Rosemary has written a long obituary of Donald, far too long for a journal to consider, but worth having on record (I think).
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Rosemary has written a long obituary of Donald, far too long for a journal to consider, but worth having on record (I think).
A Steiner system S(t,k,n) is a collection of k-subsets (called “blocks”) of an n-set of “points” with the property that any t-set of points is contained in a unique block. To avoid trivial cases, we assume that t<k<n.
Since the existence question was first posed, apparently by Woolhouse in 1844, and solved in the special case t=2, k=3 (Steiner triple systems) by Kirkman in 1847, the existence problem (for which values of the parameters do they exist) has been regarded as one of the most difficult in combinatorial design theory. In particular, only finitely many were known with t≥4, and none with t≥6.
There are necessary conditions known as the “divisibility conditions”. For each number i between 0 and t−1, the number of blocks containing a set of i points can be counted: it is
So all these numbers must be integers.
My former colleague Peter Keevash put a 56-page preprint on the arXiv last month proving that, for given k and t, if n is sufficiently large and the divisibility conditions are satisfied, then a Steiner system exists. Even though it is not clear how large “sufficiently large” is, this is a stunning result.
But he has done more. If “exactly one” is replaced by “exactly λ” in the definition, we have a t-design. There are divisibility conditions similar to the above: just put λ in the numerator. Peter shows that these exist for all sufficiently large n satisfying the divisibility conditions as well, though this is a bit harder.
In a paper entitled “Tight single-change covering designs — the inside story”, published in the Bulletin of the Institute of Combinatorics and its Applications, Donald Preece described how a piece of mathematical research actually gets done, forgotten, revived, extended, and finally published.
Content of the Bull. ICA is not easily available. I am sorry to report that my copies ended up in the skip. But thanks to the kind offices of Ian Anderson, I have my hands on a copy of the paper. The opening paragraph is well worth quoting, for the light it throws both on its subject and on its author.
“But how do you DO research in mathematics?”, people often wonder. Even a distinguished laboratory scientist may be unable to comprehend how research can be possible without test-tubes, pipettes, and bubbling flasks of evil-smelling brews. Well, you scratch your head; you scratch the back of an envelope with a pencil; you scratch a blackboard with a piece of chalk. You lie in the bath and gaze alternately at the ceiling and at your navel. You do the washing-up or go to sleep and you leave your subconscious to do your thinking for you – often astonishingly successfully. And a good old screaming-match with your colleagues can sometimes help too.
If you can get your hands on it, read the whole paper. It explains something I hadn’t known, the origins of the tight single-change covering design problem in the early days of computing, when only a few rows of a matrix could be stored in memory at a time.
Sitting on my shelves were various old notebooks in which I’d kept diaries from time to time. Four of them contained a training diary from 1987, my first year at Queen Mary and the year I ran the London Marathon for the first time. It is not my usual training diary with brief entries for mileages, etc. I had given myself a quarto page per day, and even if I didn’t run I had let my pen run on about what was happening in my life and how my training was going.
I am typing them up, though I doubt if anyone will ever want to read them: it will come to around 100 pages altogether.
But tucked in to the page for Monday 16 February 1987 was an interesting find, the programme for the third anniversary celebration of the School of Mathematical Sciences. The format of the event was: two lectures by new or newly-promoted members of the academic staff, with tea between the lectures and followed by a glass of sherry and then dinner (for which we had to pay). On this occasion, the lectures were given by me and Malcolm MacCallum: I was new, and Malcolm had just been made a professor of applied mathematics. As my diary puts it,
Without having planned it, we switched roles; I talked about my experiments on sum-free sets, phrased in terms of spectroscopy and the uncertainty principle; while he touched on Galois theory and Hilbert’s Tenth Problem as well as the differential geometry, with the physics left for a footnote at the end.
There are also approving comments in my diary about how the return in theorems-per-pound on the free tea was probably far greater than for just another library book.
But what I want to record here is the contents of the leaflet, the School’s Annual Report. This really takes me back to those far-off days, and reminds me what a great place the School of Mathematical Sciences at Queen Mary College was at the time. It’s also interesting to see how some of the same battles are still being fought! Here it is in full.
This has been the year of the UGC ratings, both Pure and Applied Mathematics were rated as “conducting research of international standing” with Pure Mathematics judged as outstanding. The School was also allocated a substantial increase in funded home postgraduates as well as some increase in undergraduates. The new UGC formula grant allocation is based on research judgements and weights postgraduates, so that extra resource came to the College under the Mathematics heading: we hope to feel the benefit of this as the College funding begins to reflect that of the UGC.
Many colleagues participated in a range of international meetings, with Malcolm MacCallum chairing the Organising Committee of the 11th International Conference on Relativity in Stockholm. The School provided members of several SERC and ESA committess (Gruenberg, Rowan-Robinson, Roxburgh and Williams). The School provided editors of four major journals: Pure and Applied Algebra (Gruenberg), Symbolic Logic (Hodges), Classical and Quantum Relativity (MacCallum) and Proceedings of the London Mathematical Society (Collins). In addition staff served on the Editorial boards of the Journal of Symbolic Computation, Journal of Combinatorial Theory, Combinatorica, European Journal of Combinatorics, Journal of Philosophical Logic, European Journal of Physics, Solar Physics, and of the Bulletin, Journal and Proceedings of the London Mathematical Society.
