Tic Tac Toe II

A Slashdot report says that Vaughan Pratt, a well known computer scientist who works in Category Theory, claimed to have found an elementary error in Smith's purported proof of the Wolfram Turing machine conjecture. The Wolfram response is available here. They claim the proof stands, although Smith did need to alter the definition of universality. Even better, Alex Smith himself replies to Pratt.

Riemann Riddles

Given a preference for surreals over the reals as arguments for the Riemann zeta function, one cannot help but wonder about the association of 2-branchings with the parity cubes in all dimensions. At the third level, for instance, there are 8 nodes on the tree correponding to (a) 8 vertices of a cube, or (b) a set of zeta values.

Does this set of zeta values combine to obey a Koide type relation, just like the primary 3 faces of the space generation cube? This sounds like an idea to generate endless hours of play, but that may have to wait until I am elsewhere! The good news is that I've found the best internet cafe in the city, which opens early and has good cheap coffee. And on the short walk from the bus station to work, there is a garden by the river where I can sit in the sun by a statue of Captain Scott, engraved with the words, "I do not regret this journey..."

Where to Now

When I was a kid back in the 1970s, I don't think the newspapers regularly carried stories with headings like Humanity At Risk, but this phrase is an apt description of the conclusions in the latest scientific report from the United Nations environment program.

Strong investments to increase supply and reduce demand, particularly through efficiency improvements, help to alleviate concerns over freshwater availability in much of the world. Still, growing populations and
economic activity continue to strain resources, particularly in the developing regions. Globally, the population living under severe water stress continues to rise, with almost all of this increase occurring in those regions exhibiting continued population growth.

The report tries very hard to be cheerful, with its colourful presentation and cartoons, but as almost all reviews indicate, the message is chilling. Wonderful! With this awareness, there is an opportunity for change.

History shows that much can change, expectedly or unexpectedly, over short periods, and it is unlikely that most trends would continue unabated for decades without changing course.

Number One Memes

From Tommaso Dorigo and The World's Fair we have the How I Get To Number One on Google meme: find 5 phrases or word-sets that put your blog at the top of a Google search! Arcadian Functor gets there with
1. twistor topos operad
2. pizza theory cheers
3. parrot smuggling alert
4. gravity monad
5. DNA functor

Tic Tac Toe

Alex Smith, a 20 year old Birmingham undergraduate, has been awarded the Wolfram prize for proving that the (2,3) Turing machine is universal. This is a very basic machine with a three letter alphabet (say 0,1 and 2) and only two states, obeying the state diagram where $m:n$ represents a substitution of the letter $m$ for the letter $n$. The third number on an arrow labels the offset of the head for that move.

Wolfram says that such a universal machine could be used as a basis for building computers from simple molecules, such as DNA.

Six Degrees

Whilst completely wasting my time on a postdoc application, I was musing over the example of Tamarkin. The 6-leaved 2-operad tree looks different as a stringy diagram. The blurring of vertices can turn the 2-operad tree into a 6-leaved 1-operad tree, associated with the 3D Stasheff associahedron, which regularly appears here. Any worldsheet will have the topological property that its boundary is just a collection of circles, so only by specifying a decomposition into punctured spheres (multi-pants) can one recover the internal circles corresponding to nodes of the 2-operad tree. Similarly, a higher level tree becomes a punctured sphere with a more complicated decomposition.

On the other hand, if we insist on associating surfaces with 2-operads, then the most symmetric choice for the 6-punctured sphere is the 2-level version of Tamarkin. Moreover, the pairing of punctures, marked by the three internal circles, is just like the Dehn map from the unpunctured genus 2 surface, whose moduli is also six dimensional (over $\mathbb{R}$).

Can we use all three twistor moduli to form a 3-operad triality? If the 6-punctured sphere is 1-operadic, and the pair just mentioned are somehow 2-operadic, is there a 3-level description including the moduli for the 3-punctured torus? Intriguingly, the 3-punctured torus is built from three copies of the 3-punctured sphere, which we can relate via Belyi maps to three different elliptic curves, the Cartesian product of which is again a nice 6-dimensional space. The genus 2 surface is usually decomposed into two 3-punctured spheres and two cylinders, whereas the 6-punctured sphere needs four 3-punctured spheres. So by gradually adding punctures to cylinders and re-gluing, we can turn the genus 2 surface into the torus into the sphere. Adding a puncture to a cylinder is the same as turning a single edge into a 2-level tree with 2 branches (the trivalent vertex), so this process can take us from 1-level trees to 2-level trees to 3-level trees. A typical 6-leaved 3-level tree (for leaves grouped as 1,1,2,2) corresponds to a polytope of dimension 12 - 3 - 1 = 8. The minimal dimension is 4, corresponding to the suspended Stasheff polytope, and an 11 dimensional polytope is obtained from the Tamarkin tree with added single edges on the top level.

