In week266 of This Week’s Finds visit Io, the volcanic moon of Jupiter:
Then read about Pythagoras, the Pythagorean tuning system, the tetractys, and the categorical groups workshop in Barcelona.
Posted at June 20, 2008 12:01 AM UTCTrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1720
I like that idea about octahedra as tops. I have no idea whether the ancient Greeks used octahedra as tops. It seems unlikely… but I wouldn’t be surprised if Scots did it in 2000 BC.
For anyone who read week266 as soon as it went up: I’ve added links to the slides from Derek Wise’s talk, and also a better explanation of the long exact sequence of homotopy 2-groups associated to a fibration.
How good to see that handwriting again, on page 14.
Well said.
Just a quick comment for now: Dave Benson’s book Music: a Mathematical Offering has a nice couple of chapters on temperaments. The book is freely available online.
Thanks for pointing that out! This book gives a nice organized introduction to the wild jungle of musical intervals with scary names like ‘syntonic comma’, ‘septimal diesis’, ‘schisma’ and ‘diaschisma’.
I only wish it included sound files. I get really frustrated reading about all sorts of subtle differences in tuning systems without being able to hear them. That’s why I included links to some websites with sound files.
I have an electronic piano that plays in a variety of tuning systems, including Pythagorean tuning and just intonation. At first I was curious as to whether I could hear the difference between these systems and the standard one for pianos — equal temperament.
It turned out to be easy: every tuning system except equal temperament sounded horribly out of tune! Very simple chords could be okay, but anything harmonically complex was a nightmarish mass of dissonant overtones.
Then, after listening to another tuning system long enough, equal temperament started sounded horribly out of tune too…
At that point I became rather upset: had I permanently poisoned my experience of listening to music? Would I need to go on a purgative diet of Gregorian chants, nothing more spicy than a major fifth?
No, the effect went away on its own after a short while.
But, I see why fans of just intonation and other tuning systems can become a bit fanatical about these issues. Imagine hearing all commercial music as horribly out of tune, with the grinning idiots listening to it happily unaware…
John said
I only wish it included sound files. I get really frustrated reading about all sorts of subtle differences in tuning systems without being able to hear them.
Indeed. That was a cause of frustration for me as well. I did learn a bit of csound (which I discovered through Benson’s book) so that I could listen to some of the temperaments myself. I can’t remember what I did with the files now…
I always meant to suggest to Dave Benson that he include some samples.
Conc. continued fraction expansions and irrationality proofs build on them, my guess is that this originated in squaring rectangles and was only much later related to the dodecahedron. An other guess is that the interest in philosophizing on that comes from the use of analogies similar to numerical proportions in rhetorics.
Here is a board game of possibly Pythagorean origin.
Here is a link conc. the reconstruction of Empedocles’ ideas acc. to Kranz (after he continued the Diels, Kranz edition of Presocratic fragments).
Wow! That board game was really complicated! The rules are baroque and mathematical:
* Common Victories:
o De Corpore: If a player captures a certain of pieces set by both players, he wins the game.
o De Bonis: If a player captures enough pieces to add up to or exceed a certain value that is set by both players, he wins the game.
o De Lite: If a player captures enough pieces to add up to or exceed a certain value that is set by both players, and the number of digits in his captured pieces’ values are less than a number set by both players, he wins the game.
o De Honore: If a player captures enough pieces to add up to or exceed a certain value that is set by both players, and the number of pieces he captured are less than a certain number set by both players, he wins the game.
o De Honore Liteque: If a player captures enough pieces to add up to or exceed a certain value that is set by both players, the number of digits in his captured pieces’ values are less than a number set by both players, he wins the game, and the number of pieces he captured are less than a certain number set by both players, he wins the game.* Proper Victories:
o Victoria Magna: This occurs when three pieces that are arranged are in an arithmetic progression.
o Victoria Major: This occurs when four pieces that are arranged have three pieces that are in a certain progression, and another three pieces are in another type of progression.
o Victoria Excellentissima: This occurs when four pieces that are arranged have all three types of mathematical progressions in three different groups.
and yet:
The game was well enough known as to justify printed treatises in Latin, French, Italian, and German, in the sixteenth century, and to have public advertisements of the sale of the board and pieces under the shadow of the old Sorbonne.
