Ted Bastin, H. Pierre Noyes, John Amson, Clive W. Kilmister: On the physical interpretation and the mathematical structure of the combinatorial hierarchy, *International Journal of Theoretical Physics* **18** (1979), 445–488; doi: 10.1007/BF00670503

Here is the abstract. See what you make of it.

The combinatorial hierarchy model for basic particle processes is based on elementary entities; any representation they may have is discrete and two-valued. We call them Schnurs to suggest their most fundamental aspect as concatenating strings. Consider a definite small number of them. Consider an elementary creation act as a result of which two different Schnurs generate a new Schnur which is again different. We speak of this process as a “discrimination.” By this process and by this process alone can the complexity of the universe be explored. By concatenations of this process we create more complex entities which are themselves Schnurs at a new level of complexity. Everything plays a dual role in which something comes in from the outside to interact, and also serves as a synopsis or concatenation of such a process. We thus incorporate the observation metaphysic at the start, rejecting Bohr’s reduction to the haptic language of common sense and classical physics. Since discriminations occur sequentially, our model is consistent with a “fixed past-uncertain future” philosophy of physics. We demonstrate that this model generates four hierarchical levels of rapidly increasing complexity. Concrete interpretation of the four levels of the hierarchy (with cardinals 3,7,127,2^{127}-1∼10^{38}) associates the three levels which map up and down with the three absolute conservation laws (charge, baryon number, lepton number) and the spin dichotomy. The first level represents +, −, and ± unit charge. The second has the quantum numbers of a baryon-antibaryon pair and associated charged meson (e.g.,n^{-}n,p^{-}n,p^{-}p,n^{-}p,π^{+},π^{0},π^{-}). The third level associates this pair, now including four spin states as well as four charge states, with a neutral lepton-antilepton pair (e^{-}e or v^{-}v), each pair in four spin states (total, 64 states) – three charged spinless, three charged spin-1, and a neutral spin-1 mesons (15 states), and a neutral vector boson associated with the leptons; this gives 3+15+3×15=63 possible boson states, so a total correct count of 63+64=127 states. Something like SU_{2}×SU_{3} and other indications of quark quantum numbers can occur as substructures at the fourth (unstable) level. Breaking into the (Bose) hierarchy by structures with the quantum numbers of a fermion, if this is an electron, allows us to understand Parker-Rhodes’ calculation of *m*_{p}/*m*_{e} =1836.1515 in terms of our interpretation of the hierarchy. A slight extension gives us the usual static approximation to the binding energy of the hydrogen atom, α^{2}*m*_{e}*c*^{2}. We also show that the cosmological implications of the theory are in accord with current experience. We conclude that we have made a promising beginning in the physical interpretation of a theory which could eventually encompass all branches of physics.

My first reaction is something along these lines. Pythagoras is thought to have believed that “all is number”, and also to have believed in reincarnation. (Of course, we know nothing about what he really believed.) So perhaps these authors are channelling the spirit of Pythagoras.

Note also that *Schnur* is German for “string”, as claimed, but is also defined by the Urban Dictionary as “the ultimate insult that means absolutely nothing”. Interesting?

Actually reading the paper didn’t clear up all my doubts. The setting is sets of binary strings. A set of strings is called a *discriminately closed subset* if it consists of the non-zero elements in a subspace of the vector space of all strings of fixed length *n*. Such a subset has cardinality 2^{j}−1, where *j* is its dimension. Now a step in the *combinatorial hierarchy* involves finding a set of 2^{j}−1 matrices which are linearly independent and have the property that each of them fixes just one vector in the subset (I think this is right, but the wording in the paper is not completely clear to me). These matrices span a DCsS of dimension 2^{j}−1 in the space of all strings of length *n*^{2}.

Of course, the exponential function grows faster than the squaring function, so the hierarchy (starting from any given DCsS) is finite. Their most important example starts with *j* = *n* = 2 (two linearly independent vectors in {0,1}^{2} and their sum), and proceeds to *j* = 2^{2}−1 = 3, *n* = 2^{2} = 4; then *j* = 2^{3}−1 = 7, *n* = 4^{2} = 16; then *j* = 2^{7}−1 = 127, *n* = 16^{2} = 256; then *j* = 2^{127}−1 ∼ 10^{38}, *n* = 256^{2} = 65536 (but this is impossible, so I am not sure what it means to continue the hierarchy to this point).

