### At home again

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

occasional meanderings in physics' brave new world

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.

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

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."

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."

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!

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!

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.

----------------------------

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.

----------------------------

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.

----------------------------

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.

----------------------------

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.

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!

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!

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.

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.

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 3x3 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.

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.

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!

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!

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.

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.

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.

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.

Matti Pitkanen now has a post about Farey sequences and the Riemann hypothesis in TGD. The idea that the hypothesis is not provable within standard mathematics appears to be gaining a foothold within physical constructions.

On the other hand, it is possible that the physical axioms could guide a concrete proof within a convenient model, such as the Jordan algebra M Theory, in which U duality is algebraically manifest. But the zeta function itself only enters here with the (operad) algebras associated to moduli integrals. So it is difficult to avoid the higher categorical framework in studying exact (eg. MHV) amplitudes, and this lands us back in the world of post ZF axioms.

After inhabiting this world for some time, it becomes difficult to look at zeta functions any other way. One simply can't help looking at the Selberg axioms and thinking of closure under products, or factorisation, as topos-like axioms, even though these are radically different things. Recall that the interplay of + and x here is thought of as a higher distributive law for monads. This suggests that the Euler relation for zeta functions is about equating invariants based on monads, or rather that the distributivity $+ \times \rightarrow \times +$ is an identity. That is, that the distributivity of complex arithmetic is somehow more responsible for Euler's product relation than the notion of primeness, which is used through the application of the fundamental theorem of arithmetic only after the product has been expanded.

This suggests that the higher dimensional versions of the Riemann zeta function should be thought of as non-commutative, non-associative and even non-distributive L-functions. Ah! So that's why Goncharov likes Shimura varieties. Note that such considerations are necessary for understanding even the values of the Riemann function, since its arguments extend throughout the heirarchy.

Update: Khalkhali has a new post on Determinants and Traces in which he notes: "... Bost and Connes in their paper Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the beginning show that the above formula (5) gives the Euler product formula for the zeta function ... In fact their paper starts by quantizing the set of prime numbers ... Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality."

Formula (6) is $Tr_{s}(\Lambda A) Tr(SA) = 1$, a relation between trace and supertrace. Hmm. I would like to understand Koszul duality better because it applies to operads, and more generally PROPS.

On the other hand, it is possible that the physical axioms could guide a concrete proof within a convenient model, such as the Jordan algebra M Theory, in which U duality is algebraically manifest. But the zeta function itself only enters here with the (operad) algebras associated to moduli integrals. So it is difficult to avoid the higher categorical framework in studying exact (eg. MHV) amplitudes, and this lands us back in the world of post ZF axioms.

After inhabiting this world for some time, it becomes difficult to look at zeta functions any other way. One simply can't help looking at the Selberg axioms and thinking of closure under products, or factorisation, as topos-like axioms, even though these are radically different things. Recall that the interplay of + and x here is thought of as a higher distributive law for monads. This suggests that the Euler relation for zeta functions is about equating invariants based on monads, or rather that the distributivity $+ \times \rightarrow \times +$ is an identity. That is, that the distributivity of complex arithmetic is somehow more responsible for Euler's product relation than the notion of primeness, which is used through the application of the fundamental theorem of arithmetic only after the product has been expanded.

This suggests that the higher dimensional versions of the Riemann zeta function should be thought of as non-commutative, non-associative and even non-distributive L-functions. Ah! So that's why Goncharov likes Shimura varieties. Note that such considerations are necessary for understanding even the values of the Riemann function, since its arguments extend throughout the heirarchy.

Update: Khalkhali has a new post on Determinants and Traces in which he notes: "... Bost and Connes in their paper Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the beginning show that the above formula (5) gives the Euler product formula for the zeta function ... In fact their paper starts by quantizing the set of prime numbers ... Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality."

Formula (6) is $Tr_{s}(\Lambda A) Tr(SA) = 1$, a relation between trace and supertrace. Hmm. I would like to understand Koszul duality better because it applies to operads, and more generally PROPS.

