Mathematical family

Officially, I retired (or strictly speaking, was made redundant) on 28 February. But I was still teaching and directing a student project until the end of May, as well as fighting against various problems caused by different parts of the University not talking to each other (so, as one congratulated me, another told me that my access to the network would be removed and my office withdrawn).

Since then I have been fantastically busy with the two extended trips I have described here, to South India and Portugal, and more recently a week in Prague. So now perhaps I can indulge some of the feelings brought on by retirement, about my mathematical family. And, although I prefer to look forward, this gives me the chance to look back at some of my mathematical interactions.

I begin with an apology. I am going to quote various figures in the past. They, like many of their contemporaries, assumed that all mathematicians are male; they had not realised the obvious fact that we are all human, and that even if gender, skin colour, religion, or sexual orientation affect many aspects of our lives, they must not interfere with the principle that everyone should have the opportunity and support to become a mathematician, if they so wish.

G. H. Hardy

Hardy expressed his views in his small book A Mathematician’s Apology. This book is widely celebrated, and contains much good. But Hardy made three mistakes:

• He thought that there is a clear dividing line between serious and trivial mathematics. This view was refuted by Persi Diaconis in his public lecture at the British Mathematical Colloquium in 2013, where he showed that there is a direct path between a puzzle about card shuffling and Artin’s conjecture on primitive roots.
• He claimed that serious mathematics is useless. This is so obviously wrong that it doesn’t need me to refute it.
• He thought that a mathematician was necessarily over the hill at 35 or 40. This is the one most relevant here. I am twice that age and I have no intention of giving up just yet.

Paul Erdős

My favourite Erdős quote is “The purpose of life is to prove and to conjecture”. (The quote goes on but this is enough for here.) He clearly thought that both of these activities were of comparable value.

As is well known, Erdős had his own idiosyncratic use of words. A mathematician is born when they are awarded their PhD degree. I am not sure how much he stuck to this rule. Many mathematicians, Erdős included, publish their first papers before finishing their PhD. But in any case, this occasion is socially marked, perhaps by a party.

A mathematician dies when they stop producing mathematics. He doesn’t set an age for this, and the occasion is not socially marked. For some, it coincides with retirement. (I recall someone telling me that they wrote to a retired mathematician asking a question about one of his results; the reply was along the lines “When I retired, I destroyed all my papers”.) But most continue to work while they are capable of it, since after all the work we do is the purpose of life. When the rest of us say that someone has died, Erdős says that they have left.

Neil Calkin

The Mathematical Genealogy website regards the bonds of family to be supervisor to student. This despite the fact that before last century, the PhD degree did not exist, and it is not at all clear in many cases whether there was a formal relationship at all. (In my own family “tree”, I believe that Henry Whitehead was not actually a student of Oswald Veblen, so my descent from Lagrange, Euler, and so on is questionable.) In its early days, a mathematician had only one supervisor; but now two supervisors are allowed, and this is rolled back into the past, so that the whole thing is a tangled web rather than a tree of ancestors. However, I should make it clear that my debts to my mathematical father Peter Neumann and grandfather Graham Higman are immense; and also, I get a lot of joy from seeing my grandchildren and great-grandchildren making their way in the mathematical world. (Sometimes I even get to write papers with my grandchildren, such as John Bamberg, David Bradley-Williams, and Simon Jurina — though, by MGP rules, John is also my nephew!)

I think that family is a wider concept than the MGP makes out, in several respects.

• Mathematicians, like other humans, have families in the usual sense. In some cases, their family members are also mathematicians. (My supervisor had two mathematical parents, and mathematical siblings.) These should also be part of their mathematical family even if not supervised in a formal way.
• Mathematicians collaborate. Nobody would deny that Hardy and Littlewood belonged to the same mathematical family.
• Finally, there are relationships which are less formal but just as important. In this sense, I regard Neil Calkin as part of my family.