I really like the suggestion that this game inspired Hermann Hesse’s “Glass Bead Game” — one of my favorite fictional games.
On page 2 of Derek Wise’s slides, he writes that any automorphism of is essentially described by automorphisms of the basis set. What changes for the case of a finite-dimensional 2-Hilbert space? Surely the only new feature is the inner products on the hom-sets… but morphisms of vector spaces are the same as morphisms of Hilbert spaces, so I don’t see how that changes things.
Jamie wrote:
On page 2 of Derek Wise’s slides, he writes that any automorphism of is essentially described by automorphisms of the basis set. What changes for the case of a finite-dimensional 2-Hilbert space?
The right person for this question is Bruce Bartlett, since he’s writing a paper on representations of groups on finite-dimensional 2-Hilbert spaces.
But: a finite-dimensional 2-Hilbert space is a finite-dimensional 2-vector space with extra structure, so I think the automorphism group gets even smaller — if we only include ‘unitary’ automorphisms, that is. A finite-dimensional 2-Hilbert space has a basis of objects, each of which has a positive real ‘norm’. I bet any unitary automorphism is equivalent to a permutation of basis elements that maps each element of the basis to another one with the same norm.
Hi Jamie,
Yes indeed one wants to work with unitary 2-representations - so one wants to work with unitary autoequivalences of a 2-Hilbert space. By that I just mean a linear -functor from the 2-Hilbert space to itself which is a unitary map at the level of hom-sets (John didn’t mention this idea in his paper on 2-Hilbert spaces… though he had considered it). As John said, when you get down to basics that just means a permutation of the simple objects which preserves scale factors. So if our 2-Hilbert space was Rep for a finite group , then a unitary autoequivalence is one which can only permute irreducible reps having the same dimension amongst each other. In practice all “geometric” autoequivalences will have this property.
Unfortunately, Lie 2-groups don’t have many representations on 2-Hilbert space of the sort I’ve secretly been talking about so far: that is, finite-dimensional ones.
So we may, perhaps, need to ponder representations of Lie 2-groups on infinite-dimensional 2-Hilbert spaces.
There is a nontrivial useful and interesting 2-representation of every strict 2-group for every rep of . This rep factors through
but not in general through the smaller
where is the 2-category of 2-vector spaces of the form , i.e. the full sub-2-category of on the algebras which are direct sums of the ground field.
I mentioned that before, but now it is also on p. 31 of arXiv:0806.1079 :-)
Yes, I like your thoughts on . Also, every strict Lie 2-group has an interesting representation in the 2-category of 2-term chain complexes: the adjoint representation.
But, thanks in large part to Derek’s talk, I’ve become convinced that representations in are also very interesting — because they’re very geometrical in flavor. All the talk of measure spaces can obscure the fact that in nice cases, these measure spaces are often manifolds! So, a 2-representation of a Lie 2-group in should be seen as a kind of hybrid of a unitary group representation and an action of the Lie group on a manifold.
The conventional theory is that the universe is made of, not 4 elements, but 4 whatchamacallits: Matter, Energy, Dark Matter, and Dark Energy. Cosmology becomes the new Alchemy. The use of the term “quintessence” in cosmological literature makes this explicit.
It’s worth noting that, in the Orient, the universe was seen as being made of 5 elements (as summarized by wikipedia):
Note that the five elements are chiefly an ancient mnemonic device for systems with 5 stages; hence the preferred translation of “Phase” over “Element”.
The elements are:
* metal (Chinese: 金, pinyin: jīn, )
* wood (Chinese: 木, pinyin: mù)
* water (Chinese: 水, pinyin: shuǐ)
* fire (Chinese: 火, pinyin: huǒ), and
* earth (Chinese: 土, pinyin: tǔ, ).
