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Professor Ron Shaw,
BA, PhD,
ScD(Cantab)
Reminiscences
Contents: 1. The years 1929-1949
2. Undergraduate years:
Cambridge
1949-1952 3. Postgraduate years:
Cambridge
1952-1955 4. The years 1955-1990
5. The years 1990-present
6. Postcript:
near-assassination of vice-chancellor-elect As an academic, I could well hold the record for lack of physical movement: 1929 Born on 5 Sep, in Tunstall, Stoke-on-Trent 1929-1947 Stoke-on-Trent 1947-1949 Derby (National Service) 1949-1955 Cambridge University 1955-1989 Hull University (assistant lecturer, lecturer, senior lecturer) 1989-1995 Hull University (personal chair in Mathematical Physics) 1995-xxxx Hull University (Emeritus Professor) When I recount this lack of movement to younger colleagues, who appear to change university, and quite often country, every 2 years, I always hope that they will say something like: "Yes, but you have had some adventurous intellectual voyages". But they never do! 1.2.
National service
and mercury poisoning
1947-1949 National service: DORA (= Dental Operating Room Assistant) in the Royal Army Dental Corps. I ended up in Derby, at a one-man Dental Centre, which, along with my bedroom, was situated immediately above a morgue. I spent my days making plaster cast of teeth, and squeezing out through muslin -- with bare hands! -- excess drops mercury from amalgam fillings. I always blame the mercury poison that I must have absorbed for my later difficulties with group cohomology and the like. 2.
Undergraduate years:
Cambridge 1949-1952
During 1949-1952 I was a Scholar at Trinity College, Cambridge, and I sat for Parts I, II, III of the Mathematical Tripos in the respective years 1950, 1951, 1952. 2.1.
Newton, Dyson,
Wittgenstein, Adams, Atiyah, MacDonald, Polkinghorne
My childhood background was very non-academic: we did not have a single book in the house. I ended up at Trinity College because my school physics teacher, due to his admiration for Isaac Newton, chose to put me in for an entrance scholarship there. (At that time I really did not know what a university was.) My first room in Trinity College was K9, in the tower in Whewell's Court which overlooks Jesus Lane. The previous occupant was Freeman J. Dyson (1923- ), and the previous occupant of the adjacent room was Ludwig Wittgenstein (1889-1951). Concerning the latter, my bedmaker used to tell stories of his strange behaviour (black decor, fan to drive away the breath of his students, ... ). I never actually met Dyson, but I was amused to read that one of his pre-university influences was, surprisingly, Piaggio's "Differential Equations", which my own Maths teacher had proof-read. I did once meet Dyson's wife, at the Zurich 1994 ICM, where she had a poster on display close to mine. Contemporaries of mine at Trinity included Frank Adams, Michael Atiyah, Ian MacDonald, Roger Phillips, John Polkinghorne, ... . Often we had coffee in John M. Brew's room on Newton's staircase, Great Court --- which fact allows me to mention that, as a result of my Royal Army Dental Corps experience, I presumably have one thing in common with Sir Isaac Newton, namely too much mercury in my system. [For from the website http://www.mercuryinschools.uwex.edu I read "Newton also was an alchemist who actually tasted the chemicals he worked with. At age 49, he became emotionally disturbed for a couple years. In 1979, hair strands from his corpse were tested for mercury and were found to contain 75 parts per million. (Normal levels are about 5 parts per million.)"] 2.2.
Who/what links
homology to Carmen Jones?
Well, for me, the answer is Michael Atiyah. One day, while a group of us were strolling after lunch around the Backs, Michael announced that the most important equation in mathematics was x^2 = 0. I did not understand this at the time, but, decades later, I dimly appreciated it whilst failing to come to grips with MacLane's book "Homology". One evening, a little later, a group of us emerged from the cinema in the Cambridge marketplace after having seen the film Carmen Jones. I myself was completely ignorant of opera and so was not aware that the film was based on Bizet's Carmen. However Michael pointed out with amusement that Escamillo (the toreador) had been transmuted in the film to Husky Miller (boxer). 2.3.
Mathematical
Tripos Part II and poetry
For a discussion of this topic, consult my very learned article "Triposland" in Eureka 14 (1951) 7-9. [Eureka is the journal of the Archimedeans, the Mathematics Society of the University of Cambridge (http://www.cam.ac.uk/societies/archim/index.html). Contents pages of all back issues of "Eureka" can be found online at: http://www.srcf.ucam.org/archim/eureka/backissues.html.] 2.4. J.A.