They point out that the numbers 127 and 2^{127}−1 are close to, respectively, the fine structure constant and the ratio of strengths of electromagnetic to gravitational force between protons. If you use cumulative sums, you get 3, 10, 137 and a number which is again about 10^{38}, and of course 137 is even closer to the target. But I am not sure why you should do this, and in any case, the fine structure constant is measureably different from 137. The non-existence of 10^{38} linearly independent matrices of order 256 supposedly has something to do with “weak decay processes”.

The paper contains some constructions of the appropriate hierarchies. One could pose the mathematical question: do the required linearly independent non-singular matrices exist for any *n* and *j* for which 2^{j}−1 ≤ *n*^{2}?

By this stage I was floundering, so I gave up my careful reading. I noted at a certain point a calculation of the ratio of proton mass to electron mass giving a value 137π/((3/14)×(1+(2/7)+(2/7)^{2})×4/5), agreeing with the experimental value to eight significant figures. (There are three terms in the geometric series because the hierarchy falls over at step 4.) Of course, putting in a more accurate value for the fine structure constant would make the agreement less good: the authors do attempt to explain this.

(If you are a mathematician looking at the paper, the mathematics is put in an appendix starting on page 480, so you can avoid the physics.)

At another point in the conversation, John told me that he ran with the fox and hunted with the hounds. On the strength of this, I can perhaps attribute to him a certain degree of scepticism about all this.

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So in the morning as I lay half asleep, these ideas assembled themselves into a pattern. Maybe not a really significant pattern, but maybe worth something. It occurred to me that perhaps someone should write a book on this …

Recently I wrote about the formulae for the number of ways of choosing a sample of *k* from a set of *n* objects:

- Order significant: with replacement , without replacement (the falling factorial)
- Order not significant: with replacement , without replacement

In my view this table illustrates the main division of combinatorics. This is perhaps clearest if we interpret the *n* objects as “colours”, and regard the task as counting colourings of a set of size *k* (which we can regard as a complete graph). The first formula is the simplest. Going right along a row, we impose a *structural* condition: the colouring should be *proper* (that is, the ends of an edge should get different colours). So the second entry in the first row is the chromatic polynomial of *K _{k}*, evaluated at

The second row is different: it counts the colourings up to *symmetry*. That is, it counts orbits of the symmetric group of degree *k* on the colourings of the two types described in the first row.

Structural combinatorics will take us from this example to graphs, matroids, and beyond. Symmetry seems only to lead to group actions and the resulting counting, but in fact it is much more widespread than that, as I want to discuss.

It is not hard to extend the above analysis to any graph. The *chromatic polynomial* of a graph *X* is a monic polynomial whose degree is equal to the number of vertices of *X*; it is defined by the property that its evaluation at a positive integer *q* is equal to the number of *q*-colourings of *X*.

What corresponds to the second entry in the second row of our introductory example, when we impose both symmetry and structure (that is, count proper colourings up to symmetry)?

If *G* is a group of automorphisms of *X*, there is a polynomial, the *orbital chromatic polynomial* of *X* and *G*: its degree is equal to the number of vertices of *X*, and its leading coefficient is 1/|*G*|; its evaluation at a positive integer *q* is equal to the number of *q*-colourings up to the action of *G*, that is, the number of orbits of *G* on colourings.

To see this, we use the *Orbit-counting Lemma*, according to which the number of orbits is equal to the average number of fixed points of elements of *G*. So we have to count the colourings fixed by an automorphism *g*. Such colourings must be constant on the cycles of *g*. So, if there are two adjacent vertices in the same cycle, the number is 0. Otherwise, form a contracted graph *X*/*g* by shrinking each cycle to a vertex; there is a bijection between colourings of *X* fixed by *g* and colourings of *X*/*g*, so the required number is the chromatic polynomial of *X*/*g* evaluated at *q*.

Thus, the orbital chromatic polynomial is an average of terms each of which is zero or a chromatic polynomial; the chromatic polynomial of the whole graph appears once in this average. So the claimed result is true.

Koko Kayibi and I started an investigation of roots of orbital chromatic polynomials. There are some similarities, and some differences, with the theory of roots of chromatic polynomials.

