A talk on discrete mathematics

The Academy for Discrete Mathematics and Applications is an Indian organisation founded in 2005 to foster and support interest in discrete mathematics. The current president, Ambat Vijayakumar, has started up a Colloquium Lecture seres, and honoured me by asking me to give the first lecture.

I decided that, rather than up-to-the-minute new research, a more reflective talk would be appropriate, and I chose three interactions between discrete mathematics and other parts of mathematics, in all of which I had some involvement:

• Using root systems to prove a stronger version of Hoffman’s conjecture about graphs whose adjacency matrix has smallest eigenvalue −2 (or greater): these must be generalized line graphs (a class invented by Hoffman) with finitely many exceptions);
• The Erdős–Rényi countable random graph and its relationship to Urysohn’s universal and homogeneous Polish space;
• a beautiful graph obtained from the smallest sporadic simple group (the Mathieu group M₁₁).

There were nearly 100 people at the on-line event, and many questions were asked afterwards; time did not permit me to answer all of them, I am afraid.

The slides are here.

One final comment. The organisers had used the Webex platform, and I only realised a day in advance that the Linux version does not permit screen sharing (if you delve into their website you find this is a new feature coming soon). So we had to use the old-fashioned “next slide please” method. So I am not recommending Webex just at the moment!

British Combinatorial Conference

The British Combinatorial Conference returns to Queen Mary Univesity of London in the first week of July. The web page is here.

Last time when the conference was there, and I was one of the organisers, the insttution was called “Queen Mary and Westfield College”. We had 319 delegates, a record which has never been broken. When I came to allocate the last contributed talk slot, there were two delegates, fortunately good friends. I offered to toss a coin for the 15 minute slot, but they asked if they could have seven and a half minutes each.

The week was beautifully sunny. The publishers moved their stalls outdoors into the library quad, which was like a street market. Unfortunately I was trapped in the conference office dealing with complaints most of the time. (One delegate didn’t like the breakfasts and wanted a refund, another had left his camera bag at the back of the lecture theatre and came back to find it gone, and so on.)

This time I will be an ordinary delegate; I will not seek re-election to the committee, and just present my contributed talk (and maybe perform at the concert if there is one).

We probably won’t break the record this time, but I hope to see many old friends there.

Graphs and groups, designs and dynamics

Four and a half year ago there was a conference and summer school on the four topics of the title (part of the G2 series of conferences, whose hiistory you can read here) at the Three Gorges Mathematics Center in Yichang, China.

Now a book containing the notes from four lecture courses on the four topics is out. (Actually not quitef – it is published at the end of this month – but the editors, Rosemary Bailey, Yaokun We and I, have received our copies.)

There were various ups and downs in the production of the book, most of which I will leave unsaid. But the editing tested my LaTeX skills in a number of ways.

• The authors had their own carefully-thought-out views on the layout of their chapters; but the publisher wanted a uniform style. Some compromise was necessary.
• The toughest challenge; two of the chapters used LaTeX packages which conflicted. If loaded one way round, they would not run; the other way, the second overwrote the commands of the first producing unwanted results. In the end I had to load one package, save a couple of commands with different names using the plain TeX \let command, and then load the other package.
• There were also a couple of problems with the way the publisher’s style file interacted with indexing commands; we let the publisher resolve that one.

Anyway, you can see details of the book (and order it) here.

A talk by Gareth Jones

Today I attended (remotely) a nice talk by Gareth Jones in the Ural Workshop on Group Theory and Combinatorics, about prime powers in permutation group theory and polynomials taking prime values in number theory.

I will give just one example result from the first part. For a prime p, a p-complement in a group G is a subgroup of order coprime to p whose index is a power of p. Philip Hall showed that a finite soluble group contains a unique conjugacy class of p-complements for all primes p. The new result is:

Theorem Given a prime p and a finite group G, there is at most one p-complement in G, up to automorphisms of G.