Neil grew up in Britain, and despite very good A-level results, was not accepted by Cambridge University. He ended up in Canada, at the University of Waterloo. In 1985, he attended the British Combinatorial Conference in Glasgow. In the problem session, I posed a question about sum-free sets. Neil got interested in this, and started working on it. Nobody at Waterloo was interested in supervising him on this topic, so (this was before the days of email) he would write me letters and put them in the post, and I would reply. I remember the letter he sent when he constructed the example I had asked for in Glasgow; it was written round the outside of the envelope. So I regard him as my “honnorary” student.

There is no standard name for this relationship. Rosemary has some people to whom she was “fairy godmother”, but this is more like fantasy than a genuine relationship.

For another reason I am proud of my relationship with Neil. He was Herb Wilf’s sidekick in setting up the Electronic Journal of Combinatorics, one of the first (and still one of the best) of what we now call “diamond open access journals”, free to authors and readers alike. On one occasion I sat in Neil’s office in Atlanta while he typed in the (to me then extremely cryptic) Unix commands to publish a paper in the journal, soon after it was set up: we had been working on sum-free sets at the time, so he is also a co-author.

Sengai

The Zen master Sengai was asked by a successful businessman to write something for the prosperity of his family. The master took up his brush and wrote, in beautiful calligraphy, “Father die; son die; grandson die.” The businessman was furious at this, but the master explained that all have to die, but if the son dies before the father, or the grandson before the son, this will bring great grief to the family; the order he suggests gives true prosperity.

This applies to mathematical families as well. I always feel it as a axe blow when one of my students dies (fortunately not a common occurrence). But rather more, it is a disappointment to me if my students don’t go beyond me, and do things which I could never have done myself.

We could interpret this in Erdős’ terminology. When a student goes into banking, or (in the case of one of my students) enters a Buddhist monastery and cuts ties with the world, then he has died according to Erdős. If my students do this, that is their business, and fine by me; but it still seems a loss.

I reckon that I have learned far more from my students than they have learned from me. A great majority of them have made successful careers as mathematicians. And this applies to my extended mathematical family as well. I have worked with some extraordinary mathematicians, and hope to go on doing so.

It is a bit invidious to name names, but since I have already named Neil, let me mention three of my students who have perhaps made the best job of going beyond what I could do: Eric Lander, Dugald Macpherson, and Colva Roney-Dougal. Eric is, by most measures, the most successful; Dugald the one who has taught me the most; and Colva has brought mathematics to life for more people than I could ever hope to do. (Other students of mine will be mentioned below as coauthors.)

Collaborators

I have been blessed with some extraordinary coauthors, who have also taught me much, and have stretched my own mathematical scope in amazing ways. What is really clear is that every collaboration is completely different from every other one. Here are a few stories.

I have mentioned Paul Erdős already, but as a coauthor he taught me a lot, about combinatorial number theory, and also about asymptotics, in our work on sum-free sets. He also, of course, showed me the side he showed everyone: his generosity, both with ideas and with love for humanity. I have talked about this collaboration (which involved me eating the full English breakfast that was brought for him in the Judge’s Room in Trinity College, Cambridge) elsewhere.

Three of my siblings, Martin Liebeck, Cheryl Praeger and Jan Saxl, worked closely together on many important papers. I have never written a paper with Martin, although we have worked together. But I have published with Cheryl and Jan, separately or together; our papers include the proof of the Sims Conjecture (with Gary Seitz), using the just announced (but not then proved) Classification of Finite Simple Groups. I knew enough logic by then to see the implications for infinite distance-transitive graphs.

I first meet Bill Kantor when I was travelling across the USA on a Greyhound bus after teaching in Ann Arbor for a semester. From him I learned, among much more, that sometimes, when a problem (such as finding the 2-transitive collineation groups of projective spaces) is too hard, it pays to tackle an even harder problem (finding the antiflag-transitive collineation groups).

Way back, when I was newly hatched, Jaap Seidel took me under his wing and I learned about strongly regular graphs, orthogonal polynomials, and many other such things from him. He was like a second mathematical father to me. I have told the story of him, me, and Jessie MacWilliams at Oberwolfach elsewhere. Also, I owe to Jaap the event that led to my interest in the ADE affair, and so to the book which was published on Thursday. Jaap was the nearest thing to a mentor that I had.