In a paper with Bill Jackson and Jason Rudd, which I secretly regard as one of my best (though not many people seem to agree), we gave a far-reaching extension of the orbital chromatic polynomial.

This is in some sense inspired by two pieces of folklore, which are not quite what they seem:

- Nowhere-zero flows in a graph (with values in a finite abelian group of order
*q*) are “dual” to*q*-colourings. - The number of nowhere-zero flows depends only on the order of the abelian group, and not on its structure.

I wanted to understand these statements, and I believe that the orbital versions given in the paper throw some light on this.

In the case of the first statement, nowhere-zero flows (or 1-coboundaries) are really dual to nowhere-zero tensions (or 1-boundaries). A *tension* on *X* with values in an abelian group *A* is a function on the (oriented) edges with the property that the (signed) sum round any cycle is zero. Thus, such a function is derived from a “potential” function on vertices, so that the value on an edge is the difference of the values at its head and its tail. The tension is nowhere-zero if and only if the potential takes different values on adjacent vertices, in other words, is a *q*-colouring of the graph. So the numbers of colourings and nowhere-zero tensions are equal (up to a simple normalising factor *q ^{c}*, where

The second statement is not really folklore, it is a theorem of Tutte. But the same comment applies. The number of orbits of an automorphism group on nowhere-zero flows with values on an abelian group *A* does depend on the structure of *A*, and can be expressed as the evaluation of a polynomial which has a variable for each element order in *A*; the required substitution for this variable is the number of elements of that order. But a variable only occurs in the polynomial if the corresponding order divides the order of the automorphism group *G*. So, if *G* is trivial, the structure of *A* has no effect.

By phrasing the result in the paper in terms of matrices over principal ideal domains, we were able to give a general theorem which specialises not only to nowhere-zero flows and tensions in graphs, but also to weight enumerators of codes.

One final point on this. The *nowhere-zero* condition is enforced by an inclusion-exclusion argument over subgraphs of the graph. This indicates a general phenomenon: an evaluation of the Tutte polynomial with one of the variables equal to −1 often is equivalent to an inclusion-exclusion argument!

I have discussed before the connection between transformation monoids and graphs. There are constructions of each type of object from the other, which are not functorial but carry a lot of structural information.

Now transformation monoids, and in particular endomorphism monoids of graphs, are closely connected with a lot of important combinatorics.

This is obscured by the fact that most graphs are *cores*: they have no proper endomorphisms. But there are many interesting graphs. I will describe here one particular family, but there are many other instances.

The *square lattice graph* *L*_{2}(*n*) has as vertex set the cells of an *n*×*n* array; two vertices in the same row or column are joined.

This graph is a *pseudocore*: it has clique number equal to chromatic number, and all its proper endomorphisms are colourings which take values in a maximum clique.

Now the *n*-cliques are easy to understand: they are just the rows and columns of the array. But the *n*-colourings are much more interesting: they are the *Latin squares* of order *n*:

So we have a transformation monoid whose elements are pairs consisting of a Latin square and a row or column of the corresponding array.

So even finding the order of this monoid requires the classification of Latin squares, which has only been achieved for *n* ≤ 11 (see here andhere).

At first, when I realised this, I was excited at the thought that Latin squares could be composed. But, of course, the way the composition works, any product is equal to its leftmost factor (up to some renormalisation), so the algebraic structure is not so interesting.

The monoid may not be interesting in its own right. But here is a case where a very important classification in semigroup theory (the *J*-classes, equivalence classes for Green’s relation *J*) coincide with an important classification of Latin squares. Two Latin squares are equivalent in this sense if they differ by permuting rows, columns and symbols, and possibly transposing rows and columns. So the equivalence classes lie between isotopy classes and main classes.

In terms of the Green’s relations, the L-classes (corresponding to multiplying by automorphisms on the left) correspond to permuting rows and columns and/or transposing; the R-classes (corresponding to multiplying on the right) correspond to permuting the entries of the square.

There are several other examples where interesting combinatorial structures and their classification problems correspond to computing the endomorphisms of a pseudocore.

For some of our recently-discovered graphs, inspired by the butterfly, things are much more complicated, and the semigroups have a rich structure which we have only begun to explore. Almost certainly there is interesting combinatorics to be discovered, but what will it be?

