements. When we're done, the point in
    the center is a cone point of order n.

    We could also mod out by the rotation *and* reflection symmetries
    of the n-gon, which form a group with 2n elements. This is harder
    to visualize, but when we're done, the point in the center is a
    "corner reflector of order 2n".

    To see one of these fancier possibilities, let's look at the
    orbifold coming from a wallpaper pattern with even more symmetries:

    . . . . . .
    . . . . . .
    . . . . . .
    . . . . . .
    . . . . . .
    . . . . . .
    . . . . . .
    . . . . . .

    These are supposed to be rectangles, not squares. So, 90-degree
    rotations are not symmetries of this pattern. But, in addition to all
    the symmetries we had last time, now we have reflections about a bunch
    of horizontal lines. We get a wallpaper group called pmm.

    What orbifold do we get now? It's a torus folded in half *twice*!
    That sounds scary, but it's not really. We can draw it as a *quarter*
    of one of the rectangles above:

    . .

    Now no points on the edges are glued together. So, it's just a
    rectangle. The points on the edges are boundary points, and the
    corners are corner reflection points of order 4.

    In a certain technical sense - soon to be explained - points on
    the edges of this rectangle count as "half points", since they lie on
    a reflection line and have been folded in half. But the corners count
    as "1/4 points", since they lie on *two* reflection lines, so they've
    been folded in half *twice*!

    This is where it gets really cool. There's a way to define an "Euler
    characteristic" for orbifolds that generalizes the usual formula for
    2d manifolds. And, it can be a fraction!

    The usual formula says to chop our 2d manifold into polygons and

    V - E + F

    That is: the number of vertices, minus the number of edges, plus the
    number of faces.

    In a 2d orbifold, we use the same formula, but with some modifications.
    First, we require that every cone point or corner reflector be one
    of our vertices. Then:

    We count edges and vertices on the boundary for 1/2 the usual amount.

    We count a cone point of order n as 1/nth of a point.

    We count a corner reflector of order 2n as 1/(2n)th of a point.

    The idea is that these features have been "folded over"
    by a certain amount, so they count for a fraction of what they
    otherwise would.

    It turns out that if we calculate the Euler characteristic of a
    2d orbifold coming from a wallpaper group, we always get zero.
    And, there are just 17 possibilities!

    In fact, wallpaper groups are secretly *the same* as 2d orbifolds
    with vanishing Euler characteristic! So, they're not just
    mathematical curiosities: they're almost as fundamental as 2d

    The torus and the cylinder, which we've already seen, are two
    examples. These are well-known to have Euler characteristic zero.
    Of course, we should be careful: now we're dealing with the cylinder
    as an orbifold, so the points on the boundary count as "half points" -
    but its Euler characteristic still vanishes. A more interesting
    example is the square we get from the group pmm. Let's chop it into
    vertices, edges, and one face, and figure out how much each of these

    | |
    | |
    1/2| 1 |1/2
    | |
    | |

    So, the Euler characteristic of this orbifold is

    (1/4 + 1/4 + 1/4 + 1/4) - (1/2 + 1/2 + 1/2 + 1/2) + 1 = 0

    This is different than the usual Euler characteristic of a

    As usual, Conway has figured out a charming way to explain all

    7) John Conway, Peter Doyle, Jane Gilman and Bill Thurston,
    Geometry and the Imagination in Minneapolis, available at

    Especially see the sections near the end entitled "Symmetry and
    orbifolds", "Names for features of symmetrical pattern", "Names
    for symmetry groups and orbifolds", "The orbifold shop" and
    "The Euler characteristic of an orbifold". He has a way of
    naming 2d orbifolds that lets you easily see how they "cost".
    Any orbifold that costs 2 dollars corresponds to a wallpaper
    group, and if you list them, you see there are exactly 17!

    (I only know two places where the number 17 played an important
    role in mathematics - the other is much more famous.)

    Why exactly 2 dollars? This is related to this formula for the Euler
    characteristic of a g-handled torus:

    V - E + F = 2 - 2g

    where g is the number of handles. At the orbifold shop, each
    handle costs 2 dollars. So, if you buy one handle, you're done:
    you get an ordinary torus, which has Euler characteristic zero.
    This is the result of taking the plane and modding out by a
    spectacularly dull wallpaper group. If you don't waste all
    your money on a handle, you can buy more interesting orbifolds.

    Devoted readers of This Week's Finds can guess why I'm talking
    about this. It's not just that I like the Alhambra. The usual Euler
    characteristic is a generalization of the cardinality of a finite set
    that allows *negative* values - but not fractional ones. There's
    also something called the "homotopy cardinality" of a space, which
    allows *fractional* values - but not negative ones!

