Archive for January, 2007

Roast Yam

One of the more common introduced mammals in the Southern Alps is the YAM (young Australian male). These mammals usually congregate in pairs, or in packs up to ten. I met an uncharacteristically charming and intelligent specimen yesterday in Christchurch. We got to talking about his recent climb of Mt Sefton and then he spotted me reading some papers and said he was very interested in physics, and he was even thinking of going back to university to study it (his background was in the Arts).

It transpired that he had been listening to radio shows on the String Wars, and in particular he remembered an episode on a certain book, the title of which he could not quite remember. Anyway, to cut a long story short, he was very keen to hear all about Category Theory! He said that there had to be something very wrong with physics when it was perfectly clear to a layman such as himself that the entrenched concepts being discussed were so plainly inadequate. We agreed that neuroscientists, for instance, had a better conceptual picture of the measurement of space than many physicists. I must confess, I had no idea this topic had become so mainstream.

Mount Cook has had a number of fatal accidents over the years, but very few lucky escapes. Last week, however, a group of three climbers headed down towards the Linda glacier from the summit. They attached themselves via a sling to a large rock, which then gave way (the rock, that is) as two of the climbers began to abseil. The guy at the top survived the fall, because a small falling rock cut cleanly through the anchor sling, separating him from his colleagues.

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M Theory Lesson 12

One of Devadoss’s many papers on the real points of the moduli of punctured spheres is Cellular Structures determined by Polygons and Trees. In this paper he considers how the real case may be extended to the compactification of the complex moduli of $n$ points.

We saw that labels on trees were not necessary for the real case, which is described by associahedra. Devadoss points out that by labelling the leaves of trees with the numbers $1,2, \cdots n$ one can describe the complex moduli. Points on $\mathbb{CP}^1$ determine non-planar trees, because diffeomorphisms of the sphere can permute any two points without collision (recall that collisions were the basis for understanding the real case). This cyclicity is allowed on branch labellings, because the leaves should really be considered identical.

Now consider that we really want 2-dimensional operads to describe complex moduli. As it happens, 2-ordinal trees are exactly like 1-level trees with an integral number of branches on each leaf of the base 1-tree. However, 2-ordinals may have any collection of numbers $a_1, a_2, a_3, \cdots a_m$ of branches. So the simple labelled trees needed for the complex moduli are only a subset of the 2-ordinals. It remains to understand, therefore, in what sense the complex moduli is really a substructure of something, in this higher dimensional setting. Recall that in topos theory thinking, number fields are never to be considered fundamental in themselves.

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Max Kelly

It is with great sadness that we hear of the sudden death of Max Kelly in Sydney, a few days ago.

Max Kelly was truly one of the founders of category theory, having worked actively in the field since the 1960s. He was knowledgable about many diverse subjects. Until the end of his life he was a keen researcher and active member of the Aus Cat group, often attending seminars despite difficulties with his sight. We will all miss him.

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The Time Machine II

In Book XI of The Confessions, St Augustine (354-430) investigates the perception of Time in his mortal attempt to understand the eternity of God. He was the first philosopher to appreciate the special significance of the present in a world of measurement, so I will quote him at some length:

“Who shall hold it and fix it, that it may rest a little, and by degrees catch the glory of that everstanding eternity, and compare it with the times which never stand, and see that it is incomparable; and that a long time cannot become long, save from the many motions that pass by, which cannot at the same instant be prolonged; but that in the Eternal nothing passeth away, but that the whole is present…

Thy years are one day, and Thy day is not daily, but today; because Thy today yields not with tomorrow, for neither doth it follow yesterday. Thy today is eternity; therefore didst Thou beget the Co-eternal, to whom Thou saidst, “This day have I begotten Thee.” Thou hast made all time; and before all times Thou art, nor in any time was there not time…

What, then, is time? If no one ask of me, I know; if I wish to explain to him who asks, I know not. Yet I say with confidence, that I know that if nothing passed away, there would not be past time; and if nothing were coming, there would not be future time; and if nothing were, there would not be present time. Those two times, therefore, past and future, how are they, when even the past now is not; and the future is not as yet? But should the present be always present, and should it not pass into time past, time truly it could not be, but eternity. If, then, time present – if it be time – only comes into existence because it passes into time past, how do we say that even this is, whose cause of being is that it shall not be, namely, so that we cannot truly say that time is, unless because it tends not to be?