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… is, if not a shepherd’s, at least a walker’s delight:

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February’s picture, called “Apocalypse on Charles Bridge”, is a composite: the silhouettes are figures from the famous Charles Bridge in Prague; the sky behind is over St Andrews, the picture was taken from just outside my front door.

Charles Bridge is on the natural route from the Old Town Square to the castle, and also on the route from the Karolinum to the former Jesuit college which now houses the Applied Mathematics department. But it is so crowded these days that often when I have to go between the last two places I prefer to detour over one of the adjacent bridges.

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Is there an easy way to generate a Steiner system S(5,8,24) for the Mathieu Group M_{24}, if a Steiner system S(5,6,12) for the Mathieu Group M_{12} is known?

I learned about the Mathieu groups when I was a student. I read Heinz Lüneburg’s book *Transitive Erweiterungen endlicher Permutationsgruppen* in the Springer Lecture Notes series, in which he constructed the Steiner system (and group) by extending three times the projective plane of order 4.

But, at about the same time, I attended a remarkable series of lectures by Graham Higman. In these lectures, he constructed the outer automorphism of the symmetric group S_{6}, and used it to construct and prove the uniqueness of the projective plane of order 4, the Moore graph of valency 7, and the Steiner system S(5,6,12), and hence deduce properties of their automorphism groups. He then gave an analogue of the “doubling” construction from S_{6} to M_{12}, starting with M_{12}, constructing its outer automorphism, and using this to build M_{24} and its Steiner system.

I don’t think he published any of this, and I doubt whether any notes are now available. I used the first part of the material in chapter 6 of my book *Designs, Graphs, Codes, and their Links* with Jack van Lint. The outer automorphism of S_{6} is described here.

So I have written out a description of the path from the small to the large Mathieu group, and put a copy here with my lecture notes. This document could be regarded as a supplement to Chapter 6 of my book with Jack. It may be of interest to someone else; let me know if you find it so.

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If you read *The Da Vinci Code*, you may remember the absurd claim by Dan Brown that public-key cryptography was invented by Leonardo Da Vinci. Well, in Gwyneth Jones’ review of *The Serpent Papers*, she almost claims that Ramon Llull invented programming languages! She actually describes Llull’s *Ars Magna* as “a form of algebraic logic expressed in complex diagrams (an ancient forerunner of all programming languages, by the way)”. I haven’t read the book, so I don’t know whether this notion is in it or not.

So I checked on Wikipedia, and found this: “Some computer scientists have adopted Llull as a sort of founding father, claiming that his system of logic was the beginning of information science,” an altogether more modest claim.

So what did Llull do? This is how it seems to me, but I disclaim any expertise on Llull. He was interested in combinatorics, without a doubt. Knuth describes a chapter of his *Ars Compendiosa Inveniendi Veritatem* which begins by enumerating sixteen attributes of God: goodness, greatness, eternity, power, wisdom, love, virtue, truth, glory, perfection, justice, generosity, mercy, humility, sovereignty, and patience. Then Llull takes each of the 120 combinations of two of these attributes, and writes a short essay on their relationship.

There is much more along the same lines in Llull’s extensive writing. But, perhaps more significantly, he invented mechanical gadgetry for generating all the combinations. His works are full of beautiful diagrams of complete graphs, rotating discs, and so forth.

He seems to have believed that any moral question could be settled by analysis of all possible combinations of attributes, virtues and vices, etc. I am not quite sure how. Neither am I sure where the logic or computer programming comes in.

There was much more to Llull than this. Before his conversion, he was a secular writer and troubador; he wrote the first novel in Catalan (which is possibly the first European novel). Afterwards he set out to convince Jews and Muslims of the truth of Christianity by rational argument (a hopeless quest, you might think – but he always believed in discussion rather than violence). His works also include Latin squares.