On the second topic, Gareth mentioned two conjectures which are well out of reach of number theorists at present. Let f_i, for i = 1,…,k, be integer polynomials. We are interested in integers t for which all the values f_i(t) are prime, and in particular, whether there are infinitely many such. There are a couple of obvious necessary conditions for this: all the polynomials should be non-constant and irreducible, with positive leading coefficients; and no positive integer should divide the value of the product of the polynomials for all possible arguments. Schinzel’s hypothesis asserts that these necessary conditions are in fact sufficient, and the Bateman–Horn conjecture goes further by providing a conjectured asymptotic for the number of values of t up to some given limit x.

It is clear that the first conjecture would show that there are infinitely many twin primes, and infinitely many Sophie Germain primes, while the second would give an asymptotic estimate of their number.

Gareth gave an application to permutation groups; I will give another.

I have mentioned here before the “theorem” of Cauchy to which Peter Neumann, Charles Sims and James Wiegold gave counterexamples. Cauchy had claimed that a primitive permutation group of degree a prime plus one is necessarily doubly transitive. The counterexamples are obtained by choosing a finite simple group S whose order is a prime plus one, and letting S×S act on S where one factor acts on the left and the other on the right. I have discussed this here.

It is not known (at least to me) whether there are infinitely many counterexamples to Cauchy’s theorem, and in particular whether the NSW construction gives infinitely many. But the conjectures just described would imply that there are, if I am not mistaken. Let us take the simple group PSL(2,p), where p is an odd prime. Putting p = 2t+1, we see that we require values of t for which the polynomials 2t+1 and 2t(t+1)(2t+1)−1 to take prime values.

Probably similar results hold for some other families of simple groups of Lie type. However, in the comments to the previous post, I noted that Stephen Glasby pointed out to me that it will not work for the groups G₂(q); the order of this group minus 1 is a reducible polynomial, with factors of degrees 6 and 8.

Ischia group theory conference 2024

The island of Ischia, about a 45-minute hydrofoil ride from Napoli, is rich in history. The museum has a Homeric drinking cup with an inscription in ancient Greek from the 8th century BCE. At the other end of history, the British composer
Sir William Walton and his Argentine wife Susana bought a desolate hilltop producing only myrtle scrub, and turned it into one of the finest gardens in Italy.

More relevant is the fact that, for the last two decades, it has hosted the biennial Ischia group theory conference, organised by mathematicians from the University of Salerno, just down the coast a bit beyond Amalfi.

Salerno, incidentally, may have a claim to be the oldest university in Italy. Around the end of the first millennium, the Schola Medica Salernitana flourished, specialising in patching up pilgrims and crusaders on the way back from Palestine, and sending them on their way with good medical advice.

The conference was dedicated to Otto Kegel, a regular at Ischia, whose 90th birthday approaches this summer. (Otto was unable to be at the meeting for health reasons, but attended many sessions on Zoom.) But as a sign of the times, half a dozen sessions were dedicated to mathematicians who have died in the last couple of years, including Bertram Huppert (d. 1 October 2023), Nikolai Vavilov (d. 14 September 2023), Zvonimir Janko (d. 12 April 2022), and Rex Dark (d. late 2022). Of course, there are others, including Richard Parker (d. 16 January 2024), Anatoly Vershik (d. 14 February 2024) and Colin Mallows (d. 4 November 2023), though perhaps the last two would not be regarded as “group theorists”.

Among many great lectures at the conference, I would nominate as my favourite the talk in memory of Zvonimir Janko by Gernot Stroth. This brought back the heady days of the 1960s and 1970s, when I was just beginning as a mathematician, and new sporadic simple groups were being found under every lamp-post. I remember on one occasion arriving at the old DPMMS building in Cambridge, when the team there had just finished the construction of J₄; the notice on Conway’s door said, simply, ∃J₄. (Janko’s fourth sporadic simple group, for the uninitiated.)