Terence Tao tells us about this paper by Guangming Pan and Wang Zhou on random $NxN$ complex matrices with entries of mean 0 and variance $\frac{1}{N}$. They claim to prove, under an assumption about moments, that the spectral distribution converges to the uniform distribution over the unit disc. This is called the circular law. In other words, the initial clustering of eigenvalues around the real line disappears as $N \rightarrow \infty$.

In would be interesting to see what this meant for honeycomb patterns in the limit of an infinite number of hexagons. Or perhaps it helps us understand the distribution of Farey numbers on the unit interval. Recall that successive terms $\frac{p}{q}$ and $\frac{r}{s}$ of a Farey sequence satisfy

$qr - ps = 1$

which is why the modular group appears when considering matrices $(r,p;s,q)$. Let $N$ be the number of terms in a Farey sequence. The Riemann hypothesis [1] is equivalent to the statement that the sum of differences between Farey terms and interval markers, namely

$\sum_{n=1}^{N} \delta_n \equiv | f_{n} - \frac{n}{N} |$

is bounded by $o (x^{\frac{1}{2} + \varepsilon})$ for all $\varepsilon > 0$ as the real number $x$ defining the sequence tends to infinity. The Farey sequences themselves are rational numbers less than 1, and fit onto the binary Farey tree described by Vepstas. The ends of the infinite tree fit onto the boundary of the Poincare disc, when the modular domain view is mapped there. Thus the interval markers above may be exchanged for roots of unity on the unit circle, and these compared to the leaves of the Farey tree.

Kauffman et al (p 51) show that this version of the Riemann hypothesis is equivalent to a question about messy unknots. They also look at DNA recombination. Unknots described by rational tangles are labelled by the pairs of adjacent rationals in a Farey sequence. So two tangles labelling two adjacent leaves of the tree at infinity can be used to construct unknots.

[1] H. M. Edwards, Riemann's Zeta Function, Academic Press (1974)

In would be interesting to see what this meant for honeycomb patterns in the limit of an infinite number of hexagons. Or perhaps it helps us understand the distribution of Farey numbers on the unit interval. Recall that successive terms $\frac{p}{q}$ and $\frac{r}{s}$ of a Farey sequence satisfy

$qr - ps = 1$

which is why the modular group appears when considering matrices $(r,p;s,q)$. Let $N$ be the number of terms in a Farey sequence. The Riemann hypothesis [1] is equivalent to the statement that the sum of differences between Farey terms and interval markers, namely

$\sum_{n=1}^{N} \delta_n \equiv | f_{n} - \frac{n}{N} |$

is bounded by $o (x^{\frac{1}{2} + \varepsilon})$ for all $\varepsilon > 0$ as the real number $x$ defining the sequence tends to infinity. The Farey sequences themselves are rational numbers less than 1, and fit onto the binary Farey tree described by Vepstas. The ends of the infinite tree fit onto the boundary of the Poincare disc, when the modular domain view is mapped there. Thus the interval markers above may be exchanged for roots of unity on the unit circle, and these compared to the leaves of the Farey tree.

Kauffman et al (p 51) show that this version of the Riemann hypothesis is equivalent to a question about messy unknots. They also look at DNA recombination. Unknots described by rational tangles are labelled by the pairs of adjacent rationals in a Farey sequence. So two tangles labelling two adjacent leaves of the tree at infinity can be used to construct unknots.

[1] H. M. Edwards, Riemann's Zeta Function, Academic Press (1974)

I apologise for moving off topic today, but I found this Smile Test very interesting. Apparently, most people are not as good at telling fake smiles as they think they are. The theory is that people are easily fooled because it is socially convenient not to know what people are thinking. Despite my expectations of doing badly, I actually did really well on this test (17/20). Now I realise that it's easier not to care what people think when one's ability to detect fakeness makes it impractical to take such things into account.