I will leave the last word to Jorge Luis Borges:

At the end of the thirteenth century, Raymond Lully was prepared to solve all arcana by means of an apparatus of concentric, revolving disks of different sizes, divided into sectors with Latin words; John Stuart Mill, at the beginning of the nineteenth, feared that some day the number of musical combinations would be exhausted and there would be no place in the future for indefinite Webers and Mozarts; Kurd Lasswitz, at the end of the nineteenth, toyed with the staggering fantasy of a universal library which would register all the variations of the twenty-odd orthographical symbols, in other words, all that it is given to express in all languages. Lully’s machine, Mill’s fear and Lasswitz’s chaotic library can be the subject of jokes, but they exaggerate a propensity which is common: making metaphysics and the arts into a kind of play with combinations.

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Last year, James Mitchell and I, talking about possible projects, wondered whether we could get a similar nice formula for the length of the longest chain of sub-semigroups in the full transformation semigroup *T _{n}*.

We didn’t succeed in getting a formula. But soon after that, Max Gadouleau came to visit, and got interested; he came up with a method which produces long chains.

Briefly, it goes like this. We look for a long chain among the semigroup of transformations of given rank *m* (with an added zero, so that if the product of two maps has rank less than *m*, we re-define it to be zero). Now in a *zero semigroup* (one with all products zero), any subset is a semigroup, so there is a chain of length one less than the order of the semigroup. So the trick is to find large zero semigroups.

This is done as follows. A map of rank *m* has an image (a subset of size *m*) and a kernel (a partition with *m* parts); the product *fg* will have rank less than *m* (so be zero in our semigroup) if and only if the image of *f* is not a transversal for the kernel of *g*. So we need a large collection of *m*-sets and a large collection of *m*-partitions such that no subset is a transversal for any of the partitions. Max invented the term *league* for this combinatorial object; the *content* of a league is the product of the numbers of subsets and partitions.

By finding leagues with large content, we were able to show that the longest chain of sub-semigroups of *T _{n}* has length at least a positive fraction 1/e

At this point the investigation broadened, and involved James’ former student Yann Peresse, as we tried our hand at various other semigroups. In some cases we were able to find exact results (either as an explicit formula, or in terms of the length of various groups involved). These cases include the inverse semigroup *I _{n}* of partial 1–1 maps on an

Our paper has just appeared on the arXiv.

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A *quasigroup* is a set *Q* with a binary operation whose “Cayley table” is a Latin square. The Cayley table has rows and columns indexed by *Q*, and the entry in row *a* and column *b* is the composition or product *ab*. The Latin square condition means that

- each element occurs once in each row; that is, given
*a*and*b*, there is a unique*x*such that*ax*=*b*. - each element occurs once in each column; that is, given
*a*and*b*, there is a unique*y*such that*ya*=*b*.

A *loop* is a quasigroup with an identity element *e* (satisfying *ex* = *xe* = *x*). If we label the first row and column with *e*, we see that the row labels coincide with the elements in the first column, and the column labels coincide with the elements in the first row.

In a quasigroup *Q*, the operatiions “left multiplication by *a*” (given by λ_{a}:*x*→*ax*) and “right multiplication by *a*” (given by ρ_{a}:*x*→*xa*) are permutations. These permutations generate a subgroup of the symmetric group on *Q*, which is clearly transitive: to map *a* to *b*, we multiply by the element given by the “division axiom”. The group generated by all the left and right multiplications is called the *multiplication group* of *Q*.

In a loop, it is also important to consider the subgroup of the multuiplication group which is the stabiliser of the identity element *e*: this is called the *inner multiplication group*.

The multiplication group of *Q* may be the full symmetric group. Indeed, this is the “uninteresting case”. (This is one of the situations in which a smaller group is more interesting.) There are a couple of motivations for this view:

- Jonathan Smith and Ken Johnson have developed a character theory of quasigroups, based on that for groups. In their theory, the character theory of
*Q*is “trivial” if and only if the multiplication group of*Q*is doubly transitive. - A class of loops defined by Bruck and Paige in 1956 consists of the
*automorphic loops*, those for which the inner multiplication group consists of automorphisms of the loop. This class includes groups, for which the inner multiplication group consists precisely of the*inner automorphisms*(conjugations)*x*→*a*^{−1}*xa*. The automorphism group of a loop is never the symmetric group except in some small cases, since there exist elements*a*and*b*for which*ab*is different from*a*and*b*(provided that |*Q*| ≥ 4), and the stabiliser of*a*and*b*in the automorphism group fixes their product.