The School ran a large number of research seminars in Pure Mathematics, Logic, Relativity, Dynamics, and Astronomy, and co-organised the London Algebra Colloquium and the London Space Plasma Physics Colloquium. We also had a lively series of postgraduate research courses as well as a range of M.Sc. courses on the London M.Sc. in Mathematics and the College M.Sc. in Astrophysics.
There has been a large number of personnel changes — upwards rather than downwards. Malcolm MacCallum was appointed to a Personal Professorship and Don Collins and John Papaloizou promoted to Readerships. Peter Cameron joined us as a Reader under a replacement scheme, and Len Soicher was appointed as a Mathematician/programmer, a new post established with special funds allocated under the Universities Academic Initiative Scheme for work in Computational Group Theory. Bernard Carr moved onto the permanent lecturing staff at the beginning of the year on completion of his SERC advanced fellowship, filling a post established with special earmarked funds from the University. Nigel Weiss and Douglas Gough, both readers in Mathematics and Astronomy at Cambridge, were appointed as honorary professors in the School. The School has also recently been awarded additional earmarked equipment monies by the University. The Secretarial staff are a source of strength and we are very pleased to have such a happy working environment.
The number of visitors and research fellows continues to grow, as does the number of postgraduates. We are especially pleased that we will have 3 SERC Advanced Fellows starting on October 1st — Peter Kropholler, Andy Lawrence and Phil Palmer. As there were only 15 such awards across the all subjects in the country, for the School to have attracted 3 of these is a considerable success. This is very desirable and contributes to the stimulating research environment we have established. But it exacerbates the very serious space shortage facing the schoolm which is now squeezed into 47% of UGC norms. We are continually pressing the College to solve this problem and are grateful to the Computer Centre for transferring to us one extra room. But we are still faced with the unpleasant task of turning away visitors for lack of space to accommodate them, and the Head of School has become an expert on “the packing problem”.
This shortage of space has inhibited the School form developing its M.Sc. programme in Mathematical Computation, an area where we have a range of expertise, we hope some satisfactory solution will soon be found to these problems.
The Astronomy Unit, together with the Astronomy Group in Physics ran the SERC Summer School for new research students in their subject, a two week residential course for some 60 students. The whole event was successful with a new generation of students starting their careers with admiration for Bernard Carr’s abilities to operate a slide projector. We were able to initiate the new Principal into the workings of our subject at a successful and enjoyable dinner at the Halls of residence.
The shortage of Mathematics teachers in Schools is a serious national problem and one that is of great concern to Universities. Ian Roxburgh joined the newly formed Steering Committee of the National Committee of Heads of Departments of Mathematics, which has been exerting pressure, and offering suggestions to the DES on how to tackle the problem. The School has responded to the UGC call for initiatives to address the problem with proposals for a series of short intensive in-service courses for both qualified and unqualified Mathematics teachers, and with a request for funds for teacher fellowships. This year the School ran a number of one day conferences for School teachers, ably organised by Dave Arrowsmith, and very well attended. All involved, lecturers and participants judged them to be highly successful and worth continuing in future years. One benefit for the School is that the applications for undergraduate places have seen a dramatic rise of 28%.
Morale in the School is high in spite of our space problems. We look forward to another year with optimism and hope.
I have been moving various files from Queen Mary to St Andrews. After the recent discussion about what constitutes publication, I started wondering about who counts as a co-author.
I believe I have 152 coauthors of published or accepted papers. But MathSciNet puts the number at 140, and the Erdős Number Project homepage only gives 129. Interesting to compare?
The first thing to notice is that both MathSciNet and ENP overcount by 1, since they don’t realise that Alejandro, Priscila P. is the same person as Kazanidis, Priscila A.
The MathSciNet rules (or what I guess are the rules, I couldn’t find this stated anywhere) are that to count as a coauthor you must have a joint paper listed in the MathSciNet database. This rules out Budd, Cooter and Spiegelhalter (with whom I wrote an article for Mathematics Today about the BBC Horizon programme on infinity), Beineke and Wilson (with whom I am jointly credited with the introductory article in Topics in Algebraic Graph Theory), Hirchfeld (joint editor of the proceedings of the first Isle of Thorns conference), and six of my nine coauthors on the paper on intricacy (see here), published under the nom de plume W. E. Opencomb. They also miss Joshua Browning, presumably because the paper is too recent (it went up on the JCT website today).
The ENP do have my coauthors from W. E. Opencomb, but miss out Robert Bailey, whom they initially confused with Rosemary Bailey; and many other people, since it is some years now since the lists were updated.
But in none of these cases do they count the arXiv as publication. Google Scholar does, however, as Robert Bailey pointed out; this would increase the number by four. However, they only found 25 of my co-authors, and offered me about the same number of suggestions of people I might wish I’d written a paper with but haven’t (such as Laszlo Lovász).
So Google Scholar may overestimate publication counts but doesn’t do so well on co-authors!
There were a few 1997 BCC T-shirts in my office when I cleared it out. Cheapskate that I am, I used them for Christmas presents. (They are very good quality!)