Ars Mathematica finally reports a retract of the claimed disproof of The Hypothesis. David Ben-Zvi has kindly provided notes on the recent Chicago conference, where Goncharov was talking about Motives, path integrals and trivalent graphs. Sounds intriguing. OK, I printed out the notes. Wow. OMG. Goncharov claims to have identified the category of mixed motives (a.k.a. the holy grail for ordinary real/complex geometry) in terms of path integrals for projective varieties. For instance, when the variety corresponds to modular subgroups indexed by $\hbar = \sqrt{N}^{-1}$, as in TGD or $N$-fold covers of moduli spaces, one gets Langlands from the cohomology. He concludes with a statement that Feynman integrals (with observables) are valued in motivic cohomology. Yeah, duh, the physicists know that. We just don't know how we're ever going to learn that much mathematics.

Ah! That means the S duality we need for the Riemann Hypothesis relies on the whole range of quantised $\hbar$, and is therefore necessarily omega-categorical. That was expected, because the surreal zeta arguments extend through the ordinals. It is fantastically exciting to have some confirmation of this link between $\hbar$ values and S duality. I wonder how string theory will deal with a variable $\hbar$. Oh, I see.

Ars Mathematica finally reports a retract of the claimed disproof of The Hypothesis. David Ben-Zvi has kindly provided notes on the recent Chicago conference, where Goncharov was talking about Motives, path integrals and trivalent graphs. Sounds intriguing. OK, I printed out the notes. Wow. OMG. Goncharov claims to have identified the category of mixed motives (a.k.a. the holy grail for ordinary real/complex geometry) in terms of path integrals for projective varieties. For instance, when the variety corresponds to modular subgroups indexed by $\hbar = \sqrt{N}^{-1}$, as in TGD or $N$-fold covers of moduli spaces, one gets Langlands from the cohomology. He concludes with a statement that Feynman integrals (with observables) are valued in motivic cohomology. Yeah, duh, the physicists know that. We just don't know how we're ever going to learn that much mathematics.

Ah! That means the S duality we need for the Riemann Hypothesis relies on the whole range of quantised $\hbar$, and is therefore necessarily omega-categorical. That was expected, because the surreal zeta arguments extend through the ordinals. It is fantastically exciting to have some confirmation of this link between $\hbar$ values and S duality. I wonder how string theory will deal with a variable $\hbar$. Oh, I see.

Recall that Joan Birman et al studied knots in the Lorenz template with two generating holes X and Y. So knots are expressed as words in X and Y. In Robert Ghrist's paper Branched two-manifolds supporting all links he shows that the template $\mathcal{V}_0$ on more letters contains an isotopic copy of every (tame) knot and link. More specifically, for a parameter range $\beta \in [6.5,10.5]$ every link appears as a periodic solution to the equation which is used to model an electric circuit. This is cool stuff. In M Theory we like ribbon diagrams which are twisted into loops like in the Lorenz template diagram. The universal template $\mathcal{V}_0$ can be embedded in an infinite sequence of more complicated templates, which in turn are embeddable in $\mathcal{V}_0$. Ghrist also considers flows arising from fibrations, such as the 1-punctured torus fibration for the figure 8 knot complement. This fibration flow is also an example of a universal flow.

I was quite intrigued when a mathematical biologist at a conference told me recently that no one really knew why DNA had four bases rather than two. Apparently it isn't clear why self-replicating molecules fail to adopt a binary code in X and Y. Somebody else muttered something about hydrogen bonds and then, inspired and ignorant, I started rambling on about knot generation in templates. After all, DNA molecules need to know how to knot themselves.

I was quite intrigued when a mathematical biologist at a conference told me recently that no one really knew why DNA had four bases rather than two. Apparently it isn't clear why self-replicating molecules fail to adopt a binary code in X and Y. Somebody else muttered something about hydrogen bonds and then, inspired and ignorant, I started rambling on about knot generation in templates. After all, DNA molecules need to know how to knot themselves.

As nosy snoopy noted, these crystal Calabi-Yau papers are really very interesting. I would like to know more about these random partitions. Okounkov has some notes here.