Todd (1908-1994)
I attended lectures given by Burkhill, Todd, Bondi, Hoyle, Dirac and others. Once, in one of his lectures, I received a sharp reprimand from Todd --- for reading the morning paper while sitting in the front row. (In my defence, he was closely following his book which I had open in front of me as well.) On the other hand, in a tutorial, I was once inordinately praised by him ("never seen it done like that before") for my answer to one of his past Part II questions. I did not quite understand this praise, as all I had done was to piece together different sections from his own book! In some recent research of mine a paper of Todd's turned out to be of relevance, namely the one entitled "The odd number 6". 2.5.
P.A.M. Dirac
(1902-1984)
[http://turnbull.mcs.st-and.ac.uk/history/Mathematicians/Dirac.html] Only once did I attempt any interaction with Dirac. By adopting his ways with bra and ket vectors (which combined together to form a bra(c)ket) I derived a contradiction. Along I think with Polkinghorne and others, I went to his room to show him the contradiction. He was not at all perturbed. He murmured something about a pure mathematician (Stone??) in the States who had looked in to such things, but he gave the impression that he himself dwelt in a higher realm, aloof from such mundane things as contradictions. I am still amazed that in Dirac's lectures no mention was made that his bra(c)ket was an inner product in a Hilbert space. On 13 Nov 1995 a one-day meeting was held at the Royal Society in commemoration of Paul Dirac. Afterwards I joined a large group of people in Westminster Abbey for a Service "Dedication of a Memorial to Paul Adrien Maurice Dirac, OM". The Memorial plaque was placed adjacent to the grave of Isaac Newton. The service included a series of readings about Dirac. The one description of Dirac that still lingers on in my mind is a quotation from Niels Bohr: "Of all physicists, Dirac has the purest soul". 2.6.
Trivial pursuits
I: Svoyi Kosiri and Professor Besicovitch
After my two years of National Service doing no mathematics, the years 1949-1952 were tremendously exciting, but also too rushed. It was a great thrill in those post-war years to be in an environment where the life of the intellect was so highly regarded. However, due to some bad advice that I should attend a double load of lectures (in order to go straight from Part I to Part II, missing out Prelim), I spent most mornings going to 4 hours of lectures, running from St. Johns to Peterhouse, often too late to find a seat in a crowded lecture room. Nevertheless I somehow found (too much!) time to indulge myself in more trivial pursuits. As a result of having Professor A.S. Besicovitch (1891-1970) as a tutor, I was sometimes invited to go round to his rooms in New Court to test my skills at his favourite chess-like Russian card game Svoyi Kosiri (in English, "One's Own Trumps"). We played it at Trinity as a 2-person game, and it then becomes a game of skill, with the opening position completely symmetrical: what A holds in Spades, B holds in Hearts, and what A holds in Diamonds, B holds in Clubs. Possibly I may be doing a great disservice to mathematics, but nevertheless let me divulge that the rules of the game, together with some hints on play, can be found in the article, which I wrote jointly with J.M. Brew, "Svoyi Kosiri is an Easy Game", Eureka 16 (1953) 8-12. Incidentally I found being tutored by Besicovitch a rather disconcerting experience. At frequent intervals he would retreat to another room, taking off or putting on another layer of clothing; often his next question to me, in his very strong Russian accent, would come through to me from this other room. Some of his questions were very tough, and sometimes I suspected he did not even know the answer himself. 2.7.
Trivial pursuits
II: Bridge, fairies and Swinnerton-Dyer (also Atiyah!)
I also developed a keen interest in playing bridge. After dinner when having a few rubbers of bridge in John Brew's room, Peter Swinnerton-Dyer, at that time a Junior Fellow at Trinity, would quite often drop in on us and pass barbed comments on the standard of our play. (Around that time he did play for England in some Helsinki Olympics (or such-like), and I believe he was later in a team that won the British Gold Cup.) Now Swinnerton-Dyer was certainly not at all responsible for the time I "wasted" playing bridge away from mathematics, but I do somewhat blame him for introducing me to fairy chess and the joys of archbishops (which bounce of the edges of the chessboard in a rather undignified fashion), knightriders (that is (n,2n) leapers) and grasshoppers. Too often I spent hours after midnight trying to decide such things as: can two knightriders force mate against a lone king? Concerning bridge, in one vacation I was captain of a team of four at a congress in Llandudno. One member of my team later became President of the Royal Society, one member later became Master of Trinity, and one member later became Director of the Isaac Newton Institute. However I have to confess that my selection abilities were not quite so outstanding as I have just made out, since all three offices just mentioned refer to one person (Sir Michael Atiyah, O.M.). 2.8. An
exceptional
year? Google provides evidence!
