Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Thursday, August 30, 2007


Tommaso Dorigo is being heavily criticised for his post about Lisa Randall. Although I understand the issue, the hypocrisy here is just running red down the walls! After two definitely-without-a-doubt on topic remarks (supporting Tommaso) on Asymptotia, I was put onto the moderation queue! So who is the worst offender here? The self righteous, politically correct heroes? Personally I'd rather discuss physics with a hot blooded Italian man who listens to what I say (and perhaps even gives constructive criticism) any day.

Sure, a world where one can have a decent conversation without worrying about anything would be nice, but let's be realistic here: that just isn't possible without some degree of equality.

Update: I'm told that moderation queues are automatic (sorry, Clifford) but if you read the comments on the relevant threads you might see my point.

M Theory Lesson 94

Both Heisenberg's honeycombs and Kapranov's Fourier transform suggest the use of non-commutative polynomial invariants. Now discussions with Matti Pitkanen about knot invariants has led to the idea of introducing non-commutativity for twisted ribbons, as in the original Mulase and Waldron symplectic matrix ensemble.

The Jones polynomial, the two variable homflypt polynomial and the Kauffman polynomial are all commutative. The most interesting invariant in this context is the Kauffman polynomial, defined in terms of the (planar isotopy) Kauffman bracket, because the latter has always been associated to the idea of diagrams as numbers. Is there a way to associate letters to (a skein relation for) twists that would extend these invariants to a non-commutative braided ribbon invariant? There is certainly an analogy between the three basic crossing diagrams ($L_{+}$, $L_{-}$ and $L_{0}$) for the homflypt polynomial and the three possible twist elements: left handed half twist, right handed half twist and flat ribbon.

As it turns out, Bar-Natan seems to be thinking about such things! As he says on his knot wiki, "If you know what this is about, good. If not, bummer."

Wednesday, August 29, 2007

M Theory Lesson 93

Back in Lesson 21 we looked at how holography suggests turning pants diagrams into little disc diagrams. The reverse process would be an operad inclusion map taking little discs to tube-like trees. This is in fact what was considered by Getzler in the paper recently discussed at The Cafe. In this paper, Getzler shows that BV algebras come from a little disc operad which can be included in an operad of moduli of genus zero surfaces (the pants pictures).

But it is not the ordinary little discs operad, where discs may be mapped into a larger disc allowing for translations and a dilation of each little disc. The BV disc operad also allows for a rotation of each little disc. That is, we allow a $\mathbb{C}^{\times}$ action on the disc. This extension of the usual little discs operad may be viewed in terms of braid groups. In fact, the little disc and framed little disc operads are just homologies:

$LD(k) = H_{\ast} (\mathbb{P}_{k})$, $FLD(k) = H_{\ast} (\mathbb{Z}^{k} \times \mathbb{P}_{k})$

where $\mathbb{P}_{k}$ is the pure braid group on $k$ strands. There is a fibration of the framed operad over the usual disc operad with a torus fibre $T^{k}$ which represents all the phases of the framed little discs. The factor of $\mathbb{Z}^{k}$ actually comes from a ribbon braid group, with generators for twisting the ribbons. This associates the rotation of a little disc with a twisting of ribbons, as previously discussed. Recall also that the connection to the topological moduli operad imbues the index $k$ (number of strands) with a dimensional meaning, since the dimension of the moduli spaces increases with the number of punctures $k$.

Tuesday, August 28, 2007

M Theory Lesson 92

In the 1960s Gian-Carlo Rota wrote a series of papers on the foundations of combinatorics, looking in particular at incidence algebras for locally finite posets. This is an algebra of functions $f(a,b)$ defined on closed intervals of the poset. Multiplication is given by convolution

$(f * g)(a,b) = \sum_{x \in [a,b]} f(a,x) g(x,b)$

This algebra includes the zeta functions $\zeta (a,b) = 1$ and their inverses the Mobius functions, as discussed in Leinster's paper on Euler characteristics for categories.

In the fourth paper in the series [1] Rota considered the analogue for vector spaces over finite fields, which are of some interest here since a category of vector spaces may be seen as a simple quantum analogue to the topos Set. The paper begins with q-analogues of binomial coefficients, namely the Gaussian coefficients, which count the rank $k$ subspaces of an $n$ dimensional vector space over the field of order $q$. It is then observed that by allowing $q$ to take a wider range of values, the limit of $q \rightarrow 1$ reduces q-identities to the classical case. The Mobius function $\mu (V,W)$ for elements of the lattice of vector subspaces is given by

$\mu (V,W) = (-1)^{k} q^{(k;2)}$

where $(k;2)$ is the binomial coefficient and $k = \textrm{dim} W - \textrm{dim} V$. The zeta function is defined by $\zeta (V,W) = 1$ when $V$ is contained in $W$, and $0$ otherwise. It is the inverse of the Mobius function in the incidence algebra for the $n$ dimensional vector space.