Recall that in Kapranov's non-commutative Fourier transform for three coordinates $x$, $y$ and $z$, it is natural to represent monomials by cubical paths traced out on such melting crystal partitions. The two coordinate case goes back to Heisenberg's original paper, as we have seen. In a modern guise, his sum rule arises in honeycombs, which look a bit like shadows of melting corners.

Recall that in Kapranov's non-commutative Fourier transform for three coordinates $x$, $y$ and $z$, it is natural to represent monomials by cubical paths traced out on such melting crystal partitions. The two coordinate case goes back to Heisenberg's original paper, as we have seen. In a modern guise, his sum rule arises in honeycombs, which look a bit like shadows of melting corners.

... the corpus of mathematics does resemble a biological entity which can only survive as a whole and would perish if separated into disjoint pieces.

Actually, it's freezing down here, but the northern summer is conference time! Later this month the northern hemisphere sees both Loops 07 in Mexico and Strings 07 in Spain. Closer to home in early July, we have GRG18 in Sydney (conference venue in photo). No doubt many other interesting conferences are also coming up soon.

In other news, I am quite saddened to see this mysterious message on Lieven Le Bruyn's blog page. Good thing I printed out his helpful maths lessons. Bloomsday is June 16, which is only next week. I wonder what Homeric epic Lieven has to tell us. Also, keen followers of this blog should definitely take a look at the wonderful links on this Cafe post. Thanks, David.

As Tommaso points out, we must not forget Pascos 07 in London in early July. Have fun, Tommaso.

In other news, I am quite saddened to see this mysterious message on Lieven Le Bruyn's blog page. Good thing I printed out his helpful maths lessons. Bloomsday is June 16, which is only next week. I wonder what Homeric epic Lieven has to tell us. Also, keen followers of this blog should definitely take a look at the wonderful links on this Cafe post. Thanks, David.

As Tommaso points out, we must not forget Pascos 07 in London in early July. Have fun, Tommaso.

Whilst on the topic of AdS/CFT, Michael Rios has an interesting post on dimension altering weak coupling phase transitions for N=4 SUSY Yang-Mills.

A continuous change in dimension from six down to five is reminiscent of Thurston's beautiful fractal 2-spheres, which are filled with a 1-dimensional curve. These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle. The punctures draw out a boundary for the manifold by tracing a knot. Now according to Matti, the fractional modular domains would fit into the domain for the once punctured torus moduli (the $n=2$ case) on the upper half plane. Perhaps the $n=5$ domain (or rather the theta functions) could be used to model a 5-sphere, much as the j-invariant Belyi map links the $n=2$ domain to $\mathbb{CP}^1$.

Note: For the new readers to this blog, our use of the term M Theory must not be confused with its more popular usage in string theory related papers. The letter M stands here possibly for Motive, or perhaps Monad. Although these terms do appear in the popular literature, they rarely correspond to the physical usage we would like to make of them.

A continuous change in dimension from six down to five is reminiscent of Thurston's beautiful fractal 2-spheres, which are filled with a 1-dimensional curve. These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle. The punctures draw out a boundary for the manifold by tracing a knot. Now according to Matti, the fractional modular domains would fit into the domain for the once punctured torus moduli (the $n=2$ case) on the upper half plane. Perhaps the $n=5$ domain (or rather the theta functions) could be used to model a 5-sphere, much as the j-invariant Belyi map links the $n=2$ domain to $\mathbb{CP}^1$.

Note: For the new readers to this blog, our use of the term M Theory must not be confused with its more popular usage in string theory related papers. The letter M stands here possibly for Motive, or perhaps Monad. Although these terms do appear in the popular literature, they rarely correspond to the physical usage we would like to make of them.

It's good to see Matti Pitkanen so happy about the fact that there are three generations in TGD. After all, it would be a poor theory that postdicted a number that contradicted observations, although not as poor as a theory that doesn't postdict anything beyond established physics. From a category theory point of view, the extension of the modular group to a series indexed by n is most easily characterised by a groupoid on the objects n. Since there is an associated sequence of subgroups of the modular group, a group theorist may wish instead to study profinite completions.

