Monsters II

In Ronan’s book on the history of the Monster group, he mentions how the Leech lattice in 24 dimensions is obtained from a light ray in a 26 dimensional space with Lorentzian signature. The light ray goes through the point with coordinates (0,1,2,3,…,23,24,70) and its length is zero precisely because the sum of squares up to 24^2 is the same as 70^2. Dixon has used the octonions to study the Leech lattice.

Integer points in higher dimensional spaces appear in nice geometric realisations of operad polytopes, as we saw for the Stasheff associahedron. To deal with a variety of lattices in 24 or 26 dimensions we would need a pretty good handle on higher operad theory. In the last post a variety of interesting polytopes was shown. Loday has worked on lattice type realisations of associahedra and permutohedra. Now there are just so many more polytopes to look at.

Recall that Borcherd’s proof of Monstrous Moonshine used a vertex algebra for the Monster group. These conjectures originated in some remarkable observations matching the coefficients of the number theoretic j-function (which turned up in our study of ribbon graphs and Grothendieck’s dessin d’enfants) to numbers in the character table for the Monster group. People who study these things are interested in double loop spaces, which are what 2-operads are about. That is, one of the requirements of a definition of weak n-category is that the groupoids be capable of modelling homotopy n-types. This is what the table in the last post is secretly about. A 2-operad component is labelled not by an ordinal n, which is a 1-level tree on n leaves, but by a 2-level tree.

Ronan also mentions the first coincidence leading to the Moonshine conjectures. Ogg was looking at certain subgroups of the modular group. The question was, when did these groups, indexed by a prime p, yield genus zero as opposed to higher genus surfaces for the action on the hyperbolic plane? He found the precise allowable primes, namely 2,3,5,7,11,13,17,19,23,29,31,41,47,59,71 which are precisely the prime factors in the size of the Monster group.

Michael Batanin has kindly given me permission to put up some of the beautiful pictures from his talk on Wednesday at AustMS 06. This resultoassociahedron has probably not appeared on the web before. Notice the indexing by a two-level tree, as mentioned yesterday.

Here is some more combinatorics from Dinner at the End of the Universe…

There are quite a number of operads floating around, such as this B operad…

The B operad is a true n-dimensional generalisation of the Stasheff one for 1-fold loop spaces. The basic K diagrams themselves do not form an operad, for a reason that we will discuss in the future. At the end of his talk, Batanin discussed the Baez-Dolan stabilisation hypothesis, for which there is now evidence.
Isn’t this great! Now we can start calculating things in M-theory using simple manipulations of polytopes.

AustMS 2006 III

The freeway 294 bus route turned out to be just as disastrous as the other options so I was lucky to make Skinner’s talk reviewing the Birch-Swinnerton-Dyer conjecture, which started out very simply explaining the genus of a curve and remained clear whilst moving rapidly into some heavy jargon.

In the Mathematical Physics session David Roberts spoke about bundle gerbes and higher Yang-Mills theory, and he was kind enough to translate occasionally for the category theorists who ran off at the end of the talk to hear Dominic Verity on complicial sets. Actually, Verity’s title was ‘Non-abelian cohomology as the raison d’etre for higher category theory‘.

In the afternoon there was another category theory session. Dorette Pronk spoke about strings of adjunctions, an analysis of which involves Temperley-Lieb type diagrams, such as discussed by Baez in week 174. That is, the planar tangles represent 2-morphisms between points labelled by arrows and their duals. Finally today, Mark Weber spoke about one of my favourite topics, namely 2-toposes, in an elementary sense. That doesn’t mean elementary as in simple, but as in axiomatic. The tricky thing is to figure out what the subobject classifier should be. It turns out that a good idea for the 2-truth arrow is the functor from pointed sets to sets!

AustMS 2006 II

Today I really tried to make the first talk, but was stumped by bad traffic after running for the bus on the last leg. Fortunately Michael Batanin’s plenary talk wasn’t until 11.30 am. It was nicely introduced by Ross Street, who told us how Batanin had moved from Russian economics back into mathematics and how after only seven months of work he completely revolutionised higher category theory. Batanin spoke about the combinatorics of higher operads.

Recall the example of the Stasheff polytope in dimension three, as realised by Loday. The collection of all such associahedra is an example of a 1-operad which characterises 1-fold loop spaces. Batanin gave a history of Stasheff’s ideas after a more general history discussing the need for a coherent theory of categorical coherence laws. The difficulty was in finding a combinatorial structure for the n-fold case.