The close analogy with poset incidence algebras suggests an extension of Leinster's weightings for a category, which are defined in terms of the Mobius function on the object set incidence algebra with $\zeta (A,B)$ counting the arrows in Hom($A,B$) (a simple extension of the single arrow counted for posets). For a category enriched in Vect the Hom space is a vector space, and it is natural to extend $\zeta (V,W)$ to the dimension of the Hom space Hom($V,W$). Euler characteristics that count dimensions of linear spaces, rather than elements of a set, hopefully bring us a little closer to the knot invariants in their categorified homological guise.

[1] J. Goldman and G-C. Rota, Stud. Appl. Math. 49 (1970) 239-258

Monday, August 27, 2007

M Theory Lesson 91

Recall that knots have primes. For example, all (m,n) torus knots are prime. Mike Sullivan has shown that although the universal Ghrist template contains all knots, there exist simpler templates (such as the Lorenz one) for which all the knots are prime. Zeta functions for templates have been defined.

Is there a zeta function for knots? The question is what to use in place of the factor $p$. Presumably the correct choice is the Jones (or homflypt) polynomial, or categorified versions thereof (a polynomial is really a kind of number, anyway). Nothing prevents one from simply defining a zeta function

$\zeta (z) = \prod_{K} (1 - J(K)^{-z})^{-1}$

over prime knots $K$, and the Jones polynomial is functorial with respect to knot composition. Then if the Jones polynomial were itself a zeta function for the knot diagram, this would be an iterated zeta function. This looks horrible, but at least the number of prime knots of $n$ crossings is bounded above by $e^n$ (!!) and convergence might be reasonable since, for torus knots, the invariant $J(K)$ looks like an ordinary ordinal when normalised to the simplest knot in the series. That is, for negative integers $z$ the corresponding Euler series would be a summation of counting numbers, just as for the Riemann zeta function.

Saturday, August 25, 2007

Quotes of The Week

There have been attempts to observe time lags in gamma flares and in gamma-ray bursts, but we have never seen something like this....
said Daniel Ferenc of U.C. Davis, discussing the new MAGIC result.

The observation of this group of galaxies that is almost devoid of dark matter flies in the face of our current understanding of the cosmos
said Arif Babul of the University of Victoria, discussing the Abell 520 cluster.

Not only has no one ever found a void this big, but we never even expected to find one this size
said Lawrence Rudnick of U. Minnesota regarding the void that corresponds to a cold spot on the WMAP map.

The Concorde cosmology is ready to crash
said Louise Riofrio on her blog.

Friday, August 24, 2007

From The Old Man

The Old Man in a Cave reports on the latest news regarding arctic ice levels. As of August 22 there is only 3.22 million square kilometres of sea ice in the north: a new record low. It is said that the (supposedly alarmist) UN Panel on Climate Change predicted this level of ice for the year 2050, not 2007.

On a lighter note: Happy 8th Birthday, Blogger!

Thursday, August 23, 2007

M Theory Lesson 90

How does one discuss a category of operads when the intention was to define categories using operads in the first place? Such self referential questions appear to lurk behind the mystery of weak n-categories. We want an $\omega$-category of $\omega$-operads such that the categories defined as algebras are precisely what we get when we take the algebras of a special Koszul monad. Gee, that's already way too much mathematics. And duality won't do for all (categorical) dimensions: in logos theory there are n-alities, so we need a concept of Koszul n-ality. Fortunately, the importance of 2 to topos theory (a la locales and schizophrenic objects) is just the place we thought about extending dualities. So we want an $\omega$-category of $\omega$-operads such that an $\omega$-monad of Koszul n-alities gives weak n-categories as n-algebras. Sigh. Maybe we should return to pictures of trees and discs.

Aside: I don't like inventing new words for things, since there are so many definitions in mathematics as it is, so I will ignore all objections to the term schizophrenic object. After all, do dwarfs object to having stars named after them? If a term is descriptive, it is a good term.

M Theory Lesson 89

In an interesting exchange on the Cafe, Schreiber made the remark, "we are understanding that all these differential graded algebras are really just the Koszul dual incarnation of the Lie version of n-groupoids of physical configurations and/or states." In physical M theory we like to avoid phrasing everything in terms of complicated algebras, because the physics does not require this and the algebras arise naturally from operads anyway. The point is that Koszul duality is a fundamental process for operads themselves, playing the role of 2-ality for the n-logos Machian correspondence.