    If we combine these ideas, we get the orbifold Euler characteristic,
    which allow both negative and fractional values. This has various
    further generalizations, like the Euler characteristic of a
    differentiable stack - and Leinster's Euler characteristic of a
    category, explained in "week244". We should be able to use these
    to categorify a lot of math involving rational numbers.

    But, it's especially cool how this game of listing all 2d orbifolds
    with Euler characteristic 0 fits together with things like the
    "Egyptian fractions" approach to ADE Dynkin diagrams - as explained in
    "week182". Here I used the Euler characteristic to list all ways ways
    to regularly tile a sphere with regular polygons. This gave the McKay
    correspondence linking Platonic solids to the simply-laced Lie
    algebras A_n, D_n, E_6, E_7 and E_8. Taking different values of the
    Euler characteristic, the same idea let us classify regular tilings of
    the plane or hyperbolic plane by regular polygons, and see how these
    correspond to "affine" or "hyperbolic" simply-laced Kac-Moody
    algebras. Even better, compact quotients of these tilings give some
    very nice modular curves, like Klein's quartic curve:

    8) John Baez, Klein's quartic curve,

    So, there a lot of connections to be made here... and I can tell I
    haven't made them all yet! Why don't you give it a try?

    To add to the fun, my friend Eugene Lerman has just written a
    very nice survey paper on orbifolds:

    9) Eugene Lerman, Orbifolds as stacks?, available as arXiv:0806.4160.

    This describes some deeper ways to think about orbifolds. For
    example, when we form an orbifold by taking the plane and modding out
    by a wallpaper group, we shouldn't really say two points on the plane
    become *equal* if there's a symmetry carrying one to the other.
    Instead, we should say they are *isomorphic* - with the symmetry being
    the isomorphism. This gives us a groupoid, whose objects are points
    on the plane and whose morphisms are symmetries taking one point to
    another. It's a "Lie groupoid", since there's a manifold of objects
    and a manifold of morphisms, and everything in sight is smooth.

    So, orbifolds can be thought of as Lie groupoids. This leads to
    the real point of Lerman's paper: orbifolds form a 2-category!
    This should be easy to believe, since there's a 2-category with

    groupoids as objects,
    functors as morphisms, and
    natural transformations as 2-morphisms.

    In "week75" and "week80" I explained the closely related 2-category
    with *categories* as objects; the same idea works for groupoids. So,
    to get a 2-category of *Lie* groupoids, we just need to take this idea
    and make everything "smooth" in a suitable sense.

    This turns out to be trickier than you might at first think - and
    that's where "differentiable stacks" come in. I should explain these
    someday, but not today. For now, try these nice introductions:

    10) J. Heinloth, Some notes on differentiable stacks, Mathematisches
    Institut Universitität Göttingen, Seminars 2004-2005, ed. Yuri Tschinkel,
    1-32. Available as http://www.math.nyu.edu/~tschinke/WS04/pdf/book.pdf
    or separately as http://www.uni-essen.de/~hm0002/stacks.pdf

    12) Kai Behrend and Ping Xu, Differentiable stacks and gerbes,
    available as arXiv:math/0605694.

    Now, besides the Alhambra, Granada also has a wonderful Department of
    Algebra. Yes - a whole department, just for algebra! And this
    department has many experts on categorical groups, also known as
    2-groups. So it's worth noting that there are 2-groups lurking in
    the Alhambra.

    Any object in any category has a group of symmetries. Similarly, any
    object in any 2-category has a 2-group of symmetries. So, any orbifold
    has a 2-group of symmetries. We should be able to get some interesting
    2-groups this way.

    The group of *all* symmetries of a manifold - its "diffeomorphism
    group" - is quite huge. That's because you can warp it and bend
    it any way you like, as long as that way is smooth. Similarly, the
    2-group of *all* symmetries of an orbifold will often be quite huge.

    To cut down the symmetry group of a manifold, we can equip it with
    a Riemannian metric - a nice distance function - and then consider
    only symmetries that preserve distances. We can get lots of nice
    groups this way, called "isometry groups". For example, the group E8,
    which has been in the news disturbingly often of late, is the
    isometry group of a 128-dimensional Riemannian manifold called the
    "octooctonionic projective plane".

    So, maybe we can get some nice 2-groups as "isometry 2-groups" of
    "Riemannian orbifolds". Of course, for this to make sense, we need
    to know what we mean by a Riemannian metric on on orbifold! I'm no
    expert on this, but I'm pretty sure the idea makes sense. And I'm
    pretty sure that the 2d orbifold we get from a specific wallpaper
    pattern has a Riemannian metric coming from the usual distance function
    on the plane.