The system of five phases was used for describing interactions and relationships between phenomena. It was employed as a device in many fields of early Chinese thought, including seemingly disparate fields such as geomancy or Feng shui, astrology, traditional Chinese medicine, music, military strategy and martial arts.
Traditional Taijiquan schools relate them to footwork and refer to them as five “steps”. The system is still used as a reference in some forms of complementary and alternative medicine and martial arts. Some claim the original foundation of these are the concept of the Five Cardinal Points.
There’s a table of diagrams in a box on the right of the wikipedia page which is missing arroes, but lists:
the 4 Greek elements (+1); Hinduism (Tattva) and Buddhism (Mahābhūta); Japanese (Godai); Tibetan (Bön); and finally the one the page was built around: Chinese (Wu Xing).
Concerning Derek Wise’s slides
How different is the 2-category from the 2-category whose
objects are function algebras (to be thought of as placeholders for the categories of modules vector bundles over )
morphisms are bimodules of these (to be thought of as vector bundles over correspondence spaces)
and 2-morphisms bimodule homomorphisms between these?
I.e. the full subcategory of on objects of the form or maybe, depending on taste, not the full subcategory but some suitably well behaved part of it?
Urs wrote:
How different is the 2-category Meas from the 2-category… ?
Morally they are quite similar — and by suitable adjustment of details concerning either Meas and/or your idea, we could probably make them the same.
One thing we’d need is a theorem that says some category of measurable spaces is equivalent to some category of commutative algebras — with the space corresponding to the algebra of the bounded measurable functions on . The classic Gelfan’d–Naimark theorem says commutative C*-algebras correspond to locally compact Hausdorff spaces… but that’s about topology, not measure theory. Here we’d probably need some sort of commutative von Neumann algebra. I vaguely remember some theorems like this from my misspent youth as an analyst.
We could, in fact, just use the category of commutative von Neumann algebras… and let the spaces be whatever sort of space is the spectrum of this kind of algebra. A Stone space? I forget.
One point of Meas is that we want to describe categories that are closed not just under direct sums but also direct integrals. A classic example is the category of unitary representations of a noncompact topological group. The current description of Meas is just a step in this direction… not, I think, the ultimate ideal thing!
My understanding is that indeed commutative von-Neumann algebras are to measure spaces just like commutative -algebras are to compact topological spaces; I got this from the wikipedia article.
We could, in fact, just use the category of commutative von Neumann algebras…
That sounds good to me: I think one should use the bicategory of comm. von Neumann algebras and bimodules between them.
For instance there is the “canonical” 2-rep of the String Lie 2-group on the vonNeumann algebra generated from any postive energy rep of , which reproduces essentially the constructions originally considered by Stolz&Teichner.
The only subtlety is here that one needs to be sure that the Connes fusion tensor product still satisfies the familiar law
for regarded as a bimodule over itself with the right action twisted by the automorphism and with denoting the Connes-fusion product.
But, apparently, one way to understand Connes fusion is to say that it is precisely such that this property holds.
and let the spaces be whatever sort of space is the spectrum of this kind of algebra.
I can second Bruce’s reply to that: von Neumann algebras are precisely the algebras of functions on measure spaces. See for instance top of p. 54 in Wiaeo?
Urs wrote:
I can second Bruce’s reply to that: von Neumann algebras are precisely the algebras of functions on measure spaces.
Thanks! Sorry for taking so long to reply — I’ve been very distracted by travel and conferences.
This sounds great: someday I’ll want to reformulate the theory of infinite-dimensional 2-Hilbert spaces in terms of the bicategory of commutative von Neumann algebras, bimodules, and bimodule homomorphisms.
I’ll need to look at some analysis books to do this correctly. For example, Dixmier’s notion of ‘field of Hilbert spaces’ on a measure space should correspond to some sort of Hilbert module over the commutative von Neumann algebra of functions on , but I’m not sure if I need to say ‘projective module’… are all modules over a commutative von Neumann algebra projective? That’s the sort of question I need help with.
Nonetheless, this approach should make everything much more elegant in the long run.