In a short paper on operads, Kapranov (who introduced modular operads with Getzler) discusses Koszul duals. Examples are the duals of the operads for associative, commutative and Lie algebras, which are respectively the associative, Lie and commutative operads. That is, associativity is self-dual but commutativity and Lie structures are interchanged. These operads are linear. More generally there is a functor $K$ on the category of operads (this is just about 1-operads) which has the property that $K^{2}$ is a kind of isomorphism. Sounds like a monad, doesn't it? Indeed, Kapranov says that one should think of $F$ as a type of cohomology theory on the category of operads. It's a bit like the power set monad for the topos Set, which is why the comment of the enigmatic David Corfield regarding duality based on the number 2 is particularly relevant here.

Another example of Koszul dual operads are the homologies of (a) uncompactified moduli spaces (actually varieties) of $n$ punctured Riemann surfaces of genus $g$ and (b) the compactified spaces.

So it would seem that the absolutely brilliant Schreiber is fast converging on the right mathematics, and he's doing it the hard way, without really thinking about the physics at all. Somebody (respectable) needs to tell him that there is no Dark Energy or no SUSY partners or no KK modes.

Wednesday, August 22, 2007

Battle Weary

Recent posts by bloggers Bergman, Mottle, Carroll and Woit suggest a mood of weariness in the physics blogosphere. The latter cannot even be bothered to write anymore and has resorted to phrases like, "blah, blah, blah, this pseudo-science is on hep-th because of blah, blah, blah." But the demise of the arxiv is nothing new and the game is just beginning! Heck, I've barely warmed up.

Tuesday, August 21, 2007

M Theory Lesson 88

The blogosphere is wonderful! In a comment at the Everything Seminar the remarkable Terence Tao points to a short article on the mathematics of Gian-Carlo Rota. On page 9 there is a brief comment on profinite combinatorics, a subject that Rota dreamed of developing. Unfortunately there is only a single obscure reference to Rota's own writing on the subject.

Given a finite field of order $q$, a continuous geometry in the sense of von Neumann is a profinite limit of (lattices of subspaces of) projective geometries

$P(1,q) \rightarrow P(2,q) \rightarrow P(4,q) \rightarrow \cdots \rightarrow P(2^{n}, q) \cdots$

and this limit contains subspaces of any dimension $d \in [0,1]$. This led Rota to consider the Riemann zeta function, and for those with library access the (two and a half page) paper to look at is:

K. S. Alexander, K. Baclawski, G-C. Rota
A stochastic interpretation of the Riemann zeta function
Proc. Nat. Acad. Sci. USA 90 (1993) 697-699

Wait, I found a reprint! A stochastic process $Z_s$ indexed by $s \in \mathbb{N}^{+}$ is found such that a probability distribution for $Z_{s} = n$ is given by $n^{-s} \zeta (s)^{-1}$. Section 3 discusses in general Mobius inversion for an infinite lattice (recall that inversion was crucial to Leinster's concept of Euler characteristic for a finite category). By specialising to the sequence of cyclic groups $\mathbb{Z}_n$ and the profinite integers (mentioned in the last lesson) one obtains the Riemann zeta correspondence for $Z_s$ a suitable random variable on the $s$-th power of the profinite integers.

Monday, August 20, 2007

The Birds and ...

Tegmark's recent arxiv paper on the mathematical universe contains some interesting insights, but I was most struck by the discussion beginning on page 3 of a comparison between the bird perspective and the frog perspective. This immediately brings to mind the figure of Aristophanes, the ancient dramatist, well known for his comedies. These plays often contained political parody. Two well known plays by Aristophanes are The Frogs (405 BC) and The Birds (414 BC). The former begins with a line from the character Xanthias, accompanying his master Dionysus to Hades:
Say, master mine, would you that I should crack one of those standing jokes upon the stage, which always make the tickled audience laugh.
Surprisingly, I have not seen this coincidence remarked upon in the physics blogosphere.

Sunday, August 19, 2007

Til A Proton Decay

The lack of suppression of spectral lines blueward of Lyman $\alpha$ for high redshift quasars indicates that most of the intergalactic hydrogen is ionised. That's a lot of protons to worry about. As Carl Brannen pointed out, protons may be thought of as mixtures of quark triples: uud or udu or duu. Similarly, a neutron is a mixture of triples such as the one drawn.
The numbers at the bottom indicate the number of twists on an (appropriately coloured) strand, each representing a charge of e/3. Thus the neutron has a total zero charge. In the full triple of diagrams there are three -1 charges to make up an electron, three 0 charges to make up an antineutrino and three +1 charges to make up a proton. On the other hand, for the proton each such diagram is labelled by charges 0,1 or 2. This cannot be split up into a positron and neutrino set because that would leave three strands with a +2 charge, which certainly could not make up a neutron.

