Archive for June, 2007

At home again

Things often turn out very differently to how one expects. There I was, furiously studying complicated theorems in category theory and having nightmares about standing for hours on end trying to reproduce them all off the top of my head. And they didn’t ask me any of that stuff. In fact, we didn’t get anywhere near it, because I was in such a panic about the questions they did ask that at the snail’s pace I was thinking we would have had to stand there for decades to cover that much ground. Anyway, today the rain cleared and all seems well in the world. Thank you, my examiners. Tomorrow I’m taking a day off to go to the hot springs with some friends. But there is still plenty of work to do when I get back.

Meanwhile, Carl threw a blog party to celebrate the mysterious growing interest in his amateur physics.

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Witten Paper

Three Dimensional Gravity Revisited, Witten’s latest paper, is now on the arxiv. So far I’ve only glanced at it and there seem to be quite a lot of references on Moonshine but not much on LQG or string theory, although later in the paper he seems keen to establish a connection with conventional strings.

Update (27/6): Distler, who actually went to Witten’s talk, says, “But the main insight, emphasized at several points by Witten in his talk, is that the gauge theory approach is wrong.” The slides make an interesting reference to a Farey tale.

And Lubos may have a point when he says, “Well, I happen to think that if Edward Witten started to work on loop quantum gravity, as defined by the existing contemporary methods and standards of the loop quantum gravity community, it wouldn’t mean that physics is undergoing a phase transition. Instead, it would simply mean that Edward Witten would be getting senile. We all admire him and love him, if you want me to say strong words, but he is still a scientist, not God.”

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Conference Calls

As the summer hots up, the gossip is flying. Reports about Strings 07 from both Peter Woit and the PF people make the claim that Witten’s talk will be about LQG type quantum gravity. Hmm. I don’t recall seeing the j-invariant in many LQG papers.

Secret Blogging Seminar just ran what was probably the first real Live Blogging session from a mathematics conference talk. Meanwhile I must busy myself preparing a poster for GRG18. I’m looking forward to that week in the sun!

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Freedom Fighters

Thanks to a poster at Tommaso Dorigo’s blog, I can now post here some excerpts from the Declaration of Academic Freedom, by which I abide. This Declaration is by Dmitri Rabounski, Editor-in-Chief of Progress in Physics.

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Article 2: Who is a scientist

A scientist is any person who does science. Any person who collaborates with a scientist in developing and propounding ideas and data in research or application is also a scientist. The holding of a formal qualification is not a prerequisite for a person to be a scientist.

Article 4: Freedom of choice of research theme

Many scientists working for higher research degrees or in other research programmes at academic institutions such as universities and colleges of advanced study, are prevented from working upon a research theme of their own choice by senior academic and/or administrative officials, not for lack of support facilities but instead because the academic hierarchy and/or other officials simply do not approve of the line of inquiry owing to its potential to upset mainstream dogma, favoured theories, or the funding of other projects that might be discredited by the proposed research. The authority of the orthodox majority is quite often evoked to scuttle a research project so that authority and budgets are not upset. This commonplace practice is a deliberate obstruction to free scientific thought, is unscientific in the extreme, and is criminal. It cannot be tolerated.

A scientist working for any academic institution, authority or agency, is to be completely free as to choice of a research theme, limited only by the material support and intellectual skills able to be offered by the educational institution, agency or authority. If a scientist carries out research as a member of a collaborative group, the research directors and team leaders shall be limited to advisory and consulting roles in relation to choice of a relevant research theme by a scientist in the group.

Article 8: Freedom to publish scientific results

A deplorable censorship of scientific papers has now become the standard practice of the editorial boards of major journals and electronic archives, and their bands of alleged expert referees. The referees are for the most part protected by anonymity so that an author cannot verify their alleged expertise. Papers are now routinely rejected if the author disagrees with or contradicts preferred theory and the mainstream orthodoxy. Many papers are now rejected automatically by virtue of the appearance in the author list of a particular scientist who has not found favour with the editors, the referees, or other expert censors, without any regard whatsoever for the contents of the paper. There is a blacklisting of dissenting scientists and this list is ommunicated between participating editorial boards. This all amounts to gross bias and a culpable suppression of free thinking, and are to be condemned by the international scientific community.