Recall that Lieven Le Bruyn was discussing the fact that

$PSL_{2}(\mathbb{Z}) \simeq B_{3} \backslash \langle ( \sigma_1 \sigma_2 \sigma_1 )^{2} \rangle$

where $B_{3}$ is the braid group on three strands and the $\sigma_{i}$ are the usual generators of the group. This came up with regard to the work of Linas Vepstas on Minkowski's devil's staircase. Vepstas has studied the fractal symmetries of this continuous function from the interval to itself via mappings of binary trees. He observes that an infinite subtree is always isomorphic to the full tree. The Minkowski map arises as a mapping from the dyadic tree with root $\frac{1}{2}$ (this is like the bottom half of the positive surreal tree which we wanted to associate with Riemann zeta arguments) to the Farey tree. As Martin Huxley says, "A nice way of stating the Riemann hypothesis is that the Farey sequence is distributed as uniformly in the interval 0 to 1 as it possibly can be."

By embedding a binary tree in the upper half plane, one naturally encounters fundamental domains for the modular group. Presumably one could play a similar game with n-ary trees in n dimensions (recall the tetractys), with categorified n-tuplet groupoids replacing the modular group. In terms of braid groups, one simply increases the number of generators. As we have seen, the restricted $B_{3}$ can describe the $n=2$ case of the (massless) fermions. Moreover, braid depth is naturally associated to the depth of MZV algebras. Note that only in the $n=2$ case does the Hurwitz doublet feature zeroes that appear to lie on the critical line, and the Hurwitz $\zeta_H (s)$ has an extra zero at $s=0$.

Aside: Now this looks cool!

Recall that Lieven Le Bruyn was discussing the fact that

$PSL_{2}(\mathbb{Z}) \simeq B_{3} \backslash \langle ( \sigma_1 \sigma_2 \sigma_1 )^{2} \rangle$

where $B_{3}$ is the braid group on three strands and the $\sigma_{i}$ are the usual generators of the group. This came up with regard to the work of Linas Vepstas on Minkowski's devil's staircase. Vepstas has studied the fractal symmetries of this continuous function from the interval to itself via mappings of binary trees. He observes that an infinite subtree is always isomorphic to the full tree. The Minkowski map arises as a mapping from the dyadic tree with root $\frac{1}{2}$ (this is like the bottom half of the positive surreal tree which we wanted to associate with Riemann zeta arguments) to the Farey tree. As Martin Huxley says, "A nice way of stating the Riemann hypothesis is that the Farey sequence is distributed as uniformly in the interval 0 to 1 as it possibly can be."

By embedding a binary tree in the upper half plane, one naturally encounters fundamental domains for the modular group. Presumably one could play a similar game with n-ary trees in n dimensions (recall the tetractys), with categorified n-tuplet groupoids replacing the modular group. In terms of braid groups, one simply increases the number of generators. As we have seen, the restricted $B_{3}$ can describe the $n=2$ case of the (massless) fermions. Moreover, braid depth is naturally associated to the depth of MZV algebras. Note that only in the $n=2$ case does the Hurwitz doublet feature zeroes that appear to lie on the critical line, and the Hurwitz $\zeta_H (s)$ has an extra zero at $s=0$.

Aside: Now this looks cool!

Can anyone provide an update on Woit's report on Witten's new ideas? There seems to be no paper online, but one may materialise closer to Strings07.

From an M Theory point of view this $SO(2,2)$ theory represents a physical 2-Time domain, not merely a 2+1D model in a naive quantization scheme. Recall that for the $c=24$ case the partition function is exactly the function $J(q) = j(q) - 744$ where $j(q)$ is the famous q-expansion of the modular j-invariant (which we have been discussing) for $q = e^{2 \pi i \tau}$.