At the time, I assumed that my contemporaries in mathematics constituted a typical year's intake for Cambridge. As the years went by, and so many of my contemporaries so soon got chairs, medals, FRS's, ... , I began to wonder if after all `my' year was a bit exceptional. Very recently (with the aid of Google!) I discovered evidence in support of this last. Usually when I do a Google search I find I get at least 5,000 hits. In the expectation that for a change I would get 0 hits, I tried typing Adams, Atiyah, Polkinghorne, Shaw, Taylor in the Google searchbox. To my surprise I discovered several hits, one being: "Though not quite as high as that prevailing in the remarkable 1952 examination when there were 40 candidates and the 11 Distinctions included the future Professors J.F. Adams, M.F. Atiyah, I.G. MacDonald, A.G. Mackie, J.C. Polkinghorne, R. Shaw, T.J. Smiley and J.C. Taylor." This turned out to be a footnote on p.6 of An Unofficial Guide to Part III, by T.W. Korner: [see http://www.dpmms.cam.ac.uk/~twk/PartIII.ps ] [Digression. I vaguely recall mention of a game played with Google where your highest score is obtained if your search term returns precisely one hit. In this connection I point out that "Svoyi Kosiri" produces just one hit, namely the Eureka article mentioned earlier. (Possibly different English transliterations may produce more hits?)] 2.9. A lack
of distinction
may well boost your career!
When the 1952 Part III results were announced I was standing on the steps of the Senate House next to another candidate who was also at Trinity, but who had arrived from a first degree at Edinburgh just for the one year of Part III. When he failed to get a distinction, he turned to me and said that he was going to switch from Mathematics to Law. I admired such a quick and decisive response and, as the years, and decades, passed by I occasionally wondered whether James had indeed switched to Law and with what outcome. More than 35 years later I was amazed one morning to recognise James in a cartoon in the Guardian newspaper! For James Mackay was now Lord Chancellor, Lord Mackay of Clashfern. I wrote to him at this point, asking him if he remembered making his decision to switch as recounted above. He replied that he remembered that day well; however he did say that prior to that day he had been seriously considering a switch to Law. 2.10.
Lord Mackay
and group characters
One day in James' room, in Whewell’s Court, Trinity, we were
discussing
group representations, and I remember him saying that he did not
understand
why such stress was placed on group characters: for do we not lose a
great
deal of information concerning a representation D of a group G by
restricting
attention solely to the traces of the various D(g), g in G? His concern
came back to me thirty years later when I was writing my 2-volume work
Linear Algebra and Group Representations (Academic Press, 1982, 1983);
see Section 13.2.2(d) of Vol. II for my attempt to respond to James's
concern. 3.
Postgraduate years:
Cambridge 1952-1955
During 1952-1955 I was a Research Scholar at Trinity College, Cambridge, and in 1992 I commenced research into the Theory of Elementary Particles. 3.1 Nicholas
Kemmer,
Roger Phillips, John Polkinghorne, John C. Taylor, Abdus Salam, Lord
Byron
My research in 1952-3 was supervised by Nicholas Kemmer, Trinity. Along with myself I recall that there were three others starting to research into the Theory of Elementary Particles, namely Roger J.N. Phillips and John C. Polkinghorne, both from Trinity , and John C. Taylor from Peterhouse. Very soon John Taylor and I developed a close and lasting friendship. For reasons explained below, my research supervisor for !954-5 was Abdus Salam, newly arrived at St. John's College. At one period in these final postgraduate months at Cambridge my rooms at Trinity were on Byron's staircase in Great Court. (When at Trinity, Byron kept a bear in his rooms on this staircase, as the more usual pets were sternly prohibited.) 3.2.