All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media. No scientist shall have their papers or reports rejected when submitted for publication in scientific journals, electronic archives, or other media, simply because their work questions current majority opinion, conflicts with the views of an editorial board, undermines the bases of other current or planned research projects by other scientists, is in conflict with any political dogma or religious creed, or the personal opinion of another, and no scientist shall be blacklisted or otherwise censured and prevented from publication by any other person whomsoever. No scientist shall block, modify, or otherwise interfere with the publication of a scientist’s work in the promise of any presents or other bribes whatsoever.

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Make sure you read the rest. Many of you clearly need to.

Update: Thanks to a comment from Matti, we now have a handy link to the papers of Carlos Castro. Just go here and search on Author=Castro under the preprint search. I have also added a button to the Archive Freedom website on my sidebar. This article is also very interesting. It includes the following chilling passage:

One of the most difficult aspects of the treatment to which Bockris was subjected was social ostracism …. Bockris’ wife Lilli felt it perhaps more than he, because she had a number of faculty wives whom she had known as friends. When she met them now in the supermarket, instead of having the usual kindly chat, they turned their backs on her. Lilli recalls that the year she spent in Vienna after the Nazis took over seemed to her less unpleasant and threatening than the isolation and nastiness which she felt in College Station, Texas from 1993 through 1995.

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Tantalising Tantrix

Let’s take a break and play a game. Whilst on a wardening stint at French Ridge hut with my kea friends, back in 2000, I met a very interesting guy: Mike McManaway, the inventor of tantrix. His wife and I promptly sat down and started thinking about the combinatorics of this game, but I don’t think we got very far.

The game (for 2 to 4 players) involves a collection of hexagonal pieces, each decorated with three strands in different colours. There are four colours in total, and each player chooses a colour, the aim being to finish up with a long strand or loop in your colour on the central board, which is slowly built up as pieces are placed, respecting colour at the edges. I can’t say I’ve played it much, but it’s fun to draw coloured knots!

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Modular Makeup

The Axiom of Choice can be a troublesome beast. It leads, for instance, to the amazing Banach-Tarski paradox, which states that one can cut up an orange and use the pieces to make two oranges. I found a wonderful book in the library on this topic, by Stan Wagon (here is the Google peek page).

A subset $U$ of a set $X$ is said to be paradoxical with respect to a group $G$ (used to rearrange pieces) if such a process can be done to $U$. The orange example comes from considering balls in $\mathbb{R}^{3}$ and translations and rotations, and it was shown by R. Robinson that only five pieces are needed to make a paradoxical orange!

Using the upper half plane and the modular group one can study similar paradoxes using Borel sets. Hausdorff showed, using the $S$ and $T$ presentation embedded in a rotation group, that the modular group is paradoxical. The relevant decomposition of hyperbolic space is three pieces $A$, $B$ and $C$ (see the pretty picture on the book cover) which are related via $TA = B$, $T^{2} A = C$ and $S A = B \cup C$.

What Tarski showed was that paradoxical decompositions are really about the non-existence of a finitely additive invariant measure.

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M Theory Lesson 71

I’m really not sure how useful this is, but since Moonshine Math is into the hexagon craze I thought I’d draw a decent picture of the annular Stasheff tiling with 6 edges on the interior and 12 on the outside. Recall that the red lines mark individual 3×3 matrix corners. By staring at this picture we see a 3D chair corner with a little cube cut out of it. Using very regular such triangles to tile a plane, and then forgetting the triangular boundaries, we obtain a regular hexagonal tiling with hexagons the size of the interior hexagons, because each triangle vertex gives one sixth of a hexagon. There is then a second hexagonal tiling with hexagons made out of 6 big triangles. The big hexagons are 27 times the area of the little ones, since there are $4 \frac{1}{2}$ hexagons to a triangle.