Anatoly Vershik approached me after I had lectured about the random graph at the ECM in Barcelona in 2000, and introduced me to the wonderful Urysohn space. As a result of his deep knowledge of its construction and my penchant for getting groups to act regularly on nice structures, we wrote a paper showing (among other things) that the Urysohn space has many abelian group structures. I was very pleased and honoured to attend both his 70th and his 80th birthday conferences. (There is a little story about paying the hotel bill at the first of these.)

Dima Fon-Der-Flass: I remember him as always happy and smiling, ranging over all of mathematics and enjoying every bit of it. We wrote two papers together which have created ripples on the mathematical pond. Dima’s bijection on the set of antichains in a poset has led to a flourishing subject now called rowmotion, while an idea of mine led to the notion of IBIS groups, those permutation groups for which all irredundant bases have the same size. Dima also devised a problem whose very surprising solution was found by Paul Glendinning: a triangular prism (the triangular cross-section arbitrary) rolls without slipping on a plane, and very slowly loses energy, eventually coming to rest. What are the probabilities that it rests on each face? (This is hard; if you want a much easier problem, imagine instead that the prism, with axis horizontal but randomly rotated, is dropped onto a plane covered with glue.)

Jack van Lint introduced me to coding theory, and so was somehow responsible for the fact that Eric Lander became my student. My book with Jack, which went through a number of changes of title and is now called Designs, Graphs, Codes, and their Links, is still the best source for some material. He also explained to me the elegantly simple proof of the formula for the number of parking functions, and told me about a construction of a certain partial geometry which led to our much simpler construction. Jack was a strong swimmer, and told me that he could give his doctor quite a shock by imagining himself on the blocks at the start of a swimming race while having his pulse taken.

I have known Jarik Nešetřil for some time. On one of my visits to Prague, we came up with a project which combined two of his interests (graph homomorphisms and homogeneous structures), and invented the notion of homomorphism-homogeneity. It also appealed to me since any graph containing the countable random graph as a spanning subgraph is homomorphism-homogeneous. I owe to Jarik many visits to the photogenic city of Prague.

João Araújo lured me into semigroup theory two decades ago, so successfully that one member of that community described me as a “card-carrying semigroup theorist”. He is now my second most frequent coauthor and has been generous to me in many ways, of which the recent Évora conference was one of the most dramatic. His questions had a very simple format: Suppose I have in mind a semigroup property P, and I want to know which permutation groups have the property that, if we add a single non-permutation, the semigroup generated has property P. This has contributed many interesting concepts to permutation group theory. Among these is the concept of synchronization, with an interesting spin-off. The Hall–Paige conjecture (the answer to the question “for which groups does the Cayley table have an orthogonal mate?”) was proved by Stewart Wilcox, Anthony Evans, and John Bray in 2009, though John didn’t publish his part. When it turned out we needed it to determine which diagonal groups are synchronizating, I was able to persuade John to re-do the calculations and write it up to put in the paper.

In view of all this, I might have expected to get more involved with the St Andrews semigroup theorists than I have done, with the exception of James Mitchell, with whom I have several papers on a variety of topics.

Dimitri Leemans works on abstract regular polytopes, geometric objects which have a group-theoretic equivalent, string C-groups. He invited me to Auckland while he was there, and we began a collaboration, also including Maria Elisa Fernandes, which culminated in a recent paper on polytopes of large rank whose automorphism group is a symmetric group. This collaboration has stretched me, not so much with new concepts but the sheer workrate of my collaborators! I visited Aveiro for a month to work on the alternating group, and only on the last day were we able to declare victory.