Wolfgang Pauli
(1898-1988) is my grandfather
My first research supervisor, Nicholas Kemmer (1911-1998), was himself a research student of Wolfgang Pauli (1898-1988). Kemmer told me that, in his first week, Pauli had given him an extremely tough problem to investigate. Kemmer was so dismayed by how difficult the problem was that he very nearly gave up Theoretical Physics completely. So, to protect me from a similar dismay, Kemmer decided not to suggest to me any problems in my first year of research. Instead he guided me through the occasional Pauli paper, and made various suggestions (Schwinger, Feynman, Dyson, ... ) of other papers to read. This suited me very well, as I liked to work on my own, following up my own ideas. In the middle of 1953 I was expecting Kemmer to make some definite suggestions, and I am not at all sure that I was going to welcome them. But in fact at this point Kemmer left Cambridge to become Tait Professor of Mathematical Physics at the Tait Institute, Edinburgh. Eventually, in early 1954, I became a research student of Abdus Salam (1926-1996). Salam's tendency was at the other extreme from Kemmer's: Salam was buzzing with research projects, often involving nuclear physics of which I was woefully ignorant. Consequently I tried to keep away from Salam as much as possible, and to carry on following up my own ideas. 3.3. My
Ph.D. dissertation
My Ph.D. dissertation was submitted in September 1955. It was entitled The Problem of Particle Types and Other Contributions to the Theory of Elementary Particles, It consisted of two parts, each self-contained with its own list of references: Part I. Representations of the Improper Lorentz Group and the Problem of Particle Types. Part II. Some Contributions to the Theory of Elementary Particles. It is easy to overlook Part II! --- since it, and its contents, are not listed at the beginning of the dissertation, but only after the end of Part I. The page numbering starts afresh in Part II. The contents of Part I are: Introduction. The contents of Part II are: Actually I wrote Part II first, completing it some time in 1954. Compared with other peoples' unified theses, Part II seemed to me to be inadequate, as it consisted of several disjoint bits. So in late 1954 and in 1955 I wrote Part I, which at least was unified. 3.4.
Yang-Mills-Shaw
theory
A footnote on p.37 of Part II of my dissertation reads: "The work described in this chapter (ch.III) was completed, except for its extension in Section 3, in January 1954, but was not published. In October 1954, Yang and Mills adopted independently the same postulate and derived similar consequences." But although their publication date was in 1954, Yang and Mills must have priority since it seems that their research was completed in 1953. The idea for chapter III of part II came to me in a flash while reading a manuscript of Schwinger's, which I found left lying around in the Philosophical Library in Cambridge. In it Schwinger showed how invariance of the Lagrangian under general gauge transformations required the introduction of the electromagnetic field. This of course was not new (though possibly it was to me at that time), but Schwinger's manuscript used real spinors, and so the usual U(1) invariance appeared instead as SO(2) invariance. Being familiar with Kemmer's work on invariance under "special" isotopic spin transformations it seemed to cry out to see what would happen if I changed Schwinger's (abelian) SO(2) to the (non-abelian) isospin SU(2). I showed my generalization to Salam in early 1954, but in a rather disparaging way, since I did not doubt at that stage that the new non-abelian gauge fields would require particles to have zero mass, and such particle did not appear to exist in nature. Later on in 1954, Salam showed me the paper by Yang and Mills. Salam still wanted me to publish my contribution, but I never did. On many occasions (the 1962 Istanbul Summer School on Group Theory in Physics, the Schrodinger Centenary Conference at Imperial College in 1987, ... ) he publicised my independent discovery. In his Nobel Prize Lecture 1979, reprinted in Rev. Modern Phys. 52 (1980), 525-538, there are (see below) several references to Yang-Mills-Shaw theory. I have also recently come across a letter from Salam to me dated 1 Oct 1988 (in connection with the submission of a paper of mine to Proc. Roy. Soc.) in which he again refers to Yang-Mills-Shaw theory, and reminds me: "I still remember asking you to publish this and you were very shy at that moment because you thought Yang-Mills had published it already although you had done the work independently." However most physicists just refer to Yang-Mills theory --- and actually I am quite glad of this! I like a quiet life, and would not have enjoyed being pestered throughout the decades by lots of queries from researchers expecting me to be up to date with latest developments. 3.5. Chapter III of Part II of my Cambridge Ph.D. dissertation Chapter 9 of the book: It includes extracts of my correspondence with Kemmer about the history of my contribution, and reproduces the relevant chapter from my Cambridge Ph.D. thesis, namely Ch.3 of Part II of the thesis. This chapter of my thesis is also reproduced in Incidentally, for a non-technical tour through the principles of physics, I recommend another of J.C. Taylor's books: Hidden Unity in Nature's Laws, New York: Cambridge University Press, 2001. 3.6.