He now understands n-operads by generalising the ordinals [m] that index the n-ary operations of a 1-operad to higher level trees, where an ordinal [m] is just a simple tree with one vertex and m leaves, and hence one level. By finding nice examples of 2-operads and higher n-operads one discovers the most astonishing range of polytopes. The collection of Stasheff associahedra, for example, form an n=1 example of a whole series K(n) of operads (related to the Getzler-Jones operads). In particular, the hexagons of the axioms for a braided monoidal category naturally show up here at n=2, whereas we saw the Mac Lane pentagon at n=1.

At the education session in the afternoon Terence Tao spoke about the measurement of distances in astronomical history, starting with the measurement of the Earth’s radius by Eratosthenes. For example, to measure the distance from the Earth to the Moon the Greeks observed that the Earth’s shadow takes a certain amount of time to pass over the Moon during a lunar eclipse. Since they already knew the radius of the Earth this gave them a fairly accurate measure of the distance using a circular orbit for the Moon about the Earth.

Aristarchus of Samos tried to measure the distance from the Earth to the Sun by observing the phases of the Moon. Since the Sun is at a finite distance, the half-Moon comes just before the halfway point in time between the times of new Moon and full Moon, because it forms a right angled triangle with the Sun and Earth.

Tao then continued increasing the scale of distance observations until he eventually mentioned briefly the poorly understood type IA supernovae. It was nice to see his great respect for Kepler’s insightful method for computing the orbit of the Earth about the Sun observationally by using Mars as a reference point. From here Kepler was able to derive the third law
GM = 4 pi^2 R^3 T^(-2).

To finish a lovely day there was a friendly reception with some delicious vegetable savouries and chicken kebabs.

AustMS 2006

Since I happen to be around at the AustMS 06 conference at Macquarie University I probably have some obligation to blog about it. We’re expecting mostly fine weather this week and a campus teeming with possums and parrots, as usual.

There are many parallel sessions. In the morning I went to the Math Physics session. Lucy Gow spoke about Yangians for Lie superalgebras and then we heard again from
Robert Coquereaux on Quantum Groupoids, but this time a powerpoint presentation which helpfully reproduced many of the graphs associated to SU(N) at level k.

The Category Theory session had four nice 25 minute talks, in particular by Panchadcharam on Mackey functors and by Pastro on the Frobeniusness of Hopf monoids in braided monoidal categories. This session was assigned one of the smallest seminar rooms, so all the seats were taken.

Terence Tao gave an astonishingly clear lecture in the afternoon on the existence of long arithmetic progressions in the primes and other matters prime. Apparently we know quite a lot about almost primes, which are integers with only a few prime factors, but getting a handle on the primes themselves is really tricky! He explained how subtle was the problem of separating structure in the primes from pseudorandomness. Tao finished the survey style talk by mentioning a best bound that is only known in terms of a 7-fold exponential: two to the two to the two to the…you get the idea…and apparently, if the Riemann hypothesis were true we would only be able to knock off one folding!

To finish the day there was anniversary cake (chocolate!) and champagne. Yum.

Avalanche

A familiarity with practical avalanche science is essential for the mountain traveller. The mountains, as Nature in general, have one simple answer for those who think they can get away with mere academic knowledge. The world’s glaciers are disappearing due to global warming, as discussed in Al Gore’s film, An Inconvenient Truth, which I highly recommend to every haughty physicist, and everybody else as well, although Al Gore appears to have great difficulty understanding how difficult it is to change one’s life whilst living in poverty. What does one care how that packet of salami was manufactured if all one sees is sustenance? Women in physics have been travelling amongst mountains all their lives, rarely finding a wide green valley in which to rest. Today the gentleman who runs the Gravity blog wrote a nice article on the work of Riofrio which has been studiously ignored by the vast majority of physicists for some time, as far as I can tell. But the truth will out, as they say, especially in an Internet age with all the wranglings recorded.

In other news, Paul Davies has a new book out, The Goldilocks Enigma. Of course I haven’t read it, but apparently he discusses at the end the possibility that the laws of physics predict that the universe should contain life. Good for him. Any other solution to the so called fine-tuning problem is clearly hogwash, given recent advances in a ribbon diagram approach to M-theory. It was quite some time ago now that M. C. Shum pointed out the clear relation between the structure of biological molecules and tortile tensor categories in her thesis.

Update 25 Sep: Take a look at Riofrio’s thread on Coral and Cosmology, about how the recession of the moon can tell us about the varying speed of light!

Measuring mass

The only basic standard unit that still requires an artefact is the kilogram. Masses are measured in comparison to the standard using an ordinary balance. Today the Hall effect can give us accurate values of Planck’s ‘constant’ h and electric charge e, and an independent definition of the Volt and Ohm. Since the metre and second are defined without an artefact, this would allow a definition of mass unit independent of an artefact using a Watt balance.