M Theory Lesson 115

As a fellow antipodean (Terence Tao) explains, the decomposition of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ into odd and even parts is the simplest example of a Fourier transform. This is the $2 \times 2$ circulant case, characterised by the identity and the Pauli swap matrix,

01
10

which are both 1-circulants. The swap is associated to reflection about zero on the real line, or rotation by $\pi$ in the complex plane. These small matrices appear too simple to be interesting, but let's consider a general $2 \times 2$ circulant

A B
B A

Constructing this matrix from row vectors, which in turn are concatenated scalars, may be compared to a construction via a vector of column vectors. We denote these two options by $(A,B)'(B,A)$ and $(A'B),(B'A)$. That these two are equal is an example of the bicategorical interchange law, where the traditional symbolic form of the matrix is literally the 2-arrow diagram! The square of this matrix takes the form

(A.A + B.B) (A.B + B.A)
(B.A + A.B) (B.B + A.A)

which in an interchange diagram subdivides each square into four little squares, introducing two new products (addition and multiplication). Assuming distributivity for the moment, the first entry would satisfy the interchange law only when $A.B + B.A = 0$ and the second entry when $A.A + B.B = 0$. For ordinary numbers this would immediately result in $A = B = 0$, so it might be more interesting to consider this second interchange law to be broken by these terms, the first being the Jordan product.

Fairy Update

A quick note: Neutrino 2008 is to be held here in Christchurch in May. Tommaso reports on the top mass, and another blogger reports on the latest news about DO constraints on SUSY, giving new mass limits for neutralinos and charginos. Meanwhile, Conway tells us that the Fairy Bump has disappeared. As Theseus would say,

More strange than true: I never may believe
These antique fables, nor these fairy toys.

Interlude

Kuhn begins his essay The Function of Measurement in Modern Physical Science with a quote from Lord Kelvin

If you cannot measure, your knowledge is meager and unsatisfactory

What Kuhn discusses, in his waffly style, is the observation that quantitative progress in physics usually requires first a lot of qualitative wandering in the dark. Moreover, it is often very difficult for proponents of a new idea to present evidence, because empirical data never matches theory exactly, and there are more subjective criteria in determining a best fit, when alternative theories are available, than many scientists would be happy to admit. He argues that all scientists, on all sides, may be considered rational, because they each use a large set of criteria in determining their position. At some times, however, when ideas begin to converge, the subjective differences become more apparent amidst the ocean of converging evidence.

A Good Light II

An anonymous commenter asks from where the school teacher got his evidence for a slowing speed of light. We can only guess! Perhaps he keeps up to date with news on Louise Riofrio's blog. Alternatively, he may have looked at wikipedia and observed that the 1926 experiment measured a value of 299796 km/sec, compared to the modern value of 299792.458 km/sec. This modern value is now the standard for $c$, which since 1983 has been used to define the metre.

Anyway, the lattes all worked well today and I'm enjoying a clear spring evening on this holiday weekend.

A Good Light

I wish all my students well in their final physics exam this year. Classes ended today with a session on Special Relativity. One bright student really surprised me with the remark, "Yes, but the speed of light is slowing down, isn't it?" I thought it best to ask her exactly what she meant by that, and she replied enthusiastically that her Dad (a school science teacher) had told her that we now have evidence that the speed of light has been slowing down. I carefully pointed out that, yes, indeed, there were physicists willing to contemplate this evidence with a critical and fair eye, but that the vast majority would entirely disagree with the idea. She said, "Yes, but Relativity is only correct until it is shown to be wrong, right?" I pointed out that, yes, a physical theory only had a certain domain of applicability, beyond which it would not apply, and she said, "Well, we've seen that already with Newtonian mechanics!" And it was true that I did have to work hard this week explaining that Newtonian clocks simply didn't work when things started moving at high speeds, so I confessed that I was actually personally fond of the idea of a slowing speed of light, until another student impatiently drew me back to the problems in hand.

Spaced Out Arrows

The source category in an ordinary topological field theory is always the 1-category n-cob of n dimensional spaces as arrows between (n-1) dimensional spaces as objects. For example, the six punctured sphere with three boundaries labelled in and three boundaries labelled out is considered as an arrow in 2-cob, a category of two dimensional spaces.

