The pictures are self-explanatory: the first is in Pittenweem, in Fife, Scotland, the second on the campus of the University of Adelaide, South Australia.
The pictures are self-explanatory: the first is in Pittenweem, in Fife, Scotland, the second on the campus of the University of Adelaide, South Australia.
This post is inspired by a nice article by Adrian Rice and Ezra Brown in the latest BSHM Bulletin, titled “Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas, Part II”.
Pappus’ Theorem states that, if alternate vertices of a hexagon are collinear, then the three intersection points of pairs of opposite edges are also collinear. In other words, if we have six points P1, …, P6 such that P1, P3, P5 are collinear, as are P2, P4, P6, and Q1 is the intersection of P2P3 and P5P6, with Q2 and Q3 similarly, then Q1, Q2, Q3 are collinear. This is a classical “configuration theorem” involving nine points and nine lines, asserting that if 8 of the 9 triples are collinear then so is the 9th.
Of course, this is a theorem of projective geometry. It is valid in the Euclidean plane, but remains valid if some of the points are at infinity (recognised in the affine plane by corresponding lines being parallel). There are quite a number of cases to consider. As in various similar situations such as the classification of conics, things are much simpler in the projective plane!
At a similar time, Diophantus discovered and studied the “two squares” identity
(a2+b2)(x2+y2) = (ax−by)2+(ay+bx)2.
The analogous four squares inequality was discovered by Euler, and the eight squares inequality by Degen, Graves and Cayley (in that order, though the last of the three somehow got his name attached to it).
The authors’ thesis is that these two pieces of work were not seen to be related until the nineteenth century. In the spirit of the work of Cauchy and Weierstrass on calculus, mathematicians turned their attention to geometry, to the task of making Euclid’s axioms more rigorous. When studying just the incidence axioms, it was discovered that (together with the usual axioms for a projective plane) Pappus’ Theorem is equivalent to the property that the plane can be coordinatised by a commutative field. On the algebraic side, the two, four and eight squares identities express the facts that there are algebras of dimenions 2, 4 and 8 over the real numbers which have multiplicative norms (and are therefore division algebras), the complex numbers, quaternions and octonions; but the last two fail to be commutative. So Diophantus’ two squares identity is the related to the only finite extension of the real numbers which coordinatises a projective plane satisfying Pappus’ theorem.
At this point, let me note that one really has to consider various “degenerate” cases of Pappus’ theorem, when some pairs of points are identified. Though tedious, this is not a serious difficulty.
After the eight squares identity was discovered, mathematicians naturally wondered whether the sequence continues. The answer is no, as was proved by Hurwitz in 1898. (His result was slightly more general; he shows it not just for sums of squares but for arbitrary non-singular quadratic forms.)
Rice and Brown also discuss finite geometries, and in particular the Steiner triple systems on 7 and 9 points; these are the projective plane over the field of two elements, and the affine plane (which can be extended to a projective plane in a standard way) over the field of three elements.
Since these fields are commutative, the Fano plane (for example) should satisfy Pappus’ theorem. Because the Fano plane only has seven points, all instances of Pappus’ theorem in it are degenerate!
It is also the case that the octonions are conveniently described by the Fano plane with arrows on some of its lines. The seven basic units (apart from the identity) are matched with the seven points of the plane, and the triples whose product is plus or minus 1 are the lines of the plane; the arrows can be put on in such a way that each line has a cyclic orientation so that the product of two of its points is the third.
There is some interesting pre-history of Steiner triple systems. As is well known, Kirkman proved the existence theorem for Steiner triple systems in 1847 in answer to a question in the Lady’s and Gentleman’s Diary. Unaware of this, Jakob Steiner posed the same question in Crelle’s Journal in 1853, and it was answered by Reiss in 1859.
The problem that had attracted Kirkman’s attention was posed by Wesley Woolhouse, the editor of the Lady’s and Gentleman’s Diary, in 1846. Woolhouse and Steiner may both have got the problem from Julius Plücker. He had discovered the Steiner triple system of order 9 as the structure of the nine inflection points of a plane cubic curve, in 1835; in the same paper he had posed the existence question for “Steiner triple systems”. (He mistakenly thought that they could exist only for orders congruent to 3 (mod 6); he corrected his mistake and added 1 (mod 6) in 1839.)