A mass spectrometer measures the mass of a charged particle travelling at known speed by noting the distance it moves in a circular arc, given by r = mv/qB for magnetic field B.

The mass of the sun is measured by applying Kepler’s Law M = (4 pi^2 R^3)/(GT^2) in terms of the period T of the Earth’s orbit and the mean distance R of the Earth to the sun. Observe that Kepler’s law is essentially the same as Riofrio’s universal law rescaled by 4 pi^2. The value of R itself is now known fairly accurately, but the mass of the sun also depends on an accurate value for the gravitational constant G.

Observe that in all of the above settings the measurement of mass actually depends on the measurement of other quantities, most noticeably distances. In gravity a mass has a characteristic Schwarzschild radius for which r = 2Gm/c^2. Similarly, the quantum mechanical Compton wavelength is given by l = h/mc, but this indicates an inverse relation between mass and length. As is well known, these lengths agree at the Planck mass. This suggests that any theory attempting to explain the meaning of quantised mass ought to incorporate some form of T-duality, a correspondence between lengths and their inverses, up to a scale factor.

Trails and Triality

This little hill is a popular walk for humans near a town down south. Of course I stole the photo from one of them. The humans provide much entertainment for us keas.

On his octonion webpages John Baez discusses triality, namely non-degenerate triple maps t(u,v,w) into the reals. These are closely associated with the division algebras. When the triple takes the form (vector, L spinor, R spinor), each of which is acted upon by Spin(8), triality takes on a physical significance. The Dynkin diagram for Spin(8) looks like a trivalent vertex (4 nodes). In M-theory one also considers open and closed string vertices.

Many people seem to think I’ve gone completely crazy. Cool! Something happening. Make sure to look out for Carl Brannen’s upcoming paper on the derivation of the lepton masses! And remember: what can be said in a line is not always trivial.

Trippy Trefoils

Loday found a realisation of all the Stasheff polytopes, including the K4 one shown here. The vertices in 4 dimensions are permutations on 4 letters, so the polytope lies in the 3 dimensional plane x + y + z + w = 10 and the vertices are on the integer lattice.
Loday also noticed that by tracing a curve around the faces, once through the pentagons and twice through the squares, one obtains a trefoil knot by judiciously choosing the crossings at the centre of the squares.

The classical Alexander knot invariant for the trefoil is obtained as follows. Let (t,-t,1,-1) be labels for the 4 region types: such as before undercrossing on left etc. Make a 3×5 matrix for the 3 crossings and the 5 regions of the trefoil knot, such as

-t t -1 0 1
-1 t 0 -t 1
0 1 -1 -t t

Now ignore 2 columns, take the determinant, and scale by powers of t appropriately, to obtain A(trefoil) = (t^-1) – 1 + t. The Jones polynomial can distinguish handedness for the trefoil knot. The left Jones polynomial is J(trefoil) = t + t^3 – t^4. Recall that the trefoil knot appeared recently in a heuristic discussion of the Koide formula for mass ratios. Here one uses the variable t=exp(a), which is the usual change of variables for Vassiliev invariants.

Space Filling

In 1890 Peano showed that there existed a 1-dimensional curve (with endpoints) filling the square [0,1]x[0,1]. One can similarly fill any hypercube with a curve. This works because the cardinality of the continuum is a fixed number, no matter what the dimension.

In the 1990s, Thurston studied 3-dimensional geometry and found some other beautiful space-filling curves, in particular curves filling a sphere. The example shown on page 3 of the paper comes from a fibration of a 3-manifold over a circle with fibre the punctured torus, where the puncture is bounded by a figure8 knot. Such manifolds have tori as boundaries, namely a neighbourhood of the knot. Geometry in 3 dimensions is very rich. For example, the set of volumes of finite volume hyperbolic manifolds is of cardinality omega^omega. Here is a helpful outline of notes about Thurston’s work.

This is nice, because in three dimensions we can still picture what is going on. What about higher dimensions? The fact is that higher dimensions are actually a lot simpler than dimension three. It was only recently that the Poincare conjecture was proven for dimension 3, whereas it has been known for dimensions > 4 since the early 1960s (Smale) and for dimension 4 since 1982 (Freedman).

The importance of knot theory in understanding 3-manifolds begs the question: is there a categorical way to understand this complexity? Knots, after all, come from diagrams describing the structure of braided tensor categories. Ribbons appear with the addition of extra structure. Computations involving moving knots and ribbons about are much, much simpler than more traditional analogues.