Since it is no secret that there are a huge number of quasigroups, most not very interesting, the following result is not a surprise:

**Theorem** Almost all quasigroups have the property that their multiplication group is the symmetric group.

“Almost all” here means that the proportion of *n*-element quasigroups with this property tends to 1 as *n* tends to infinity.

(There is a diversion necessary here, and a small story. As I have set it up, we are counting “labelled” quasigroups, in which the elements are numbered *q*_{1}, …, *q _{n}*. It would be more natural to count them up to isomorphism. In fact this is the same, since almost all quasigroups have trivial automorphism group and so have

The fact that almost all Latin squares have trivial automorphism group was proved independently by Laci Babai and me at about the same time; the proof also works for Steiner triple systems and 1-factorisations of complete graphs. Laci got the result for Steiner triple systems into print first, and the rest became “folklore” for a while. It depended on the proof of the van der Waerden permanent conjecture by Egorychev and Falikman — more precisely, a weaker version proved earlier by Bang suffices.)

The proof is interesting, as it relies on a lovely theorem of Łuczak and Pyber:

**Theorem** Almost all elements of the symmetric group *S _{n}* lie in no proper transitive subgroup of

Now the multiplication group of a quasigroup is transitive, as we noted, and it contains the first row of the Latin square (regarded as a permutation); for a random quasigroup, this is a random permutation. So it is almost always the symmetric or alternating group. The final step uses a theorem of Häggkvist and Janssen, according to which the proportion of Latin squares whose rows are all even permutations is exponentially small.

Now the niggling problem is:

**Problem** Is it also true that almost all loops have multiplication group the symmetric group?

The above argument doesn’t work, because the first row of the Cayley table of a loop is the identity. However, the second row is a derangement. One of the oldest results in probability theory is that a positive proportion of elements of the symmetric group (in the limit, 1/e) are derangements; and since almost none of them lie in “small” proper subgroups of *S _{n}*, the argument concludes as before.

But this is not correct. The second row of a random Latin square whose first row is the identity is not a *uniform* random derangement, since the fist two rows can be extended to a Latin square in different numbers of ways depending on the chosen derangement.

For example, of the nine derangements in *S*_{4}, three of them are products of two transpositions, and have four extensions to a Latin square; the other six are 4-cycles, and have only two extensions. So each derangement of the first type has probability 1/6, whereas one of the second type has probability 1/12.

Remarkably, it turns out (empirically) that this non-uniformity rapidly washes out as *n* increases:

**Conjecture** The probability distribution on derangements defined as above (where the probability of a derangement is the probability that a random Latin square with first row the identity has that derangement as its second row) tends *very rapidly* to the uniform distribution as *n* tends to infinity.

For example, for *n* = 7, the numbers of extensions of the first two rows to a Latin square for the various cycle structures of derangements are

- (7): 6566400
- (2,5): 6604800
- (2,2,3): 6635520
- (3,4): 6543360

So, from differing by a factor of 2 for *n* = 4, the probabilities agree to within a little over 1% for *n* = 7. For larger values, the agreement is even more striking.

There are some results by Nick Cavenagh, Catherine Greenhill and Ian Wanless on bounding the ratios of these probabilities, but they are very far even from showing that the ratios tend to 1.

If it could be shown that this distribution tends to uniform faster than the error term in the Łuczak–Pyber theorem, we would have solved the problem. But there may be other ways to attack it which I haven’t thought of yet.

While I am on the subject, I would like to say a bit more about the Häggkvist–Janssen theorem.

**Problem** What can be said about the distribution of the number of rows of a random Latin square which (as permutations) have even parity?

Since the message of the preceding conjecture is that there is a lot of near-independence in quite large pieces of a random Latin square, the natural conjecture would be that the distribution is close to binomial with parameter 1/2, at least for large *n*.

Evidence supports this conjecture fairly well, but the tails of the distribution seem a little heavier than it would predict. The exponential bound in the Häggkvist–Janssen theorem is substantially larger than 1/2^{n}, and examination of small Latin squares suggests that this is with good reason.

But my feeling is that this is an effect of the small sizes which we can deal with. “Interesting” Latin squares are likely to be far from typical in this respect. For example, in the Cayley table of a group, either all rows have the same parity, or there are equally many even and odd rows. (The first alternative holds if the Sylow 2-subgroups of the group in question are cyclic.) These and other interesting quasigroups may be biasing the statistics. I expect it to hold for large squares.