Saturday, August 18, 2007

M Theory Lesson 87

In a 2002 paper, Kennison proves the following result. First, let Stone be the category of Stone spaces and Bool be the category of Boolean algebras. There is a categorical equivalence $P$ from Bool to Stone which takes the space of all points in an object $B$. An arrow $t: X \rightarrow X$ in Stone is called Boolean cyclic if the corresponding arrow in Bool is cyclic, which means that the supremum over all equalisers of $t^{n}$ and $1_{X}$ is just $X$. Intuitively, these represent dynamical processes that eventually cycle. Now let $\mathbb{N}$ be the ordinals. An action is a map $\mathbb{N} \times X \rightarrow X$ given by $(n,x) \mapsto (t^{n}(x))$. Kennison showed that the property of being Boolean cyclic was equivalent to actually having an action by Z, the profinite integers, namely the product over ordinal primes $\prod \mathbb{Z}_{p}$ of the p-adic integers.

This is interesting because J. Borger has been looking at finite sets with Z actions in order to characterise $\Lambda$-rings in terms of finite sets with actions of $G \times \mathbb{N}^{+}$ for $G$ the absolute Galois group for the rationals. Recall that this group acts on Grothendieck's ribbon graphs. A $\Lambda$-ring structure is a series of arrows $f_{p}: R \rightarrow R$ for a ring $R$. For example, one may take

$R = \frac{\mathbb{Z} [x]}{x^r - 1}$

along with the Frobenius maps $\psi_{p}: x \mapsto x^p$. Borger et al show that certain nice $\Lambda$-rings can only be a field if they are in fact the rationals. It turns out that all nice $\Lambda$-rings are subrings of products of the cyclotomic example above, for some $r$. This seems to be important somehow...

Thursday, August 16, 2007

M Theory Lesson 86

Smolin's slides from Loops07 are now available. Skip the stuff about The Dark Force and look in particular at slide number 38. The important thing to note here is that (a) The Loopies have enlisted the help of none other than Louis Kauffman, an absolutely brilliant knot theorist, and (b) Kauffman has invented something called the Kauffman numbers for three stranded braids, which do this: turning elementary braids into codes of the kind that appear in Carl Brannen's version of the Standard Model (eg. see Carl's comment here). Thus it appears there is a growing consensus that the three generations arise not from more complicated knots, as originally proposed, but rather from the kind of combinatorics that appear in M Theory. Category Theory is not mentioned at all in this work, despite the increasing usage of both knot theory and quantum information language.

Update: Carl Brannen points out that his scheme for the generations is far more advanced than the one outlined in Smolin's talk in later slides. I would have to agree.

Towards the Light

Jason D. Padgett, a student at Tommaso Dorigo's blog, who has to my knowledge previously been ignored, left an interesting comment, an edited version of which I shall post here.
Hello again, thanks for the comment. Some were not as kind ... The equation I am referring to comes from the Planck constant and the pure geometry of space time. What I did initially was to try to draw ... the structure of space time keeping in mind Planck’s constant. In other words ... every time you move through space you must move exactly one or whole multiples of one Planck constant.
Don't take this literally to mean spacetime, but think of it as a local model for spacetime, in which there is a proper notion of local Planck scale. Carrying on:
Planck constants also vibrate at the speed of light. What causes the Planck constants to vibrate is uncertainty ... Anyway, once you have a grid drawn with Planck lengths and you start them vibrating (from uncertainty) you will see that at specific points the Plancks will collide with each other at the speed of light ...
OK, so this is kind of fun, but is it telling us something interesting? Next we have:
When you draw this diagram the only shape that space time can take is a 2 dimensional hexagon or 3 dimensional cube ...
Heisenberg's hexagons from fractals! Now that's cool. I'm hoping to take a closer look at his work.

Wednesday, August 15, 2007

M Theory Lesson 85

It would be interesting to look at Zagier's conjecture

$d_n = d_{n - 2} + d_{n - 3}$

using associahedra. After all, the relations between MZV values come from the decomposition of an associahedron face as a product of lower dimensional associahedra. For example, the square faces of the 14 vertex K4 polytope arise as products of the K2 intervals, which represent a basic associator. Thus 1-operad combinatorics gives a way to count MZV relations. The dimension of an MZV space of weight $n$ should be related to the difference between the total number of possible arguments (the ordered partitions of n) and the conditions imposed by the combinatorics.

Euler considered a generating function for the partition function, namely $P^{-1}$ for

$P(x) = \prod_{1}^{\infty} (1 - x^n)$

For $n = 3$ we have $P_3 = 3$ since 3 may be written as 1+1+1 or 1+2 or 3. Subtracting 2 kinds of relation for the K4 faces leads us to suspect that $d_3$ is in fact one. That was very rough, but it is nice to think about how generating functions for MZV spaces relate to generating functions from operads.

The Dark Side

Louise Riofrio offers yet another reality check on The Dark Side, inspired by the reappearance of Lunsford in the blogosphere, and Carl Brannen continues with the ABC of the Standard Model. Meanwhile I have been a little lazy, having just discovered an amusing puzzle (diverting even despite its repulsive popularity).

