Archive for October, 2007

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.

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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…”

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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