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This and other examples show that large collaborations in mathematics, though not yet common, are not a new thing. But it is true that technology, and the leadership of people like Tim Gowers and Terry Tao, have changed the character of such collaborations.

Gowers proposed an experiment in collaboration on his blog in 2007, and receiving an enthusiastic response, set up the first Polymath project. The aim was to find an “elementary” combinatorial proof of the density version of the Hales–Jewett theorem. Success was declared in six weeks, though the write-up took a bit longer. The collaborators chose the name D. H. J. Polymath to publish the result: the initials for the theorem they had proved, and Polymath for the generic name of the collaboration, which used technological facilities such as blogs and wikis to enhance the collaboration.

There have been several Polymath projects since, not all of them so successful. But Polymath 8 has achieved much mathematics, and much notoriety, and is described by an article by D. H. J. Polymath in the current *Newsletter of the European Mathematical Society*, which I recommend.

The impetus for the project was the astonishing result by Yitang Zhang, that there are infinitely many prime pairs differing by at most 70000000. Tao fairly soon decided that improving the bound would make a good Polymath project, since there are many places where Zhang’s estimates could be refined, and these would involve a variety of mathematical and programming skills. Many people contributed, and eventually the bound was reduced to 4680.

Then James Maynard came up with a different argument which reduced Zhang’s bound to 600. (He has just been awarded the 2014 Ramanujan Prize for this and other work.) So it was decided to enlist him in the second phase of Polymath 8, and see how far this bound could be improved. They have got it down to 246, and there are various generalisations and conditional results as well.

One of the reasons for the wide interest created is that there was a “headline” figure, namely the best value so far achieved, and mathematicians on the sideline could watch this figure decreasing as the project continued, and read and understand the arguments used.

The *EMS Newsletter* article collects the perspectives of ten of the participants (Tao, Andrew Gibson, Pace Nielsen, Maynard, Gergely Harcos, David Roberts, Andrew Sutherland, Wouter Castryck, Emmanuel Kowalski, and Philippe Michel) on their involvement in the project. Some common themes run through their accounts:

- One of the hardest things was to get used to making mistakes in public, with no chance to suppress them; but all the participants who commented on this felt that it was crucial for the success of the project that everyone was prepared to contribute and risk looking foolish.
- Several participants, especially the younger ones, remarked on the danger for a non-tenured mathematician taking part in a Polymath collaboration, since no proper academic credit can be given; but none of them regretted their own involvement.
- There was a lot of praise for the way that Tao as “ringmaster” handled the project, which was felt to have contributed to its success.
- Everyone spoke of the excitement of the collaboration, and the exhaustion which it produced!

(In the Opencomb collaboration, the first point was not an issue, since we could stick our necks out and only appear foolish to the other nine participants; in Polymath 8, if not all the world, then a good crowd of spectators was watching.)

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Seventeen of us spoke, the brief being to explain to the whole school what we are excited about. In addition there were a number of posters from students and others.

I learned a great deal about what my colleagues in various parts of Applied Maths and Statistics are up to: solitary waves in the interface between liquids of different densities, the interaction of the solar wind with the earth’s magnetosphere, analysis of spatially distributed data (or, do koalas have favourite trees?), and much more.

But perhaps the highlight for me was the prizewinning student poster, by Jonathan Hodgson, on doing magnetohydrodynamics in seven dimensions. It is clear how to generalise the scalar product, but what about the vector product? His approach was to use the purely imaginary octonions (the next step up from an interpretation in three dimensions using the purely imaginary quaternions, and so this struck a chord after the talks at the LMS event last Friday). He gave a very clear description of the structure (starting from the Fano plane with arrows on it which gives the simplest definition of the octonions) and of the problems which arise with this project. I have seen in a couple of places that some brave physicists wonder whether the octonions have a place in the “theory of everything”: this seemed a sober attempt to put the groundwork in place.

The down side of the day was that the university caterers hadn’t managed to deliver most of the sandwiches that had been ordered, so we had to sit through the afternoon talks less than fully satisfied. Perhaps this helped keep us awake.

I enjoyed myself, and I very much hope that this becomes a tradition!

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