Apart from the conference dinner, the memorable social events were a recital of traditional Neapolitan songs by a singer/guitarist and a mandolin player (it struck me that there is a parallel between this music and Portuguese fado, especially Coimbra fado, the mandolin substituting for the Portuguese guitar); a concert of baroque flute and cello music in Chieso Santa Maria di Portosalvo on the harbour (my favourite, of course, the first movement of Bach’s cello suite in G); and a visit to the Waltons’ Giardin La Mortella, on its hilltop at the north of the island, with stunning flowers at this time of year.

The main thing detracting from my pleasure was the fact that I was suffering from a miserable cold the whole time, in common with my family, students, and probably a good part of the population of Britain.

West Virginia University

Anthony Hilton, my former colleague at Queen Mary, spent some time as an Eberly Professor at West Virginia University. Now he has passed on to me the news that the University has decided to close down Pure Mathematics research and put the money into “computer related activities”, whatever that means.

One of the last mathematicians standing, John Goldwasser, who has been there since 1988, is organising a small conference (I suppose I could say a wake) to mark the demise of pure mathematics at the university. As I expected, when I looked at the web page of the School of Mathematics and Data Sciences (slogan: The Revolution Starts Here), I found no mention of this conference, or indeed of the closure.

I am afraid I do not have the address of anyone you could write to if you want to express an opinion on this (which in any case seems to be a done deal). If you have such an address, please let us know.

Cauchy numbers: job done

After recruiting Scott Harper to the team, we have finished the job of determining all the Cauchy numbers (these are the positive integers n for which there exists a finite list F of finite groups so that a finite group has order divisible by n if and only if some member of F is embeddable in it).

The answer is: n is a Cauchy number if and only if one of the following holds:

• n is a prime power;
• n = 6;
• n = 2p^a, where p is a Fermat prime greater than 3 and a is a positive integer.

In the second and third cases, we can tell you the list F: for example, for n = 6, the list consists of the cyclic and dihedral groups of order 6 and the alternating group A₄. (Of course, in the first case, the list consists of all groups of order n.)

The paper will appear on the arXiv on Tuesday, with the same number as before.

Anatoly Vershik

Anatoly Vershik died the day before yesterday.

As I have told here, he was the person who told me about the Urysohn space. I had given a talk at the ECM in Barcelona on the countable random graph, and after it he approached me and asked “Do you know about the Urysohn space?” We wrote a paper on it, extending some of my results on the random graph. Indeed, the Urysohn space has many different Abelian group structures.

He also made my two trips to St Petersburg possible, by having birthdays in 2004 and 2014 (actually his birthdays were in late December the previous year). I learned so much at these conferences, and had a small adventure when I came to leave after the first one.

Richard Parker

Richard Parker died last month. Now only two of the authors of the ATLAS of finite groups remain, the two Robs.

I knew Richard, but perhaps not well enough to write anything appropriate as a tribute. But I recommend you take a look at Rob Wilson’s account on his blog.

More on Cauchy numbers

Following on from the earlier post, the new version of the paper has just gone on the arXiv: 2311.15652 (version 2).

If we say that n is a Cauchy number if there is a finite set F of finite groups, all with orders divisible by n, such that every group with order divisible by n must contain a group in F as a subgroup, then our result is as follows:

Let n be the product of two distinct primes q and r. Then n is a Cauchy number if and only if one of the primes is 2 and the other is a Fermat prime.

This means that, in all other cases, there are infinitely many groups which are minimal with respect to containing the cyclic groups of orders q and r. The arguments fall into two quite different cases. Apart from the pair {3,5}, there are infinitely many simple groups with this property. But for {3,5}, there is only one simple group, namely A₅. So instead we use a theorem of Gaschütz which guarantees that we can build an infinite sequence of extensions, each a Frattini extension of the one before, starting with A₅. (This means that each group after the first has a normal 2-subgroup contained in its Frattini subgroup such that the quotient is the preceding group.) These do the job.

I learned of this theorem from my co-authors. So collaboration, and putting a paper on the arXiv, have very positive results!