Monday, August 13, 2007

M Theory Lesson 84

I was hoping somebody could point me to papers on 2-categories and double shuffle relations for MZV algebras. These arise from the two shuffle products, on series or on integral forms, of zeta values. Both of these products may be thought to arise from associahedra combinatorics, as discussed in the work of Brown. A recent paper, by Zagier et al, extends these relations to a set that can characterise the full algebra of MZVs. It does so by introducing an infinite series

$A(x) = exp(\sum_{2}^{\infty} \frac{-1}{n} \zeta (n) x^n)$

with zeta value coefficients. New results include a proof that the weight 3 and 4 zeta values form one dimensional algebras. For example,

$4 \zeta (3,1) = \zeta (4) = \zeta (3,1) + \zeta (2,2) = \zeta (2,1,1)$

There is a conjecture due to Zagier which states that the dimensions of the $\mathbb{Q}$ vector spaces for weight $n$ obey the recurrence relation

$d_n = d_{n - 2} + d_{n - 3}$

with $d_{0} = 1$ and $d_{1} = 0$. The appendix discusses the weight $n$ and depth $d$ generalisation due to Broadhurst and Kreimer, which I believe appeared in the Feynman diagram paper listed as a reference.

Sunday, August 12, 2007

Quote of the Fortnight

From D. R. Lunsford on Not Even Wrong:
Anthropocentrism is easily understood as the natural expression of the narcissistic era we inhabit.
It's good to see Lunsford back in the blogosphere after a long absence. Here is his 2003 paper on a (3,3) space (often cited by visitors to this blog, a number of whom are of course banned by the arxiv preprint server). The abstract states that the cosmological constant must vanish. Very sensible. When I told D. Wiltshire about this possibility back in 2002, in terms of varying $c$ and $\hbar$, he immediately scoffed and said that concordance nucleosynthesis ruled out such possibilities. Now he constantly receives the same reaction from many colleagues as he tries to convince them that Dark Energy is a false road (but of course I'm still crazy). As Louise Riofrio often tells us: concordes are prone to crash.

Saturday, August 11, 2007

Edge of The Ice

I haven't written about climate change for a while since our colleague Mottle is doing such a fine job of promoting awareness of this issue. Now a New York Times story about the current area of arctic ice has prompted a round of posts from blogs like Real Climate, Weather Underground, Old Man in a Cave or Only In It For The Gold.

A record low ice area was recorded on August 9. The remark in the news that caught my attention though was the following: last week the Russians planted a flag on the seabed at the North Pole. The Canadian prime minister then noted that, "international interest in the region [was] increasing." Great! So we're all screwed folks, but don't worry, there's plenty more oil under the melting ice!

Of course the issue is not about longer summers or more oil: it is about water. If the arctic ice melts, the land locked ice probably will also. A one millimetre rise in sea levels probably won't radically alter the real estate value of Joe's waterfront mansion, but that's already a lot of fresh water. And if the Greenland ice sheet melts, Joe's house will be way under water. There's water in the oceans, lakes, rivers, atmosphere and ice: take some from somewhere and it goes somewhere else. Pretty simple. Then again, as one cheerful commenter pointed out, there might be a worldwide famine in 3 years time when the farmers all plant the wrong crops due to the unpredictability of the weather for the season ahead (or the fact that the farm turned into a desert) so we might not need so much water (because a lot of us will be dead).

image source

Friday, August 10, 2007

End of The Universe

The latest comment by U. Schreiber in regard to the Crans paper is, "It rather neatly provides the right formal language for holography, it seems. I am enchanted."

Now Schreiber's view of the underlying physics still appears, mysteriously, to be quite conventional. He talks about stringy holography as if the boundary field idea is a principle in its own right. Instead, we originally considered the Gray tensor product in the context of a heirarchical quantum Machian principle (completely independently of gauge theories) where boundaries arise through a consideration of twistor causality. Anyway, there is a growing consensus (at least up to two people) that this dimension raising aspect of weak n-categories is crucial to quantum gravity.

It will be interesting to see how the mathematics converges: a calm process from a mathematical point of view, but perhaps a cataclysmic process from a physical one. Meanwhile I received a very polite letter from McDonalds declining my job application, so fortunately I am again in gainful employment marking large piles of assignments on Fluids and Thermodynamics.