And I must mention Rosemary Bailey in this context. I think that my paper on diagonal groups with her, Cheryl Praeger, and Csaba Schneider is one of my best, and it was Rosemary who did the hard work by proving the three-dimensional case; my contribution was just the inductive step to deal with all larger dimensions. Another thing I am proud of is getting the outer automorphism of the symmetric group S₆ into an agricultural statistics journal. (Rosemary has herself smuggled the Fano plane into an ecology journal.)

Also, I learned from her that what I had been taught to think of as a block design was quite different from what statisticians thought. This confusion had further consequences. I first met Donald Preece at the BCC in Aberystwyth in 1973. In his talk, he had some matrix equations that I recognised from my work with Jaap Seidel on quadratic forms. So I sat next to him on the excursion coach and told him what I knew. He did some magic on my examples, and we wrote a paper. It took me a quarter of a century to realise what he was doing; then, while walking through the rain forest in the Lamington National Park, I was able to use some group theory to construct infinitely many examples of his designs.

In a sense, Sudoku is closer to the statisticians’ viewpoint than the mathematicians’. By 1960, statisticians Behrens and Nelder had invented gerechte designs and critical sets in Latin squares, so had all the ingredients of Sudoku; but they didn’t put them together. Anyway, Robert Connelly came to Queen Mary to give a talk about rigidity of structures, an interest he shared with my colleague and coauthor Bill Jackson. He said to Rosemary and me after the talk, “Are you interested in Sudoku? I have something to show you”. What he showed us was his invention, symmetric Sudoku, for which there are just two non-isomorphic solutions, and this can be shown by a beautiful argument involving perfect codes over a 3-letter alphabet and the card game SET. This led to a long paper in the American Mathematical Monthly. David Spiegelhalter, one of whose hobbies is stained glass, made a beautiful window out of a symmetric Sudoku solution. (He is a coauthor of mine, but not of a research paper: we were both on the BBC Horizon programme “To Infinity and Beyond”, and together with the producer and Chris Budd we wrote an article about it for the IMA Bulletin.) Symmetric Sudoku was later and I believe independently invented by Vaughan Jones.

Shamik Ghosh had, perhaps inadvertently, a huge influence on my career, by asking a question which I was able to answer about the power graph of a group. This led, by devious routes, to Ambat Vijayakumar asking me to lead a research discussion about graphs and groups. This discussion was so successful that I now have a huge number of Indian coauthors, so many that I am afraid my responses to them are sometimes shamefully slow. (To name just a few, T. Tamizh Chelvam, Angsuman Das, Hiranya Kishore Dey, Vinayak Joshi, G. Arun Kumar, Ranjit Mehatari, and Rajat Kanti Nath.) My other debt to Vijay is that he suggested writing a book, with his colleague and my coauthor Aparna Lakshmanan S, about the Shrikhande graph, which is more than just that; it is a textbook on discrete mathematics using the Shrikhande graph as a focus.

Many of these graphs on groups have a connection to the Gruenberg–Kegel graph of a group (formerly called the prime graph). It was Natalia Maslova who drew me into the circle of research on these, and became a valued coauthor.

When I arrived in St Andrews, Collin Bleak lured me into the rich world of Thompson groups and their relatives, full of exciting connections between Cantor space, synchronizing automata, and dynamical systems. (These synchronizing automata resemble those in my work with João Araújo, but have a stronger property: instead of some word being a reset word, in the sense of taking the automaton to a known state, there is a number k such that every word of length k is a reset word. These automata are quotients of de Bruijn graphs.)

Misha Klin is my coauthor on one paper having seven authors, which arose because four different teams (three doubles and one single) were working on the same problem. (The London Mathematical Society had just introduced a LaTeX style file, which had great difficulty coping with seven authors.) But he is one of my mathematical family for other reasons. In 1990 he posed a problem which Laci Babai and I were able to solve, but he turned down our invitation to be a coauthor. Laci and I, incidentally, have a poor record for promptness in publishing our results. But I remember very well one poignant moment. We were sitting in an office in Montréal working on a paper. My copy was in London, and probably few readers remember how slow transatlantic Telnet was in those days: you would type a line blind, press the RETURN key, and make a cup of tea while waiting for it to be echoed back to you. Anyway, I finished my part a little before Laci, and tried to read my email; I found news of the death of Paul Erdős.