In M Theory, on the other hand, we often consider such diagrams to be arrows in a multicategory, since an operad is a multicategory on one object. That is, each little circle boundary element is a separate object in the multicategory, which is hence built up entirely of connected spaces. Allowing for multiple outputs takes us beyond ordinary operads to structures incorporating duals, but starting with the operad case, one might ask for an operad map into a target operad, instead of an ordinary (modular) functor. When the target is based on a vector space, such a map yields an algebra for an operad. Since two dimensional TFTs hinge on the assignment of one (Hilbert) space $V$ to the circle object, it appears there is no loss of information in considering the $\textrm{End}(V)$ operad instead of the entire category of vector spaces.

But 1-categories are not really that interesting. In a 2-category of two dimensional spaces, the one object is a point, and there are two 1-arrows: the arc and the circle (open and closed string). So the Hom 1-category has two objects to map into an algebraic target. In three dimensions the top level has an infinite set of surface objects, labelled by genus and boundary circles. By not allowing arcs, in this case there is one object (the point) and only one 1-arrow (the circle). Such tricategories are better known as braided monoidal categories. If the braiding is to be represented concretely by the knotting of pieces of a three dimensional space, then the product acts on the 2-arrows, namely the surface pieces, so that the simplest $\Sigma_1 \otimes \Sigma_2$ is a concatenation of cylinders and the braiding comes from interacting cylinders which cross over. Arrow composition should be gluing, as usual, but to obtain two compositions (horizontal and vertical) one would now like boundaries marked both in and out and also left and right. This requires surface pieces with at least four circle boundary elements (4 punctured spheres). Such gluings clearly give a well-defined surface, satisfying an interchange law. Such a composition of four surface pieces has a new hole in the centre, contributing to the genus of the composition. Note that this is an attempt to discuss concrete braidings with 2-surfaces bounding 3-spaces, in contrast to the usual situation of knots lying in a fixed 3-space.

It's A Boy

Congratulations to my third nephew, Aidan Barry Moore, who was born yesterday in Melbourne on his mother's birthday. Oh, and congratulations to my sister and hubby, too!

M Theory Lesson 114

The observant M theorist will have noticed the large variety of hexagons that keep popping up. In particular, there is the broken pentagon hexagon, labelled by elements of $S_3$, and also the Yang-Baxter hexagon, with its alternating associator and braiding maps. But if the permutation maps (eg. $ABC \rightarrow BAC$) are considered as braid generators then these two hexagons look very similar. When the four-leaved trees undergo a flip from outside edge to outside edge, we may take that to be an associator on three elements.

Generalising permutations to braidings is very natural in category theory. Ordinary algebras are vector spaces, associated with the symmetric monoidal category Vect, but the symmetry arises as an additional constraint on the underlying braiding of a braided monoidal category. The Loday-Ronco maps between $A_n$ and $S_n$, and their Hopf algebras, should therefore have a generalisation to braided monoidal objects, such as the weak Hopf algebras of Robert Coquereaux, considered recently by Street and Pastro.

M Theory Lesson 113

Another new paper of interest to M theorists is Eugenia Cheng's Iterated Distributive Laws. Recall that a distributive law is a map (natural transformation) between two monads $ST \Rightarrow TS$ such that, in particular, $TS$ is again a monad. Cheng points out that whether or not $S(TU)$ is a monad, given monads $S$, $T$ and $U$, depends on the hexagon rule, aka the Yang-Baxter equation. Later this is associated to a hexagon for the Gray tensor product.

One example of a distributive law is for the category of 2-globular sets, which are two dimensional globule diagrams of sources and targets $S_2 \Rightarrow S_1 \Rightarrow S_0$ in Set such that $ss = st$ and $ts = tt$. One monad is vertical composition, and the other is horizontal composition. The distributive law is just the interchange rule for bicategories. Iterating this idea, Cheng considers composition in n-categories using n-globular sets, a la Batanin.

Links II

Tommaso Dorigo has two great guest posts by Alejandro Rivero and Louise Riofrio. A post lest we forget Archimedes. Meanwhile John Baez provides this awkward but tantalising link for devotees of Kapranov papers. This one's about number theory!

M Theory Lesson 112

Recall that a lattice is a poset such that each pair of elements has a least upper bound and greatest lower bound. In topos theory we also assume that the whole lattice has a largest element 1 and smallest element 0. For example, in Set the lattice of subsets of a set $S$ has union (join) and intersection (meet). The greatest element is $S$ and the zero is the empty set.

Consider ordering a set of numbers by size. The property of the rational numbers that leads to the reals is the following: consider the set of rationals $q$ defined by $q^2 < 2$. This set is bounded above in $\mathbb{Q}$ but contains no least upper bound in $\mathbb{Q}$. Only the real numbers have the property that this least upper bound exists. The interval $[0,1]$ in the (extended) reals is an example of a complete lattice, as is the power set example above.