Should the concept be named after the person who first proposed it (so they would be Plücker triple systems), or the person who first constructed it (so they would be Kirkman triple systems)? This problem arises in many other parts of mathematics; Lyons discovered evidence for a sporadic simple group which was constructed by Sims, for example. The matter is further complicated in the present case by the fact that the term Kirkman triple system is used with a different meaning, based on Kirkman’s schoolgirls problem.
But there is one further connection which Rice and Brown don’t seem to mention.
The Pappus configuration has nine points and nine lines. These lines cover 27 of the 36 pairs of distinct points. The nine pairs not covered correspond to a partition of the nine points into three sets of three points each containing no collinear pairs. If we add these three sets as new “lines”, we obtain the affine plane of order 3. The uniqueness of the Pappus configuration is thus related to the fact that the parallel classes of lines in the affine plane are equivalent under symmetries of the plane.
I’m now in Australia for nearly two months, but before I talk about that, I would like to draw attention to an issue relevant to people in England and Wales who enjoy the countryside.
The system of rights of way is one of the great glories of the English countryside. Any Ordnance Survey map will show a network of walks and rides through the country; you are entitled to walk or ride along them. They come in several flavours: footpaths, bridleways, RUPPs (roads used as public paths) and BOATs (byways open to all traffic): I am not quite sure of the difference between the last two, and in any case there has been some recent change in the status.
(As I have said before, in Scotland it is different; we have a right to roam anywhere, which sounds great but in practice means that it is much harder to plan a walk since there will be no guarantee that you can get through the fences or hedges.)
Many of these rights of way date back to the early days of the rambling movement which did so much to get people out into the country. In a lot of cases, the documented use of a path as a public highway in the past was used to have it registered as a right of way.
This is the system which is about to change. On 1 January 2026, the definitive maps will be closed to the addition of rights of way on the basis of historic evidence.
I am not sure what this means for rights of way as a whole. They can be extinguished, for reasons good and bad: maybe because a nuclear power station is being built over the path, or because a rich foreigner has bought the property and doesn’t want the public walking across his land. Will it be possible for new rights of way to be created? I don’t know; but it seems that one important mechanism for this will be lost.
This is not just a triumph of evil landowners. Indeed the committee that drew up the legislation had walkers represented, and attempted to find a compromise between vested interests. What it does mean, however, is that the time to establish rights of way on the basis of historic use is running out, and a lot has to be done in ten years.
If this matters to you, see this webpage maintained by the Open Spaces Society, and support them in their action.
Combinatorics is flourishing, at least in the lists of the European Mathematical Society publishing house.
A few years ago, I mentioned the new journal Combinatorics, Physics and their Interactions, aka Annales de l’Institut Henri Poincaré D.
Now they have announced a new journal starting up next year, Journal of Combinatorial Algebra, with Mark Sapir as editor-in-chief. The “Aims and Scope” say,
Its domain is the rich and deep interplay between combinatorics and algebra. Its scope includes combinatorial aspects of group, semigroup and ring theory, representation theory, commutative algebra, algebraic geometry and dynamical systems.
I am not completely certain how this scope differs from that of the Springer Journal of Algebraic Combinatorics, but no doubt this will become clearer in time; a journal is defined by its contents more than by its aims and scope.
It is particularly pleasing that both these journals are specifically about combinatorics and something else. This is combinatorics doing the job it grew up to do, in my opinion.
And also, of course, the first issue of the EMS Surveys in Mathematical Sciences in 2014 was devoted to a survey article by Terence Tao on “Algebraic combinatorial geometry”.
St Andrews is currently hosting a big photography festival. Because Fox Talbot’s patent on his photographic process was not valid in Scotland, local St Andrews photographers were able to have a go themselves, and develop the process. The festival covers both the historic and the modern.
We haven’t had time to see much of the exhibition; a trip to the antipodes looms, and it will mostly be over when we get back. But we made time for a few things.