Now Matti Pitkanen has just been looking at how the Hurwitz zeta function (at the special point $a = 0.5$) naturally arises on imposing modular invariance on theta series. By definition, the j-invariant is invariant under modular transformations, so perhaps there is a simple relation between its theta function components and the components for the (let us call it) SUSY doublet of Hurwitz $\zeta (s,\frac{1}{2})$ and Riemann zeta functions. Starting with $\theta (0, \tau)$ we see that under $\tau \rightarrow \tau + 1$ this becomes $\theta_{01}(0, \tau)$. This in turn becomes $\theta (0, \tau)$ under a second transformation and both functions are part of the j-invariant triality. The third component $\theta_{10} (0, \tau)$ transforms to $(\sqrt{2}+ \sqrt{2} i) \theta_{10} (0, \tau)$ which is an 8th root of unity. This particular Hurwitz function is simply $\zeta_{H}(s) = (2^s - 1) \zeta (s)$, a simple multiple of the Riemann zeta function. According to Mathworld it has the interesting functional relation

$\zeta_{H}(s) = 2 (4 \pi)^{s-1} \Gamma (1-s) (sin \pi (1 + \frac{s}{2}). \zeta_{H}(1-s) + sin \pi (\frac{s}{2}). \zeta (1 - s))$

but this doesn't seem quite right. This should be the usual functional relation for the Riemann zeta function. Anyway, the doublet $(\zeta, \zeta_H)$ describes the two term duality of the j-invariant.

From an M Theory point of view this $SO(2,2)$ theory represents a physical 2-Time domain, not merely a 2+1D model in a naive quantization scheme. Recall that for the $c=24$ case the partition function is exactly the function $J(q) = j(q) - 744$ where $j(q)$ is the famous q-expansion of the modular j-invariant (which we have been discussing) for $q = e^{2 \pi i \tau}$.

Now Matti Pitkanen has just been looking at how the Hurwitz zeta function (at the special point $a = 0.5$) naturally arises on imposing modular invariance on theta series. By definition, the j-invariant is invariant under modular transformations, so perhaps there is a simple relation between its theta function components and the components for the (let us call it) SUSY doublet of Hurwitz $\zeta (s,\frac{1}{2})$ and Riemann zeta functions. Starting with $\theta (0, \tau)$ we see that under $\tau \rightarrow \tau + 1$ this becomes $\theta_{01}(0, \tau)$. This in turn becomes $\theta (0, \tau)$ under a second transformation and both functions are part of the j-invariant triality. The third component $\theta_{10} (0, \tau)$ transforms to $(\sqrt{2}+ \sqrt{2} i) \theta_{10} (0, \tau)$ which is an 8th root of unity. This particular Hurwitz function is simply $\zeta_{H}(s) = (2^s - 1) \zeta (s)$, a simple multiple of the Riemann zeta function. According to Mathworld it has the interesting functional relation

$\zeta_{H}(s) = 2 (4 \pi)^{s-1} \Gamma (1-s) (sin \pi (1 + \frac{s}{2}). \zeta_{H}(1-s) + sin \pi (\frac{s}{2}). \zeta (1 - s))$

but this doesn't seem quite right. This should be the usual functional relation for the Riemann zeta function. Anyway, the doublet $(\zeta, \zeta_H)$ describes the two term duality of the j-invariant.

Carl Brannen has a new blog post refuting Mottle's opinion piece on variable speed of light theories. Meanwhile Matti Pitkanen has been posting interesting updates on TGD and the zero energy ontology. As good old Einstein once said, "We can't solve problems by using the same kind of thinking we used when we created them."

In Lesson 59 we saw how the functional equation for the Riemann zeta function depended on a theta series at two different arguments, $\tau$ and $\frac{-1}{\tau}$. The interchange of these two complex parameters is familiar as an order 2 operation $S$ for the modular group. Perhaps the order 3 operation, $TS$, should give us a triality relation for the zeta function, analogous to the triality satisfied by the j-invariant. Recall that the operation $T$ sends $\tau$ to $\tau + 1$, which is why a fundamental region in the upper half plane has width 1. $TS$ fixes the cubed root of unity $\omega$. Complex numbers that get sent to the unit circle under $TS$ take the form $(\frac{1}{2} + iy)$.