Thursday, August 09, 2007

Blogrolling On IV

Now the function $J(1,y)$ really likes to be a complex number. Since it is always of norm 1 when the discriminant is non-positive, it is only real when $J = \pm 1$. Note that $J(1,y)$ may be generalised yet again using the rule

$xn J^2 + xy J + n = J$

which is also designed to give a norm 1 complex number at $x = 1$. In this case, the condition for $J(1,y) = \pm 1$ is

$y = 1 \pm Q$

for $n^2 = Q^2 - N^2$ where $Q$ and $N$ are whole numbers. When $n = 1$ this reduces to the previous case of $N = 0$, but a general Pythagorean triple gives a solution for $y$. Note that each such choice of $y$ leads to a new Abel series with coefficients somehow depending on $n$ and all of these series sum to $\pm 1$ although the substitution rule is different for each choice of $n$. Other choices for $y$ give roots of unity. For example, when $y = 6$ and $n = 3$ we find that $J = \frac{1}{6} [- 5 \pm \sqrt{11} i]$.

Update: Here is the original post on the Everything Seminar, with nice examples of series summing to -1.

Wednesday, August 08, 2007

M Theory Lesson 83

Underlying the Abel summation game is the concept of stuff type, discussed in the context of quantum mechanics by Baez and Morton.

Now as Tony Smith points out, we are not necessarily interested in groupoids as a basis for cardinality. With types as functors into Set, the source is the category FinSetB of finite sets and bijections, which looks a lot like the ordinals $\mathbb{N}$ except that there are many sets representing each $n \in \mathbb{N}$. Is this a nice category? Not really. In contrast, the category FinSet of finite sets and all functions is a topos: it is closed under Cartesian product, function sets Hom(A,B) act as exponential objects, and the usual characterising squares for Boolean logic are still pullbacks. A functor $G$ from FinSet to Set is a functor between toposes, albeit not a special topos kind of functor.

Could we replace a functor $F$ from FinSetB to Set with a functor $G$? It doesn't look like there is any nice way to do this: FinSet has an awful lot more arrows in it than FinSetB, because there are a lot of functions between two sets that don't happen to give a bijection. Nonetheless, it looks like an interesting question about the universality of a functor $I$ from FinSetB to FinSet, and if we view this question from the undefined world of higher topos (or logos) theory then perhaps there is a way to axiomatise a category of categories so that such a special functor exists.

Note also that a major motivation for the higher logos theory was the chance to view categorical dimension itself as cardinality in an abstract sense, which means a constructive approach to weak n-category theory as a whole. I think the Crans paper (which Schreiber is suddenly keen on) is one of few interesting attempts to understand higher categories in this way.

Tuesday, August 07, 2007

From the Front

Well, Abel sums don't seem to be too popular, so how about some real news from the front: Tommaso Dorigo reports on new D0 results at odds with non-relativistic QCD, namely upsilon polarisation. Since I have no problem with filching pictures, the relevant graph is this one: The new black data points for the Y(1S) state (9460 MeV $c^{-2}$), along with some older CDF points, show a behaviour for the polarisation parameter $\alpha$ which is very different from the yellow NRQCD band. The Y(2S) state is consistent with NRQCD. Note that the fourth CDF point is consistent with the D0 result, but not simultaneously with NRQCD, whereas the central two CDF points appear to be at odds with the D0 results, which show a clear momentum dependence for polarisation. The large negative offset is consistent with a significant longitudinal polarisation.

In case people are wondering, the relative locations of CDF and D0 are shown on this Fermilab map. The nearby town is situated to the West. I'd like to visit one day.

Monday, August 06, 2007

Blogrolling On III

Note that if we always set $x = 1$, $J(x,y)$ has a non-zero imaginary part when $-1 < y < 3$. If $y = 2$, and thus $z = 2x$, the coefficients take the form

$C_{i} = 2^i S_i$

for the Schroeder numbers $S_i$. Then $J$ is a cubed root of unity $\omega$, and the generating function corresponds to the rule

$J = 1 + 2J + J^2$

Can we prove a law $J^4 = J$? Similarly, if $y = 3$ and $J = -1$ the rule would be $J = 1 + 3J + J^2$. Can we interpret these rules in terms of trees? Let's write the first rule as $J = 1 + J + J + J^2$. This looks a lot like the Motzkin rule, except for the extra factor of $J$. What if we distinguished left and right branches for Motzkin trees? That is, take a full binary rooted tree template and count whole trees with 0, 1 or 2 branches that may be fitted to the template. Then the desired rule works by differentiating left and right unary branches from the root. Instead of a fivefold bijection of the set of Motzkin trees, there is now a fourfold mapping, and the series

$1 + 4 + 24 + 176 + 1440 + \cdots = \omega$

Motzkin numbers $M_i$ are also given in terms of trinomial coefficients $T(n,1)$. The $T(n,0)$ coefficients go back to Euler. These are the number of permutations of $n$ ternary symbols (-1, 0, or 1) which sum to 0. A general $T(n,k)$ is the number of permutations of these symbols that sum to $k$.