On the railings on the Scores, the work of four Scottish photographers had been blown up and printed on waterproof material to stay out for the whole festival. These photographers were consciously documenting Scotland: one travelled along the English border photographing the scene on the Scottish side, another photographed women farming in the most extreme conditions in the highlands and islands.
In the Old Union Coffee Shop in North Street, there was an exhibition of work by Franki Raffles, a feminist and photographer who left a large body of work depicting women in various situations, mostly work. The most memorable image for me was a shepherdess on a bleak mountainside in Georgia.
We managed to catch an exhibition by the St Andrews Photographic Society at the Town Hall, although the hours were variable: a mixed bag, as one might expect. But we really wanted to see the work of the St Andrews Photographers, a group which may have some connection with the St Andrews Photographic Society (I am not sure exactly), but includes our colleague Richard Cormack. There was claimed to be an exhibition in Holy Trinity Church, but we found the church firmly locked, perhaps because of Lammas Fair (even though, by then, the fair was over).
However, in Pittenweem (on the south coast of Fife), the art festival has been on for a week, and this was the closing weekend. This festival draws artists from Fife and well beyond, who rent houses, front rooms and garages in the town to display their work, and is always worth a visit. So we went yesterday, and found the St Andrews Photographers running an exhibition in a garage in the High Street. So we did after all get to see their work (and buy a couple of prints).
The whole festival was far too large to take all of it in, with over a hundred artists on show around the town and outside. We went to maybe twenty exhibits, and saw some remarkable work. Some of the best, to my eye, were reflections of the Forth Rail Bridge by Karen Trotter, and abstract landscapes by Lynn McGregor.
But the most notable thing about the day occurred elsewhere. The weather was glorious, one of the few really beautiful days we have had in this rather dreary summer. The blue of the sea and the red of the flowers were so intense that the art did slightly pall by comparison. After we were “arted out”, we decided to walk a short stretch of the coastal path, westward to St Monans and Elie.
As we were walking along the beach at the East Links of Elie, we heard an extraordinary, out-of-the-world, music. I wondered if it might have been seals singing. On a rocky point stretching into the sea I saw some black shapes sticking up. With my camera on maximum zoom, the viewfinder revealed them as cormorants, which were certainly not responsible for the sound. We tried to imagine that it was caused by the wind moaning through the ruined tower on the point. But I took a picture anyway.
When we got home, I looked at it on the computer screen, and saw that the rocky point was indeed covered with seals, who were so well camouflaged that I simply hadn’t seen them while we were there. I am certain that it was indeed singing seals that we heard. A memorable experience!
The Young Researchers in Mathematics conference took place this week in St Andrews. Since I was a plenary speaker, I was signed up as a proper participant, so I felt entitled to go to as many talks as I wanted without feeling I was gatecrashing. As expected, it was a lively conference, with plenty of enthusiasm and unexpected connections. Below are comments on some of the young researchers’ talks. If you trouble to read it through, note the threads connecting talks in quite different areas.
A couple of talks by St Andrews students involved transducers, finite machines which read input, change state, and write output. Casey Donoven explained that the second world war Enigma machine is an example; it is synchronous, that is, it writes one output symbol for every input symbol it reads. A transducer can read an infinite input string, and so defines a map on the set of such strings, which is contninuous (in the product topology induced from the discrete topology on the alphabet).
The set of binary strings corresponds naturally to the “middle third” Cantor set in the unit interval, with 0 and 1 corresponding to taking the left or right subinterval at each step. There is also a map from the set of binary strings to the unit interval, given by the “binary decimal” representation, which is not quite bijective since countably many reals have two binary representations. A transducer which preserves these pairs of strings induces a continuous map on the unit interval, as Casey explained.
Shayo Olukoya considered the group defined by synchronous transducers which are also bi-synchronizing. This means that there is a number k such that, after reading k symbols, the machine is in a state which depends only on the symbols read, and not on the initial state, and the same holds for a transducer realising the inverse function. He explained how to solve algorithmically the problem of whether the induced map has finite order.