…Behold, the present time, which alone we found could be called long, is abridged to the space scarcely of one day. But let us discuss even that, for there is not one day present as a whole. For it is made up of four-and-twenty hours of night and day, whereof the first hath the rest future, the last hath them past, but any one of the intervening hath those before it past, those after it future. And that one hour passeth away in fleeting particles. Whatever of it hath flown away is past, whatever remaineth is future. If any portion of time be conceived which cannot now be divided into even the minutest particles of moments, this only is that which may be called present; which, however, flies so rapidly from future to past, that it cannot be extended by any delay. For if it be extended, it is divided into the past and future; but the present hath no space…

And yet, O Lord, we perceive intervals of times, and we compare them with themselves, and we say some are longer, others shorter. We even measure by how much shorter or longer this time may be than that; and we answer, “That this is double or treble, while that is but once, or only as much as that.” But we measure times passing when we measure them by perceiving them; but past times, which now are not, or future times, which as yet are not, who can measure them? Unless, perchance, any one will dare to say, that that can be measured which is not. When, therefore, time is passing, it can be perceived and measured; but when it has passed, it cannot, since it is not.

…In whatever manner, therefore, this secret preconception of future things may be, nothing can be seen, save what is. But what now is is not future, but present.”

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The Time Machine

I immediately thought of Louise Riofrio when I saw this post on light cones at Asymptotia. In 1895, H. G. Wells wrote The Time Machine.

“Scientific people,” proceeded the Time Traveller, after the pause required for the proper assimilation of this, “know very well that Time is only a kind of Space”.

Rod Taylor then promptly dives into the leather seat and pushes the lever to accelerate into Time. Popular conceptions of time travel have evolved little since H. G. Wells’ story. The fact is that 20th century physics, after Relativity, did little to alter our notions of time. Moreover, even Relativity could not cure us of this charming delusion, in which The Time Traveller is almost always depicted hopping into a futuristic vehicle, as if about to drive down the road. Is this not itself a hint that GR might fail when it comes to cosmological scales?

In M-Theory, the picture must change. When quantum information content defines Epoch, there is the idea that humans (or things of a similar complexity) can only have arisen in this environment, in this now. Now must become a time, a time we understand better than a distant time, like a Planck time we cannot see.

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Nelson to Battle

Ernest Rutherford was born in Nelson in 1871 and graduated with a degree in the physical sciences from Canterbury College in 1893. He took a second degree there in geology and chemistry. Rutherford left New Zealand in 1895 to work for J. J. Thompson at the Cavendish Laboratory. In 1911, after working at McGill and Manchester, Rutherford deduced from the work of Marsden and Geiger that most of an atom’s mass is contained in the nucleus, thousands of times smaller. Such a discrepancy between the masses and characteristic lengths of the basic building blocks of nature had never been observed, or even remotely conceived, before.

These days physicists blithely discuss interactions from the Planck scale, to nuclear scales, to planetary scales, to the size of the Universe itself! One might be forgiven for thinking that they have been a little cursory in their consideration of some of the scales in between. After all, chemists know quite a lot about the molecular scale, but we know absolutely nothing whatsoever about what happens at the Planck scale, or anywhere near it.

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Parrot Party

Sorry, Carl, but if you keep sending me cool photos I’m just going to have to post some of them, such as this shot of you with Escher the Grey Parrot.
…and here is a photo from home:

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007 Marches On

Just a quick hello, today. Those experimental guys have actually been busy over the holidays. Tommaso reports on a new Higgs prediction from recent improved EW data. Take a look!

The search terms on this blog are getting more interesting. For example, who would be googling Broken Pentagon or Operad + Jordan Algebra? But my all time favourite would have to be M-Theory hogwash. There appear to be a few of brands of M-Theory out there these days. Who knows what will show up on google?

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M Theory Lesson 11

Let’s try another picture formatting option on squashed cube diagrams, which are what we get when we start thinking about higher dimensional categories. A natural transformation is an arrow between functors between 1-categories. What if we had a kind of 2-functor between 2-categories? These can have pseudonatural transformations between them, and then of course there has to be yet another level of arrow, and these are called modifications.

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Upcoming Events

On Thursday Jan 25, John Conway will be speaking at the University of Auckland about his Game of Life. He’s a good speaker, so make it if you can.

A new regular Kiwi event is the Victoria-Canterbury Gravity workshop, which will be held on Feb 7-8 this year at the University of Canterbury. There will be a talk entitled Gravitational Charge in Ribbon Graph M-Theory and other goodies. If you’re thinking of coming, I promise that the summer weather is generally much better than the weather we had for the Kerrfest, namely three days of wind and freezing rain.

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