Considering Vect as a quantum logic category, it is still true that lattices of subspaces are complete, even for the rationals! In logos theory the logic of lattices may be weakened. Notions of meet and join are still useful but may not be required to exist for all objects. A ternary analogue of lattice is something else entirely.

Links

I've just spotted another excellent physics blog for the roll. Thanks to a Seminar commenter for this link, and Universe Today for a fascinating report on complex dust in quasar winds.

M Theory Lesson 111

A commenter at Carl Brannen's blog has noted the similarity between the snuark mass computations and the Fano plane, which we recall describes the octonions via a cube with corners $1, e_1, e_2, \cdots , e_7$. In Carl's notation, the correspondence is

$e_1 = +y$
$e_2 = +z$
$e_3 = -y$
$e_4 = +x$
$e_5 = -x$
$e_6 = -z$
$e_7 = 0$

where $0$ is the fictitious vacuum and the $1$ is hidden at the rear of his diagrams. On the octonion cube, this gives a source of $0$ and a target of $1$, with $x$, $y$ and $z$ axes running through the other three full diagonals. Note that a projection of this cube onto a plane results in a hexagon with vertices $+x, -z, +y, -x, +z, -y$, which has a selected triple of nodes $(x,y,z)$ as previously noted. The basic simplex formed from the source in these directions is the area marked with the phase $\frac{\pi}{24}$ in Carl's computation.

Aside: Note also the new paper by Yidun Wan on 3-braids, which discusses Veneziano bubbles and rotations by $\frac{\pi}{3}$ and $\frac{2 \pi}{3}$, the symmetries of a triangle.

M Theory Lesson 110

The connection between mass matrices and Fourier transforms leads to endless musing over the important work of Kapranov! Recall that a path on a cubical lattice is represented by a monomial in three letters, such as $X(-Y)Z$. To begin with we may consider positive paths in the positive octant (ie. 3d Young diagrams, which don't backtrack) and forget the minus signs. If the path taken to the target point was irrelevant, a commutative monomial $XYZ$ would denote the target point without its path history. Only non-commutative monomials describe all possible paths from the source to the target. On considering $3 \times 3 \times 3$ cubes, homogeneous monomials of total degree 9 describe the paths. The homogeneity follows from the symmetry of subdivision.

A path integral for a fixed source and target may have contributions from paths of varying length, depending on the number of subdivisions. The shortest possible path here is of length 3. Recall that in one dimension, the number of subdivisions $n$ is associated to the size of a circulant matrix, associated to the position space $\mathbb{Z}_n$. But this almost trivial one dimensional case somehow encoded the three dimensions of space! A truly three dimensional Fourier transform, with a six dimensional phase space, would contain much more physical information.

Aside: Carl Brannen's new site has a sidebar button.

There She Goes

A photo from the news today of the new lake near home, just now overflowing:

Updates

Well, it took me quite a while, but I finally came across a Kapranov paper on the non-commutative Fourier transform! Meanwhile Carl Brannen has been updating his blog with fascinating calculations in snuark QFT. Thanks to Tommaso Dorigo for joining the Category Theory for Physics outreach program. Things here will probably be quieter next week, since my schedule is quite full with both latte serving and teaching.

Ere Riemann

One of the best books I ever picked up second hand [1] begins with the very old problem of the vibrating string, as studied by Euler and contemporaries. In the author's historical opinion, the young Lagrange took on board both (i) Euler's preference for introducing the concept of non-differentiable functions and (ii) Euler's confusion over, and lack of enthusiasm for, expressing general solutions in terms of differentiable periodic functions. In other words, Lagrange was happy to work with generalised functions, but unhappy about Euler's Leibnizian infinitesimals. This prompted Lagrange to reinterpret calculus using Taylor series, in his own words "independent of all metaphysics". Nobody can know exactly what he meant by this, but perhaps it is a reference to the philosophy of Leibniz.

These events must strike a chord with anyone educated in physics in the 20th century, heavily influenced by the thinking of Lagrange. Of course, modern physicists also have Fourier analysis, but this was a later development. In fact, Fourier's first paper on the heat equation (1807) was never published because of Lagrange's objections to it.