Experts will see Richard Thompson’s groups lurking in the background here. Nayab Khalid described a more general situation, a “substitution and glueing” procedure on directed graphs, which produces Thompson-like groups acting on interesting fractals like the basilica Julia set, and proving things about these groups.
Returning to Casey’s almost-bijection between Cantor space 2N and the unit interval, there is also a famous almost-bijection between Baire space NN (the set of infinite sequences of positive integers) and the unit interval, given by the continued fraction. The Gauss map T: x → 1/x (mod 1) on the unit interval corresponds to the left shift on NN. Natalia Jurga generalised this situation to maps with some of the properties of the Gauss map. Any probability vector on N induces a product measure on Baire space, which can be pulled back to the unit interval using the map T. Natalia was interested in the maximum Hausdorff dimension of such a measure.
Jason Dungca also used the Gauss map, with the probability of an integer n proportional to 1/n2; he was interested in phase changes in the multifractal spectrum of Gibbs measures. I won’t attempt further explanation.
Wojciech Ożański started his talk about singularities of the Navier–Stokes equation in fluid mechanics by reminding us of the definitions of Hausdorff and box dimension. He was interested in bounding these dimensions of the sets of singularities and blow-up times for a solution.
Douglas Howroyd defined these two dimensions and also Assouad dimension, the one he was chiefly interested in. He showed us how to compute the dimension of fractals defined by certain sets of affine contractions. These map the unit square onto various rectangular regions obtained from a dissection of the square, and go by the name of carpets.
Demi Allen used Hausdorff dimension to refine a result of Khintchine on diophantine approximation. Given a function ψ, let A(ψ) be the set of real numbers x in the unit interval which satisfy |x−p/q| < ψ(q)/q for infinitely many rationals p/q. Khintchine showed that the Lebesgue measure of A(ψ) is equal to 1 if ∑ψ(q) diverges, 0 if it converges (and ψ is monotonic). So, for example, if ψ(q) = q−t, the Lebesgue measure is 0 if t > 1. In this case, its Hausdorff dimension turns out to be 2/(t+1). Demi was interested in multivariable versions of this. The key turned out to be a version of the “mass transference principle” relating Lebesgue measure on a ball to Hausdorff meaasure on a larger ball, extended from balls to neighbourhoods of hyperplanes.
Khavlah Mustafa considered the Möbius (linear fractional) groups over certain rings such as the complex, dual, and double numbers. (These are obtained from the real numbers by adjoining square roots of −1, 0 and 1 respectively.) She described fixed points and orbits of 1-parameter subgroups, and gave a dynamical classification of the fixed points.
For classical groups over finite fields, Daniel Rogers described how to find maximal subgroups in the catch-all Aschbacher class C9, which are themselves classical groups over fields of the same characteristic. (For ease of exposition he restricted his attention to the special linear groups.) He was particularly interested in dimension 16, the smallest for which the classification of maximal subgroups is not yet done, and showed us some interesting examples. He used results of Steinberg which made it all seem much easier than in fact it is.
Scott Harper talked about 2-generated groups (a class including the finite simple groups, as we know from the classification of these groups). His basic question goes back to Netto, who conjectured in 1882 that two random permutations generate the symmetric or alternating group with high probability (tending to 1 as n→∞). The spread of such a group is the maximum k for which, given any k non-identity elements x1,…xk, there is an element y with the property that xi and y generate the group for all i. Thus “spread at least 1” means that any non-identity element is in a 2-element generating set, a property also called “3/2-generation”. Remarkably, no finite group having spread precisely 1 is known, and it is conjectured that none exists.
I was delighted by Alex Rogers’ talk, which took me back to things from my distant past, forty years ago. She was calculating the PI-degree (the smallest degree of a polynomial identity) for quantum matrix algebras. (For ordinary matrix algebras of n×n matrices, the answer is 2n, given by the Amitsur–Levitski Theorem.) My first real job was at Bedford College (a part of the University of London which no longer exists); the head of department was Paul Cohn, and his big idea was to develop “non-commutative algebraic geometry”, not so far from modern quantum groups, and PI-degree played a role in this. Then, at the end, Alex revealed that the formula for PI-degree in some cases depends on the invariant factors of a specific matrix, and these are powers of 2. This suggested to me the chains of binary codes that my second DPhil student, Eric Lander, invented and studied.