Salam's 1979
Nobel Prize Lecture
In 1979 Sheldon Lee Glashow, Abdus Salam and Steven Weinberg were jointly awarded the Nobel Prize in Physics "for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current". Salam's Nobel Prize Lecture can be found at: http://www.nobel.se/physics/laureates/1979/salam-lecture.pdf In it he generously acknowledged my (unpublished!) contribution, as the following extracts show. "Now the fact that I was using gauge ideas similar to the Yang - Mills (non-Abelian SU(2)-invariant) gauge theory was no news to me. This was because the Yang - Mills theory [9] (which married gauge ideas of Maxwell with the internal symmetry SU(2) of which the proton-neutron system constituted a doublet had been independently invented by a Ph. D. pupil of mine, Ronald Shaw, [10] at Cambridge at the same time as Yang and Mills had written. Shaw's work is relatively unknown; it remains buried in his Cambridge thesis. ... " "And finally, though the use of a Yang-Mills-Shaw (non-Abelian) gauge theory for describing spin-one intermediate charged mesons was suggested already in 1957, ... " "Once the Yang-Mills-Shaw ideas were accepted as relevant ... " "... a theory which looks like Yang-Mills-Shaw's." "Regarding spontaneously broken Yang-Mills-Shaw theories in general ... " "... and we learnt of his (Oscar Klein's) anticipation of a theory similar to Yang-Mills-Shaw's long before these authors." For me the Pauli Exclusion Principle will
always read:
"Square roots are excluded in the mathematical expression of Nature's laws." This came about as follows. In 1958(?) John C. Taylor and myself were invited to Salam's rooms in Imperial College to see Pauli. (I had previously come across Pauli only once, at a Padua/Venice conference in 1957.) Pauli asked us what we were currently interested in. Luckily I did not talk of my own work, but instead, on the blackboard, I sketched out some details of a recent paper that had interested me, written by someone else (Fronsdal??). It involved expressions for the infinitesimal generators of the inhomogeneous Lorentz group in a momentum space representation, and in the denominator of one expression there occurred the square root of (E+m). At this point Pauli became convulsed, and would not let me proceed any further! Apparently square roots (but perhaps only those in denominators?) just could not be tolerated. I have never been able think of an explanation of Pauli's wrath. (I consulted John Taylor to check my memory of this event, and he thinks that he remembers Pauli's exact words as being "I do not like square roots".) I think Pauli did have a tendency to rush in and immediately reject ideas not to his immediate taste. I give the following extracts from Salam's Nobel Lecture. "He returned the next day with a message from the Oracle: `Give my regards to my friend Salam and tell him to think of something better' ". "Pauli's reaction was swift and terrible." " ... he concludes his letter with the remark: `Every reader will realize that you deliberately conceal here something and will ask you the same questions'." "I must admit I was taken aback by Pauli's fierce prejudice against universalism ... " 4.2.
Mind, and
Bertrand Russell
In 1958 I was gratified to have a paper "The Paradox of the Unexpected Examination" accepted by the philosophical journal Mind (see Mind 67 (1958) 382-384) --- especially as it was next to a paper by Bertrand Russell. Extract from B. Medlin, Amer. Phil. Quart. 1 (1964) 66-72: "Several philosophers have discussed this problem in Mind. Of these, we must put Mr. Shaw first and the rest nowhere." 4.3.