    (Warning: the same wallpaper group can arise as symmetries of
    wallpaper patterns that are different enough to give different
    Riemannian orbifolds!)

    So, here's a potentially fun question: what 2-groups show up as
    isometry 2-groups of Riemannian orbifolds coming from wallpaper
    patterns? Try to work out some examples. I don't expect the answer
    to be staggeringly profound - but it sets up a link between the Alhambra
    and 2-groups, and that's cool enough for me!

    By the way, I obtained copies of some very interesting theses in Granada:

    13) Antonio Martinez Ceggara, Cohomologia Varietal, Ph.D. thesis,
    Departamento de Algebra y Fundamentos, Universidad de Santiago de Compostela.

    14) Pilar Carrasco, Complejos Hipercruzados: Cohomologia y Extensiones,
    Ph. D. theis, Cuadernos de Algebra 6, Departamento de Algebra y
    Fundamentos, Universidad de Granada, 1987.

    Antonio Cegarra was the one who brought 2-group theory to Granada,
    and Pilar Carrasco was his student. It's unfortunate that these
    theses come from the day before electronic typesetting. Luckily,
    Carrasco's was later turned into a paper:

    15) Pilar Carrasco and Antonio Martinez Ceggara, Group-theoretic
    algebraic models for homotopy types, Jour. Pure Appl. Algebra 75
    (1991), 195-235. Also available as
    http://www.ugr.es/~acegarra/Group [Broken] theoretic algebraic models for homotopy types.pdf

    This tackles the ever-fascinating, ever-elusive problem of taking the
    information in the homotopy type of a topological space and packaging
    it in some manageable way. If our space is connected, with a chosen
    basepoint, and it has vanishing homotopy groups above the 2nd, a
    2-group will do the job quite nicely! The same idea should work for
    numbers larger than 2, but n-groups get more and more elaborate as n
    increases. Carrasco and Cegarra package all the information into a
    "hypercrossed complex", and I would really like to understand this

    Carrasco and Cegarra's paper is quite dense. So, I'm very happy to
    hear that Carrasco plans to translate her thesis into English!

    Before I finish, let me mention one more paper about 2-groups:

    16) João Faria Martins, The fundamental crossed module of the
    complement of a knotted surface, available as arXiv:0801.3921.

    Martins was unable to attend the Barcelona workshop on 2-groups,
    but I met him later in Lisbon, and he explained some of the ideas
    here to me.

    A crossed module is just another way of thinking about a 2-group.
    So, translating the language a bit, the basic concept behind this
    paper is the "fundamental 2-group" of (X,A,*). Here X is
    a topological space that contains a subspace A that contains a point *.

    Here's how it goes. A 2-group is a 2-category with one object:


    a bunch of morphisms:


    (which must all be invertible), and a bunch of 2-morphisms:

    / \
    * T *
    \ /

    (which must also be invertible).

    So, we get the fundamental 2-group of (X,A,*) as follows:

    Let the only object be the point *.

    Let the morphisms be paths in A starting and ending at *.

    Let the 2-morphisms be homotopy classes of paths of paths in X.

    If you let A be all of X, you get the fundamental 2-group of (X,*),
    and this is what people mean when they say connected pointed homotopy
    2-types are classified by 2-groups. But the generalization is also
    quite nice.

    In the above paper, Martins uses this generalization, and a bunch of
    other ideas, to give an explicit presentation of the fundamental
    2-group of the complement of a 2-knot (a sphere embedded in 4d
    Euclidean space). In a certain sense this generalizes the usual
    "Wirtinger presentation" of the fundamental group of the complement
    of a knot. But, it's a bit different.


    Addenda: I should mention the definition of a wallpaper group: it's a
    discrete subgroup of the isometry group of the plane that includes
    translations in two linearly independent directions. We need the
    right equivalence relation on these groups to get just 17 of them:
    they're equivalent if you can conjugate one by an affine
    transformation of the plane and get the other.

    Alas, over at the n-Category Cafe Richard Hepworth has shown that all
    the isometry 2-groups of orbifolds coming from wallpaper groups are
    equivalent to mere groups. It's a pity! But, at least his remarks
    shed a lot of light on the general theory of isometry 2-groups.

    First he wrote:

    Here is a recipe for computing the isometry 2-groups of a Riemannian
    orbifold X//G, with G a discrete group acting on a connected simply
    connected manifold X. I think these are precisely the sorts of
    orbifolds you are interested in. Apologies for the nasty

    There is one object.