But from M Theory one would guess that a triality relation arises from a triple of moduli, rather than the 1-punctured torus case. Is there a natural way to view the modular generators in terms of higher dimensional Belyi maps? This is presumably the sort of thing that nice Calabi-Yau 3-folds can do.

In Lesson 59 we saw how the functional equation for the Riemann zeta function depended on a theta series at two different arguments, $\tau$ and $\frac{-1}{\tau}$. The interchange of these two complex parameters is familiar as an order 2 operation $S$ for the modular group. Perhaps the order 3 operation, $TS$, should give us a triality relation for the zeta function, analogous to the triality satisfied by the j-invariant. Recall that the operation $T$ sends $\tau$ to $\tau + 1$, which is why a fundamental region in the upper half plane has width 1. $TS$ fixes the cubed root of unity $\omega$. Complex numbers that get sent to the unit circle under $TS$ take the form $(\frac{1}{2} + iy)$.

But from M Theory one would guess that a triality relation arises from a triple of moduli, rather than the 1-punctured torus case. Is there a natural way to view the modular generators in terms of higher dimensional Belyi maps? This is presumably the sort of thing that nice Calabi-Yau 3-folds can do.

Professor Tao has kindly made available some colourful slides of his Cosmic Distance Ladder talk, which he gave in Australia last year. The slides nicely summarise some high points in the history of distance measurements across the cosmos.

At a seminar here yesterday we heard that as recently as the 1920s people still believed that the Milky Way was the entire universe, with the sun sitting at its centre. A mapping of globular clusters in the Milky Way soon indicated that the sun lay on the periphery of the galaxy, and it was finally recognised that the strange spiral nebulae were in fact distant galaxies, not unlike our own. The same seminar concluded, on the observation that accelerated publication rates had not produced as many major breakthroughs in the last two decades, that technology had finally caught up with observations over the electromagnetic spectrum and that we may well have seen most of what there is to see. Naturally there was some dissent. To believe that in a mere 80 years humanity can go from a relatively trivial understanding of the cosmos to complete comprehension is hubris.

Today we are just beginning to observe the so-called Dark Matter in the skies. The luminous baryonic matter is a small fraction ($\frac{\pi - 3}{\pi}$) of the matter we imagine is out there. No doubt this is only another small step in the evolving vision of our mind. A Theory of Everything can never be more than a theory of what little we have known before.

At a seminar here yesterday we heard that as recently as the 1920s people still believed that the Milky Way was the entire universe, with the sun sitting at its centre. A mapping of globular clusters in the Milky Way soon indicated that the sun lay on the periphery of the galaxy, and it was finally recognised that the strange spiral nebulae were in fact distant galaxies, not unlike our own. The same seminar concluded, on the observation that accelerated publication rates had not produced as many major breakthroughs in the last two decades, that technology had finally caught up with observations over the electromagnetic spectrum and that we may well have seen most of what there is to see. Naturally there was some dissent. To believe that in a mere 80 years humanity can go from a relatively trivial understanding of the cosmos to complete comprehension is hubris.

Today we are just beginning to observe the so-called Dark Matter in the skies. The luminous baryonic matter is a small fraction ($\frac{\pi - 3}{\pi}$) of the matter we imagine is out there. No doubt this is only another small step in the evolving vision of our mind. A Theory of Everything can never be more than a theory of what little we have known before.

Ellis's recent critique of varying speed of light theories has caused a flurry of blog posts. A summary of his five main points, in his own words, runs as follows.

Point 1: Any VSL theory involving variable speed of photon travel must of necessity be based on some other method of measuring spatial distances than radar.

Point 2: Any VSL theory based on changes in the metric tensor components must explain how it differs from GR and how time and space measurements are related to the metric tensor.

Point 3: Any VSL theory involving a change in the limiting speed will not be Lorentz invariant; the way Lorentz invariance is broken must be made explicit.

Point 4: Any VSL theory involving a change in the speed of the photon travel must eventually propose some other equations than standard Maxwell's equations to govern electromagnetism.

Point 5: Any VSL theory must be done consistently in terms of its effects on the whole set of physical equations.