Sunday, August 05, 2007

Blogrolling On II

Recall that binary trees satisfied the Catalan relation $T = 1 + T^{2}$, whereas Schroeder numbers, or Motzkin numbers, satisfy the substitution rule $X = 1 + X + X^{2}$. In this paper, Cossali studies a generalisation of the generating function for the Catalan numbers $C_{n}$. Let

$L(x,z) = \sum_n \sum_m C_{m} B(2m + n, n) x^m z^n = J(x, z/x)$

where $B(i,j)$ is the binomial coefficient. The substitution rules above are all united by the function $J(x,y)$ which satisfies the rule

$xJ^{2} + xy J + 1 = J$

When $x = 1$ and $y = 0$ this reduces to $J = 1 + J^{2}$, the rule for counting the number of vertices on an associahedron, but when $x = y = 1$ it reduces to the rule $J^{2} + J + 1 = J$. In general, the generating function takes the form

$J(x,y) = \frac{1}{2x} [(1 - yx) \pm \sqrt{(1 - yx)^{2} - 4x}]$

Note that the coefficients $g(m,n) = C_{m} B(2m + n, n)$ of the double series $L(x,z)$ could only give the Schroeder numbers when $x = z$, so by summing the diagonals of the table on page 4 (that is, summing the $g(m,n)$ of weight $i$, starting with $g(0,0) = 1$ for $i = 0$) the Schroeder numbers are obtained. This proves that the Schroeder numbers $S_{i}$ are given by

$S_{i} = \sum_{m + n = i} C_m B(2m + n, n)$

which is a little nicer than the hypergeometric expression $2F1 (1 - i, 2 + i ; 2;-1)$. By setting $y = -1$ the coefficients are alternating sums of the $g(m,n)$ and this leads to the sequence $1,0,0,0,0,0,...$ The Motzkin rule follows from setting $y = \sqrt{x}^{-1}$ and then working with $z = \sqrt{x}$, which one may verify by looking at the generating function. Can we generate more cute Abel sums using $J(x,y)$?

Light Speed Returns

Our poor friend turbo-1 at PF tried to support the consideration of a varying speed of light by quoting Einstein as follows:

"In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position."

Alas, nobody agreed with turbo-1 (or Einstein for that matter) and it was said that there is no evidence in favour of a varying $c$ in a cosmological setting. As Tony Smith says on Louise's blog, "Nature spreads the Big Lie" (that's the magazine, not the bitch, Tommaso).

On a lighter note, Urs Schreiber has discovered the virtues of dimension raising Gray tensor products in particle physics (skip the stuff about membranes) and there are now too many mathematicians blogging about Category Theory to keep up with them. Gone are the days when a physicist could greet the term functor with a blank expression. I'm going to have to find a new branch of mathematics so I can be a rebel again.

Saturday, August 04, 2007

M Theory Lesson 82

Remember the Lorenz template? Now Terence Tao has a fantastic post bringing our attention to the work of Etienne Ghys.

Ghys does amazing things with knot dynamics for the space of (unimodular) lattices in $\mathbb{R}^{2}$, which is the complement of the trefoil knot in the 3-sphere. Elements of the modular group define a periodic orbit in this space, and Ghys shows that this class of knots is the same as the Lorenz knots! And the linking number between such a knot and the trefoil is the Rademacher function $R$ from SL(2,$\mathbb{Z}$) to $\mathbb{Z}$ defined by

$2 \pi i R(A) = 24 (A(\tau) - \tau) log \eta - 6 log (- (c \tau + d)^{2})$

where $A = (a,b;c,d)$ and $\eta$ is the Dedekind function! Carl will like those 12th roots of unity again. The proof of similarity to the Lorenz system uses hexagonal and rhombic tilings which are deformed towards the Lorenz template.

Friday, August 03, 2007

Naughty Numbers

The Everything Seminar continues its series on infinite sums of ordinals using either analytic or zeta function regularisation. Meanwhile Matti Pitkanen had the cool idea of generalising the categorical interpretation (at p=1) to p-adic series.

So what are we actually doing here? The problem is that in n-logos theory we have to be really, really careful about defining numbers. A finite ordinal is the number of elements of a finite set. In other words, a finite ordinal is the equivalence class of all finite sets. But replacing an equivalence class by a single object is a very categorical operation. Moreover, the disjoint union of two finite sets becomes the sum of two positive integers, and the Cartesian product of two sets becomes multiplication, so this tells us something about what numbers are. Essentially, the category of finite sets has been decategorified, since we don't attach arrows $n \rightarrow m$ to the collection of numbers. That is, somehow the numbers $n \in \mathbb{N}$ are each (-1)-categories, and their collection is a 0-category, or set.

Most of the time the elements $n \in \mathbb{N}$ are taken to be objects in a 1-category, but that might just be because one has taken a functor selecting the set $\mathbb{N}$ as an identity arrow in the topos of sets. It's all very confusing. Basically, numbers need to be in many places at once, just like distance measurements.