Christian Bean demonstrated the program Struct which he and his colleagues are developing for studying (and finding patterns in) sets of permutations, and if possible enumerating these sets. As he said, the program is not yet completely user-friendly.
Waring’s problem concerns expressing positive integers as sums of kth powers: how many powers are required, and how many solutions are there? Kirsti Biggs extended this to Waring’s problem with shifts, where we are given small irrational numbers θi and a number η, and we want to make the difference between the given n and a sum of s terms (xi−θi)k smaller than η. The tool was replacing the count by an integral and using the Davenport–Heilbronn method.
Adelina Mânzățeanu found rational curves on a smooth cubic hypersurface over a finite field, passing through two given points. She made much use of one of my favourite theorems, the Chevalley–Warning Theorem. Her methods involved analysis in Fq(t) in some sense parallel to analogous arguments over Q, and she gave us a dictionary for comparing the two situations. (What is the analogue of a torus, of the Fourier transform, in the completion of Fq(t)?)
Thanatkrit Kaewtem talked about γ-Banach spaces (these are like Banach spaces but the unit ball need not be convex; a typical example is lγ for γ < 1). He was considering inner and outer entropy measures on a bounded operator between two such spaces, involving covering the image of a unit ball by translates of a small ball, an idea going back to Kolmogorov. Various inequalities connect these numbers to the operator norm and (in the case of an operator from a space to itself) its eigenvalues. He showed us why some of these inequalities are tight.
The first time I went to the YRM (in Warwick in 2011), there were no Queen Mary students there. Things were a little better this time. The sole QM student, Wan Nur Fairuz Alwani Wan Rozali, was considering a discrete dynamical system analogoes to the pendulum, given by iteration of a function on Z2 (I didn’t catch the exact formula for the function), proving a conjecture about its first return time to the positive X-axis.
With great self-restraint (there were some great talks, authoritative and exciting), I won’t say much about the plenary and keynote talks, except for one remarkable coincidence. Both Clément Mouhout, talking about partial differential equations, and Philip Welch, talking about axiomatic set theory, saw fit to tell us the story of the famous “Scottish Book” in which mathematicians of the Lwów school such as Banach, Kuratowski, and Ulam recorded problems. It has no direct connection with the location of the conference, but was named after the Scottish Café in in Lwów which the mathematicians used to meet. There could hardly be a better demonstration of the unity of mathematics than this!
And here are a couple of quotes from keynote speakers, brazenly taken out of context:
For me it was an excellent conference, with a higher than usual proportion of talks which completed a circuit in my brain by connecting to something quite different. I have plenty of new things to think about as a result. So thanks to Oliver, Daniel, Tom, Cristina, Zoë, and Sascha for putting on such a good show.
Mathematics is in good hands!
Last week, Neill and family came to visit us, as part of their Scottish holiday.
We had a very pleasant time, with breakfast in the White Chimneys in Pitscottie, a walk through Dura Den and over Blebocraigs, an afternoon at the St Andrews Highland Games, and a taste test comparing the two Scottish/Italian ice cream shops in St Andrews, Jannettas and Nardini (I am pleased to say the local concern got the thumbs up).
At the Highland Games, three things impressed us: tossing the caber (I had never seen this done before, I am lost in admiration), the tug of war (some epic battles in which nothing seemed to happen for ten minutes until a small crack in one team’s defence led to a quick finish), and, maybe best of all, our local MSP and leader of the Scottish Liberal Democrats, Willie Rennie, giving a good account of himself in the open 1600 metres race. Incidentally, I wondered why the distances run were 90 metres and 1600 metres, until I realised that these are just 100 yards and 1 mile.
We talked about Pokémon GO. Neill’s view is that, at last, Life is catching up with Art. (He did capture a few of the little critters in St Andrews, including winning his first fight.)