I am very happy that V. Arvind is now a co-author of mine. Jordan proved that any finite transitive permutation group of degree greater than 1 must contain a derangement; with Arjeh Cohen I was able to prove that there are many derangements, and with Rosemary, Michael Giudici and Gordon Royle I investigated the subgroup generated by the derangements. This is taken further in a recent paper with N. Gavioli and C. M. Scoppola, who will also soon be my coauthors. But some time ago, I realised that my result with Arjeh gives a randomized polynomial-time algorithm to find a derangement: simply choose random group elements until you find one! I asked whether the algorithm can be derandomized. Emil Vaughan found an algorithm, based on an idea by Bill Kantor, but it is very complicated and depends on the Classification of Finite Simple Groups for the proof of correctness. Subsequently, I found that there is an elementary and very elegant proof in a paper by Arvind. But our forthcoming paper, with Xuanlong Ma and Natalia Maslova, is on something different: the commuting graph of a group.

Other students

Why do we restrict ourselves to PhD students in working out our families? I have had a number of students who have taken courses from me or been supervised on projects or internships. Among the most recent are Bea Adam-Day, Kamilla Rekvényi, Marina Anagnostopoulou-Merkouri and Dan Roebuck. But there have been many over the years.

I am particularly pleased that Scott Harper is speaking at our retirement do next Tuesday; he will say something about our joint work, in particular on Cauchy numbers. (I had suggested that rather than old codgers like us, we should have people we had taught at St Andrews who are now making successful academic careers. Retirement should be a time to look forward, not back!) And in a sense I owe the start of this collaboration to Hamid Reza Dorbidi, who proposed similar problems to me as a joint research project.

So who is my family?

So I count ancestors and descendants (in the MGP sense) as well as other students, and coauthors, as being part of my mathematical family. But I think there are more. Here are a couple of examples of what I mean.

Tim Burness recruited three very strong teams to give the proof of my conjecture about base size for non-standard actions of almost simple primitive groups. So, even if he were not already in my family (he is my mathematical nephew), I would include him. I encouraged both Scott and Marina to do their PhD with him. (Colva Roney-Dougal and her students and collaborators have taken up very successfully the further investigation of bases.)

I first got interested in relational structures (in the sense of Roland Fraïssé) after reading the paper by Robert Woodrow classifying countable homogeneous triangle-free graphs. One of the starring characters in my thesis was the Higman–Sims graph (now renamed the Mesner graph); it is vertex-transitive and triangle free, and the stabiliser of a vertex is 3-transitive on its neighbours. Henson’s triangle-free graph is a kind of infinite analogue. I talked about the former at the BCC in 1977, and the latter at the BCC in 1987 (on the other side of London): it was at the second of these that Paul Erdős became interested in one of my problems on sum-free sets. (A Cayley graph is triangle-free if and only if the connection set is sum-free.) Later Robert became a co-author, but our work was not really collaborative on that occasion: what Sam Tarzi and I were doing in London overlapped with what Robert and his colleagues Claude Laflamme and Maurice Pouzet had done in Calgary. (Incidentally, Maurice proved a conjecture of mine about zero-divisors in a graded algebra associated with an infinite permutation group with no finite orbits. Perhaps anyone who settled a published conjecture of mine is family?)

So I would have counted Tim and Robert as family even without the direct connection.

And earlier I mentioned Persi Diaconis, who doesn’t fit any of the formal categories, but without doubt is part of my family. I think Eamonn O’Brien belongs here as well. And perhaps also the external examiner of my doctoral thesis, Donald Higman.

So I can’t lay down hard and fast rules, but I have been amazingly blessed in having the mathematical family that I have.

And clearly there are many more people who should be cited here; my apologies to them, but I have to stop somewhere! Even just mentioning all my coauthors would make this a very long post. (For the record, lists of my students, coauthors, and conjectures are available on my web page.)