    The arrows are pairs (f,phi) where f is an isometry of X, phi an
    automorphism of G, and f is equivariant with respect to phi, i.e.
    f(gx) = phi(g)f(x).

    The 2-arrows from (f,phi) to (k,kappa) are the elements g of G for
    which k = gf and kappa = g phi g^{-1}.

    I'll leave you to work out the various composition maps.

    Of course, I haven't told you what a Riemannian metric on an
    orbifold (or groupoid or stack) is. You can find definitions in
    various places, but in the case in point these things are all
    equivalent to putting a G-invariant metric on X. (Does one always
    exist? Yes, but the fact that G acts with finite stabilizers is
    essential here.)

    I also haven't said how I get the above answer. If you already knew
    the answer for X = point then the above would be your first guess. The
    reason the guess is correct is thanks to the assumption that X is
    connected and simply connected: all G-valued functions on X are
    constant, and all G-bundles are trivial.

    Maybe if you're interested I can go into more detail.

    Then he wrote:

    Hmm, John and David's discussion has reminded me of a result I once

    Suppose we are given a morphism f:X -> Y of orbifolds and a
    2-automorphism phi:f => f. Then phi is trivial if and only if its
    restriction to any point of X is trivial.

    One consequence of this is the following:

    Let X be an "effective" orbifold. Then any self-equivalence of X
    has no nontrivial automorphisms.

    And in particular:

    The isometry 2-group of an effective orbifold is equivalent to a group.

    What does "effective" mean? It means that the automorphism group of
    any point in X acts effectively on its tangent space, and
    consequently that almost all points of the orbifold have no inertia
    at all. This includes all of the orbifolds you are discussing. The
    fact that the isometry 2-groups are all equivalent to groups is
    apparent in the description I gave earlier: if we have a 2-arrow
    g:(f,phi) => (f,phi) then gf = f, so that g is the identity.

    Of course, there are many interesting noneffective orbifolds. Some
    of them are called gerbes. (Gerbes with band a finite group.) Maybe
    you want to compute their isometry 2-groups? Here's a fact for you:

    The isometry 2-group of the nontrivial Z/2-gerbe over S^2 is itself
    a Z/2-gerbe over O(3), the isometry group of S^2. In particular, it
    is not equivalent to any group.

    The gerbe over O(3), however, is trivial. But is there a trivialization
    that respects the 2-group structure?

    Actually not *all* the orbifolds coming from wallpaper groups are
    "effective" in the above sense. For the orbifold to be effective,
    I believe the corresponding group must include eitherreflections or
    glide reflections across two different axes, or a rotation by less
    than 180 degrees. For example, the torus and "cylinder" described
    above are noneffective orbifolds. But, most of the interesting
    examples are covered by Hepworth's results, and the others seem also
    to have isometry 2-groups that are equivalent to mere groups.

    For details try:

    17) Richard Hepworth, The age grading and the Chen-Ruan cup product,
    available as arXiv:0706.4326.

    Richard Hepworth, Morse inequalities for orbifold cohomology,
    available as arXiv:0712.2432.

    According to Hepworth, the first paper "contains the little fact that
    implies that the automorphism 2-group of an effective orbifold is
    equivalent to a group". Of course it contains lots of other stuff,
    too! The second discusses Morse theory on differentiable Deligne-
    Mumford stacks (these are the proper etale ones). It defines Morse
    functions, vector fields and Riemannian metrics on differentiable DM
    stacks. It also proves that Morse functions are generic and that
    vector fields can be integrated.

    You can see more discussion of this Week's Finds at the n-Category


    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


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    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Jul 31, 2008 #2
    In sci.physics.research message <g6n52k$9u9$1@glue.ucr.edu>, Tue, 29 Jul
    2008 16:22:53, John Baez <baez@math.removethis.ucr.andthis.edu> posted:
    >Also available at http://math.ucr.edu/home/baez/week267.html
    >July 23, 2008
    >This Week's Finds in Mathematical Physics (Week 267)
    >John Baez
    >After the workshop on categorical groups in Barcelona, I went to
    >Granada - the world capital of categorical groups! Pilar Carrasco,
    >an expert on this subject, had kindly invited me to spend a week
    >there and give some talks. Even more kindly, she put me in a hotel
    >right next to the Alhambra. So, this week I'll tell you about some
    >categorical groups I saw in the Alhambra.
    > ...

    Du Sautoy, Marcus Finding moonshine : a mathematician's journey through
    symmetry / Marcus du S . - London : Fourth Estate, 2007
    includes his observations on Alhambra groups, IIRC.

    (c) John Stockton, near London. *@merlyn.demon.co.uk/?.?.Stockton@physics.org
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