The critique takes a glaringly classical view of the cosmos, but one can hardly argue with the last four points. As far as popular VSL theories are concerned, I mostly agree with the critique. The first point, however, makes no sense at all! It is of course true that the very definition of the metre defines the speed of light locally, but this does not imply that we should compare measurements over cosmological distances using the same speed of light. How do we compare measurements over cosmological distances at all? The only work I am aware of which considers real measurements over large scales is Louise Riofrio's analysis of the speed of light on an older Earth. In fact, sensible approaches to quantum gravity take the measurement of distance via photon travel far more seriously than it is considered in classical gravity. Connes has expressed this nicely in his motivation for a new spectral physics. A varying speed of light is introduced by Riofrio as a useful picture for what we observe in the cosmos from here. Ellis can hardly argue with the observed expansion of space, or rather the observed increase in distance between stationary objects over cosmic time, which is based very much upon the spectral concept of distance.

Ellis mentions Penrose's approach to causality, so he probably won't argue with anybody who wants to replace Maxwell's equations immediately with sheaf cohomology in twistor theory. As Penrose himself says, twistor theory seems to have a bearing on quantum gravity, rather than general relativity itself. But it is well known that mass generation is a difficult question in twistor theory, although Hughston and Hurd made interesting progress in the late 1980s by combining two massless solutions to obtain an $H^2$ massive state. This work only stresses the importance of understanding higher dimensional non-Abelian cohomology, and in this framework a varying speed of light is really a minor concern.

The wording of Ellis's points betrays some further prejudices about the mathematics being used in VSL investigations. For example, the whole set of physical equations is far too restrictive a notion for a category theorist. Causality takes us beyond the realm of mere set theory, as we have seen.

Point 1: Any VSL theory involving variable speed of photon travel must of necessity be based on some other method of measuring spatial distances than radar.

Point 2: Any VSL theory based on changes in the metric tensor components must explain how it differs from GR and how time and space measurements are related to the metric tensor.

Point 3: Any VSL theory involving a change in the limiting speed will not be Lorentz invariant; the way Lorentz invariance is broken must be made explicit.

Point 4: Any VSL theory involving a change in the speed of the photon travel must eventually propose some other equations than standard Maxwell's equations to govern electromagnetism.

Point 5: Any VSL theory must be done consistently in terms of its effects on the whole set of physical equations.

The critique takes a glaringly classical view of the cosmos, but one can hardly argue with the last four points. As far as popular VSL theories are concerned, I mostly agree with the critique. The first point, however, makes no sense at all! It is of course true that the very definition of the metre defines the speed of light locally, but this does not imply that we should compare measurements over cosmological distances using the same speed of light. How do we compare measurements over cosmological distances at all? The only work I am aware of which considers real measurements over large scales is Louise Riofrio's analysis of the speed of light on an older Earth. In fact, sensible approaches to quantum gravity take the measurement of distance via photon travel far more seriously than it is considered in classical gravity. Connes has expressed this nicely in his motivation for a new spectral physics. A varying speed of light is introduced by Riofrio as a useful picture for what we observe in the cosmos from here. Ellis can hardly argue with the observed expansion of space, or rather the observed increase in distance between stationary objects over cosmic time, which is based very much upon the spectral concept of distance.

Ellis mentions Penrose's approach to causality, so he probably won't argue with anybody who wants to replace Maxwell's equations immediately with sheaf cohomology in twistor theory. As Penrose himself says, twistor theory seems to have a bearing on quantum gravity, rather than general relativity itself. But it is well known that mass generation is a difficult question in twistor theory, although Hughston and Hurd made interesting progress in the late 1980s by combining two massless solutions to obtain an $H^2$ massive state. This work only stresses the importance of understanding higher dimensional non-Abelian cohomology, and in this framework a varying speed of light is really a minor concern.

The wording of Ellis's points betrays some further prejudices about the mathematics being used in VSL investigations. For example, the whole set of physical equations is far too restrictive a notion for a category theorist. Causality takes us beyond the realm of mere set theory, as we have seen.