One nice thing about the 2-group conference in Barcelona is that we established a close connection between Bruce Bartlett’s talk on weak 2-representations of finite groups on finite-dimensional 2-Hilbert spaces and Derek Wise’s talk on strict representations of arbitrary groups on arbitrary-dimensional 2-Hilbert spaces. Bruce has gone much further in understanding the geometrical aspects of this theory, but Derek and his coworkers have gone deeper into the analysis of infinite-dimensional 2-Hilbert spaces. It should all fit together very nicely in a beautiful big theory!
I’m happy that neither Bruce nor Derek expressed any objection to the idea of joining forces and working on this big theory. Of course, we’re all quite busy, so it will be a long time — possibly infinite — before we actually write a joint paper. But, it’s a nice prospect to contemplate.
This could rebuild the foundations of Quantum Mechanics, unifying the von Neumann approach with n-Categorification. Nice!
“a long time — possibly infinite — before we actually write a joint paper” – consistent with there being many people with Erdos number = infinity. The issue becomes what is one’s Baez number. John Baez has that set at 0. His coauthors, such as Greg Egan, have that as 1. Their coauthors have that as 2, and so, by induction…
A couple of typos:
“a 2-group is a 2-group with …”
“…its cokernel says us if it’s onto”
Fixed! Thanks!
I also added a mention of what we need besides the ‘kernel’ and ‘cokernel’ of a group homomorphism: the ‘pip’ and ‘copip’. Just some tantalizing jargon…
Hi John, thanks a lot for the blurb; just one quick point, where it says
…is that the category of representations of a finite group G is equivalent to some category where an object X is a Kaehler manifold on which G acts…
There isn’t a requirement for the complex manifold to be Kaehler - just that it be compact and that it comes equipped with an arbitrary hermitian metric. One needs the metric simply for integration purposes - so that the space of sections is a Hilbert space, and that one can compose kernels.
Here icosahedral news.
Conc. Pythagoras’ strangeness, Dodds “The Greeks and the Irrational” contains a very interesting chapter connecting it to shamanism. Personality traits related to shamanism and their possible postioning in human development are analyzed by Yuri I. Manin in several of his Essays.
Wow, so glass is made of icosahedra! That’s interesting news.
Now that I’ve stopped going to conferences I finally have time for some addenda to “week266”:
First, the slides for Aurora del Río’s nice talk are — as for many of the talks —- now available from the website of the Barcelona categorical group workshop:
25) Aurora del Río, Algebraic K-theory for categorical groups, http://mat.uab.cat/~kock/crm/hocat/cat-groups/slides/delRio.pdf
I also found a link to the paper she and Antonio Garzón wrote on this topic:
26) Antonio Garzón and Aurora del Río, On algebraic K-theory categorical groups, http://www.ugr.es/~agarzon/K-thCG.pdf
Second, Bruce Bartlett pointed out that to get something equivalent to the category of representations of a finite group G, he doesn’t require that his manifolds be Kähler: it’s enough that they be complex manifolds equipped with a Riemann metric. Personally I greatly prefer Kähler manifolds, so I hope it’s enough to use those!
Third, writing “week266” caused me to miss Behrang Noohi’s talk on using diagrams called “butterflies” to efficiently describe weak homomorphisms between strict 2-groups (in the guise of crossed modules). Bad move! Luckily Tim Porter summarized it here at the n-Category Café:
27) Timothy Porter, Behrang Noohi on butterflies and weak morphisms between 2-groups, available at http://golem.ph.utexas.edu/category/2008/06/behrang_noohi_on_butterflies_a.html
For more details, you can’t beat the original paper:
28) Behrang Noohi, On weak maps between 2-groups, available as arXiv:math/0506313.