Note that a perfectly regular honeycomb has a representative eigenvalue set of $(1,0,-1), (2,1,0)$ and $(0,-1,-2)$. This corresponds to the top vertex labelling $[ \frac{1}{2}, 1, - \frac{3}{2} ]$ which obeys the zero sum rule.

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M Theory Lesson 70

Since it seems to be Mad Moonshine blogging week, let’s take a quick look at Gannon’s paper The algebraic meaning of genus 0, mentioned back in Week 233.

After a nice review (for physicists) of the Moonshine theorem, Gannon gets around to discussing the braid group $B_{3}$. Recall that $B_{3}$ gives the modular group $PSL_{2}(\mathbb{Z})$ when quotiented by its centre. Now $B_{3}$ is the fundamental group for the space $SL_{2}(\mathbb{Z}) \backslash SL_{2}(\mathbb{R})$, which looks like the complement of the trefoil knot.

Even more amazing, $B_{3}$ is the mapping class group for an extended moduli space $M_{1,1}^{ext}$ of 1-punctured tori marked with a state $v$, which naturally appears in rational CFT. For conformal weight $k$, this group acts on (some convenient) characters $\chi (\tau, v)$ via

$\sigma_1 . \chi (\tau, v) = e^{\frac{-2 \pi i k}{12}} \chi (\tau + 1, v)$

$\sigma_2 . \chi (\tau, v) = e^{\frac{-2 \pi i k}{12}} \chi ( \frac{\tau}{1 – \tau} , \frac{v}{(1 – \tau)^{k}})$

where we recognise the usual action of modular generators $T$ and $S$ on $\tau$. Carl Brannen will just love those 12th roots. Naturally, we would like to compare all this to Loday’s trefoil on the K4 polytope, with crossings on the three squares. Since this polytope is dual to Mulase’s 6-valent ribbon vertex cell decomposition of $\mathbb{R}^{3}$, it must somehow describe the trefoil complement space, and the triality of the j-invariant would be lifted to a triality for these squares. Oh, perhaps we could use the torus that we made out of two such polytopes as cylinders based on honeycomb geometries. That is, draw the trefoil on this torus, which is two glued copies of the planar annulus (replacing the 2-sphere), each bounded inside by a central hexagon. On each annulus, the square tiles correspond to three principal directions in the honeycomb plane. Or maybe not!

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M Theory Lesson 69

Recall that when replacing trees by dual polygons, one can distinguish the type of the associahedron face by the kind of diagonals for the polygon. For example, the K4 Stasheff polytope has 6 pentagonal faces and 3 squares. These are distinguished by the chorded hexagons where a diagonal that splits a hexagon in two corresponds to a square. This shows how pentagons may be paired, by taking dual diagonals, but squares are at best self dual. Labelled trees may be replaced by labelled polygons.

The description of trees as clusters of polygons, used by Devadoss in tiling moduli spaces, is better known to category theorists as the theory of 2-opetopes. The dimension 2 describes the planar nature of polygons, but this may be generalised. On that note, David Corfield points out a wonderful new paper on the arxiv.

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Happy New Year

Lieven Le Bruyn has created a new blog, Moonshine Math (yes, the j-invariant!), born on Bloomsday as promised. Today also signals the coming of the new year, as Matariki rises in the sky. Maybe this year I’ll get less spam telling me I don’t need to be an average man any longer. Meanwhile, yesterday Mottle had an interesting post on a new paper by Dvali which considers a large number of copies of SM particle species as a route to explaining the heirarchy problem. Sounds a bit familiar.

Tommaso Dorigo posts about the D0 and CDF discovery of the $\Xi_{b}$ particle with a mass of $5.774 \pm 0.019$ GeV/$c^2$, which is about six times the proton mass. This particle is made of one down, one strange and one bottom quark.

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