As Baez points out in TWF 191, a structure type is a functor from the category of finite sets (and bijections) to the category of sets. Here the category of finite sets may be thought of as the category with objects $n$ and morphisms $n \rightarrow n$ the permutation group $S_{n}$. The category of structure types (and natural transformations) is then a categorified version of the polynomials $\mathbb{N} [x]$, which form a rig because they don't have negatives. That is, it is a kind of 2-rig. Commutativity of addition has been replaced by a symmetric monoidal structure. And moreover, as Baez points out, a monoid object in the category of structure types is an operad (and we like those).

Now we want to categorify everything again, because numbers are really dimensions of weak n-categories rather than counting numbers. This is analogous to categorifying Betti numbers to cohomology, or Jones polynomials to Khovanov homology. That would bring us to tricategorical rigs, whereupon the right tensor structure is now dimension raising, so somehow numbers come from all dimensions at once!

Thursday, August 02, 2007

M Theory Lesson 81

Any linear code over a finite field with a circulant generator matrix, such as the Hamming code, is called a cyclic code. For example, the weight 2 code words (1,1,0), (0,1,1) and (1,0,1) form a cyclic binary code along with the zero word (0,0,0). This code corresponds to an ideal in the field

$\frac{\mathbb{F}_2 [x]}{(x^3 - 1)}$

which is a finite version of the rational field of Lesson 38, which looked at simple rational $3 \times 3$ circulants.

Recall that Carl Brannen's circulants are instead of multiplicative type since the conjugate off-diagonal elements multiply, as complex numbers, to give 1. This suggests a setting of exponentiated linear circulants, and hence an infinite series representation for the complex elements.

Wednesday, August 01, 2007

M Theory Lesson 80

Recall that 3x3 complex circulants are useful for building mass operators. A basic 7x7 circulant is the incidence matrix for the plane shown. This matrix is also a generator matrix [1] for the linear binary Hamming code which contains 16 binary words of length 7. The error correction for the Hamming code works by adjoining three bits $x_5, x_6$ and $x_7$ to an element $(x_1, x_2, x_3, x_4)$ of $\mathbb{F}_{2}^{4}$ such that if one labels the Fano points by the $x_{i}$, with the error bits at the centre of the outer edges, the sum of four vertices on the edge triangles equals zero. For example, a codeword 1000110 satisfies

$x_1 + x_2 + x_4 + x_6 = 1 + 0 + 0 + 1 = 0$

The weight 3 codewords appear in the matrix above. This says that PSL(2,7) is isomorphic to GL(3,2), the automorphism group of the projective plane on $\mathbb{F}_{2}$.

We saw this a while ago while pondering octonion products and their possible relation to non-associative triple ribbon diagrams. Any binary code gives a lattice via the group homomorphism $\mathbb{Z}^{n} \rightarrow \mathbb{F}_{2}^{n}$ for which the inverse image of a code defines a lattice. For convenience one multiplies the lattice coordinates by a factor of $\sqrt{2}^{-1}$. Thus there is a way to assign a theta series to any binary code. By adding a check bit to the Hamming code (an eighth bit which is the sum of the others) one obtains a code which maps to an even unimodular lattice in $\mathbb{R}^{8}$, otherwise known as the E8 lattice.

[1] Lattices and Codes, W. Ebeling (1994) (based on lectures by F. Hirzebruch)

Blogrolling On

The consistently intriguing posts of The Everything Seminar has landed it on the sidebar. The latest post about summing divergent series includes the generating function

$T(z) = 1 + z + 2 z^2 + 5 z^3 + 14 z^4 + 42 z^5 + \cdots$

for the Catalan numbers. M theorists will recognise the coefficients as the number of vertices on an associahedron: two vertices for a line segment, five for a pentagon, and so on. And lest there be any doubt he is thinking about trees, he refers to this paper, which maps 7-tuples of (planar rooted binary) trees to trees. A commenter (with a blog called God Plays Dice) pointed out that a different set of trees gave another isomorphism between 5-tuples and singlets.

Apparently the reason the number 7 works for general trees is because 7 = 6 mod 1 and by dividing trees into left and right halves we get an equation $T = 1 + T^2$, where the 1 stands for the root. So planar rooted trees are associated to a sixth root of unity, and the fourth root case is about trees with either 0, 1 or 2 children at each vertex. Its generating function $M$ yields the series

$M(1) = 1 + 2 + 4 + 9 + 21 + 51 + · · · = −i$

It seems there are lots of cute ways of writing down complex numbers as infinite sums, so long as one uses series derived from trees! Here's a cool TWF on this stuff, with a link to this helpful, and seminal, paper by Fiore and Leinster. Oh, I can't wait to go and play some more...