Also here at the n-Category Café, Bruce Bartlett discussed Tim Porter’s talk at the categorical groups workshop:
29) Bruce Bartlett, Tim Porter on formal homotopy quantum field theories and 2-groups, available at http://golem.ph.utexas.edu/category/2008/06/tim_porter_on_formal_homotopy.html
Actually Porter gave two talks. The first was an introduction to simplicial methods and crossed complexes, but Bartlett didn’t summarize that, and no slides are available. So for that, you should get ahold of the following free book:
30) Timothy Porter, The Crossed Menagerie: an Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology, available at http://www.informatics.bangor.ac.uk/~tporter/menagerie.pdf
and (harder) this review article highly recommended by Porter:
31) E. Curtis, Simplicial homotopy theory, Adv. Math. 6 (1971), 107-209.
The second talk by Porter, the one Bartlett blogged about, can be found at the workshop’s website:
32) Timothy Porter, Formal homotopy quantum field theories and 2-groups, available at http://mat.uab.cat/~kock/crm/hocat/cat-groups/slides/Porter.pdf
This talk covered the papers by Martins, Porter and Turaev mentioned in “week266”.
I apologize to everyone whose talks I still have not mentioned!
Finally, just for fun, here are a few photos from the workshop:
Chenchang Zhu, with Carles Cascabuerta talking to student
in background
Timothy Porter and Bruce Bartlett
Aurora del Río, Alain Rousseau, Pilar Carrasco and André Joyal
(photo by Tim Porter)
Derek Wise, John Baez and Bruce Bartlett
(photo by Mathieu Anel)
Now that I’ve stopped going to conferences
Good to see you back here at the Café finally! :-)
And nice photos, too.
Second, Bruce Bartlett pointed out that to get something equivalent to the category of representations of a finite group G, he doesn’t require that his manifolds be Kähler: it’s enough that they be complex manifolds equipped with a Riemann metric. Personally I greatly prefer Kähler manifolds, so I hope it’s enough to use those!
By the way, this paper is now on the arXiv: The geometry of unitary 2-representations of finite groups and their 2-characters. After trawling the net for some time, let me offer my current understanding on the point you raised above.
My understanding is that you only need the manifolds to be Kähler when you actually want to explicitly compute something. That’s because it seems that the only game in town as far as computations go is the index theorem, and for this you need the manifold to be Kähler. Also if you want to work just at the level of the line bundle (as I do) and not at the level of the entire vector bundle of differential forms, I think you also need the curvature of the line bundle to be sufficiently positive so that the higher comology vanishes.
In other words: on the general level, to avoid cluttering up the simple underlying principle that “representations of groups are the same as equivariant line bundles”, one might prefer to stick with all line bundles over compact complex manifolds. By the way, at this level the statement is almost a big tautology (of the kind category theorists know and love!)
If you actually want to roll up your sleeves and do some real work - i.e. get out concrete formulas in terms of characteristic classes, etc. - then you need to restrict yourself to Kähler manifolds.
Unfortunately I think this means there is a small error in the introduction of my paper: only in the nice Kähler case can I actually prove that the ‘geometric character’ localizes over the fixed points - because then it’s just the Atiyah-Singer fixed point theorem. Nevertheless I think all the commutative diagrams I drew still hold in the most general setting.
Re: This Week’s Finds in Mathematical Physics (Week 266)
It is not difficult to use an octahedron as a spinning top. It “floats” like air. I did it with an 8-sided dice while playing RPG.
Maybe old greek children liked to play with spinning tops?
Kylix of Hermes and Youth Spinning Top
c. 480-470 B.C.E
Baltimore, The Johns Hopkins University, Archaeological Collection, B9
Ex Collection Hartwig, Rome
In Coming of Age in Ancient Greece, cat. 16
On this red-figure kylix, the painter Douris shows Hermes teaching a youth how to spin a top. Hermes (left) is not shown here with the kerykeion, the traditional wand, so it is a bit difficult to identify him for certain, but this image does resemble several in which he is shown with his traditional attribute.
As Hermes served the patron deity of the gymnasium, he was thought to have a particularly close association with young boys and was said to have invented the spinning top. Accordingly, many Greek male youth brought tops and other toys to sanctuaries to be dedicated to Hermes when they came of age and it became time for them to lay away these childish things.