[1] I. Grattan-Guinness, The development of the foundations of mathematical analysis from Euler to Riemann, MIT Press (1970)

Riemann Returns

Alain Connes is probably best known for playing with connections between physics and the Riemann hypothesis. Actually, there is an easy physical way of looking at the zeta function, using the free Riemann gas. Imagine a system with eigenstates labelled by the prime numbers, and energy eigenvalues $E_p = E_0 \textrm{log} p$. Multiparticle states are given by ordinals $n$ which decompose into prime factors, $n = p_1 p_2 p_3 \cdots p_i$. Setting $s = \frac{E_0}{k T}$ one can write the partition function

$Z(T) = \sum_n e^{- \frac{E_n}{kT} } = \sum_n e^{- s \textrm{log} n} $
$= \sum_n \frac{1}{n^s} = \zeta (s)$

The Hagedorn temperature corresponds to the $\zeta$ pole at $s = 1$. What would happen if we considered knot zeta functions instead? States would be labelled not by integers, but by knots, which similarly factor into primes. Instead of prime factors $p$ there are polynomials $J(K)$, and logarithms of $J(K)$ are associated with energy levels.

As it happens, such logarithms (for the normalised colored Jones polynomial) occur in the well known volume conjecture, which states that the hyperbolic volume of a knot complement is given by

$V(K) = 2 \pi \textrm{lim}_{N \rightarrow \infty} \frac{1}{N} \textrm{log} |J(K)|$

where $J(K)$ is evaluated at the root of unity $q = e^{\frac{2 \pi i}{N}}$. These volumes can be associated to singular spaces appearing in 4d spin foam models, where there is in fact a partition function interpretation. Note that as $N \rightarrow \infty$ the so-called cosmological constant often associated with a quantum symmetry should disappear, as one expects (and observes) for classical spacetimes.

Change of Season

Since I moved to Christchurch five years ago I have, while penniless, moved abode 16 times, been through 5 thesis supervisors, been hospitalised 3 times and once missing presumed dead (and the geniuses in my department seemed a bit surprised to discover I was stressed). Now after 6 months in the same home I feel positively settled! And today it mysteriously turns out that I'm a prominent blogger of some kind, at least according to Math Bloggers. But must run: I have an interview for a waitressing job in town.

M Theory Lesson 109

In the easy book Special matrices of Mathematical Physics the chapter on circulant matrices begins with the statement: making a Fourier transform is equivalent to diagonalising a circulant matrix, and an inverse Fourier transform takes a diagonal matrix into a circulant matrix. In fact, for a circulant built from the numbers $X_1 , X_2 , \cdots , X_n$ the eigenvalues $\lambda_k$ are related by the following pair of $\mathbb{Z}_n$ (Abelian) transforms

$\lambda_k = \sum_{j} e^{\frac{2 \pi i}{n} jk} X_j$
$X_j = \frac{1}{n} \sum_{k} e^{- \frac{2 \pi i}{n} jk} \lambda_k$

Perhaps non-commutative Fourier transforms should be about circulants of circulants! In fact, examples of quantum groups (Hopf algebras) in the book are built from matrices of matrices. The simple quantum analogue of phase space is viewed as the space underlying the circulant/diagonal matrix space, such as the discrete torus $\mathbb{Z}_n \otimes \mathbb{Z}_n$, where the cyclic group $\mathbb{Z}_n$ belongs to the self-dual schizophrenic object $U(1)$ of Stone (Pontrjagin) duality.

A circulant $C$ is diagonalised via $M^{-1} C M$ with $M_{ij} = \frac{\omega^{- ij}}{\sqrt{n}}$ for $\omega$ the primitive $n$-th root of unity. Given that the eigenspace projectors are circulant, a circulant matrix can be written in the basis of projectors. It can also be written as a degree $n$ polynomial in a nice circulant $S$ representing a basic shift operator. In the $3 \times 3$ case this is the familiar

001
100
010

corresponding to the permutation $(312)$, and a circulant takes the form $X_1 + X_2 (231) + X_3 (312)$. A spectral function $f$ on a circulant is defined to be the circulant

$f(C) = \frac{1}{n} \sum_j (\sum_k f(\lambda_k) \omega^{-jk}) S^j$

The Fourier transform pair is particularly simple in terms of the matrices $S^j$ and the projectors. Thus it is useful to define a matrix product via convolution, $G.H = \sum_j (G*H)_j S^j$ where

$(G*H)_j = \sum_k G_k H_{j - k}$

Now let the diagonal matrix with entries $\delta_{ij} \omega^{i}$ be denoted $D$. Then one has $DS = \omega SD$, a simple non-commutativity condition, which extends to the Weyl rule $D^m S^n = \omega^{mn} S^n D^m$ for a quantum torus. This setup has the feature that the $n \rightarrow \infty$ process is associated with a continuum limit, which is just what we want to do with n-categories. To quote Weyl: The problems of mathematics are not problems in a vacuum ...