Unorthodox
route from Hull to the 1962 ICM, Stockholm
In 1962 I decided to drive to the ICM at Stockholm in my Mini car. The resulting distance on my odometer might seem a trifle excessive, since it recorded > 4,000 miles! This came about because I was in an unusually adventuresome mood and decided to drive to Stockholm via Istanbul. For in that year there was an Istanbul Summer School of Theoretical Physics, which took place in Robert College, Istanbul. I remember picking up two other participants in Strasbourg, and I believe we made it to Istanbul in just over 4 days and 4 nights. I particularly recall an interesting stopover in Skopje (before the devastating later earthquake) and also one the next night in Sofia. In Sofia some Bulgarians came to our rescue while we were attempting to decipher the menu in a restaurant. They turned out to be journalists who delighted in showing us around their printing works. Later that evening we joined up with them again and they brought along with them two highly intelligent literary companions, who were much better versed in English literature than we were. I should perhaps have said "English and Scottish literature", since one of them, a poet, said he had just been translating into Bulgarian some of Burn's poetry, and he proceeded to declaim some verses to us. The 1962 Istanbul Summer School can be considered to be the progenitor of the ICGTMP series of conferences (ICGTMP = Intern. Colloq. on Group Theoretical Methods in Physics), many of which I attended in the period 1974-1992. (The XXV ICGTMP is scheduled to take place in Cocoyoc, Mexico, 2-6 Aug 2004.) 4.4.
Eugene P.
Wigner (1902-1995)
At the 1962 Istanbul Summer School I recall lectures, or lecture courses, given by Racah, Speiser, Nambu, Salam, and several others; however the two contributions which had most impact upon my own future research were undoubtedly the lectures given by Eugene Wigner on Representations of the Inhomogeneous Lorentz Group including Reflections, and the course of lectures delivered by Louis Michel on Invariance in Quantum Mechanics and Group Extension. [See Group Theoretical Concepts and Methods in Elementary Particle Physics, F. Gürsey (ed.), Gordon and Breach, 1964.] My Mini car came in useful at the Summer School, for on several occasions I gave Wigner a lift in it back up to Robert College from restaurants on the side of the Bosphorus. I also recall conversations with Wigner on the conference excursion to the Prince's Isles in the Sea of Marmara. At one point Wigner asked me about my journey from England to Istanbul, but when I told him how much I had enjoyed the interaction with the Bulgarian journalists and literary people, I recall his disapproval, presumably because he was strongly anticommunist. In later years I had the occasional rare interchange with Wigner. Recently I opened some old files and discovered a letter from Wigner, dated 1 Oct 1970, which included "You are absolutely right in what you say and I cannot understand how we could make such a mistake." Unfortunately there was no copy of my own letter, and I can not now remember what was Wigner's mistake! More than 10 years after Wigner's Istanbul lectures I took another look at his treatment of spacetime reflections, and saw how to simplify his treatment, essentially by incorporating the reflections at the level of the "little group", and then using "generalized inducing", as explained in my 1974 papers. When I communicated my simplified treatment to Wigner, he replied (12 Sep 1974) with "Thank you for letting me see your article. I really enjoyed it." I met Wigner again at the 13th ICGTMP, Maryland 1984. (At this conference the Wigner medal was presented to Louis Michel.) I was somewhat startled, but suitably gratified, when it was he, Wigner, who sought information from me concerning the history (via Clebsch and Gordan) of his own 3j-coefficients! I did once give a talk at Oxford (1972) entitled "A century of 3j: 1872-1972" --- but I could hardly believe that Wigner was aware of this. I had also recently included material in Vol. II of my "Linear Algebra and Group Representations" which was somewhat relevant, see sections 10.5, 11.4.1, 11.5 and 12.3. No doubt I burbled on about ancient books on Classical Invariant Theory, such as Grace and Young's "Algebra of Invariants", but I imagine Wigner already knew about these. On my return to Hull I did remember a correspondence I had had (~1971-2) with Lady Jeffreys, and so I was able to send Wigner a copy of her "The history of the Clebsch-Gordan series". A Wigner anecdote. Wigner's rooms at the 13th ICGTMP were in the conference centre, and for the first few days he passed without mishap through a certain door, on his way between his rooms and the lecture halls. But on one day the Pure Mathematician in him appeared to have taken over from the Physicist, since he was observed standing motionless facing the door for a long period of time, clearly thinking that the notice on the door "This door must be kept closed at all times" did not seem to allow any exceptions. 4.5. Semester at METU, Ankara, 1963 As I remarked previously, I have a rather boring CV, remaining for 40 years at the University of Hull. However I did spend 5 months away from Hull in 1963, when I accepted an invitation to go for one semester to the Middle East Technical University (METU) Ankara, as Associate Professor. My head of department at METU was Erdal Inönü, and on his staff was Feza Gürsey who at that time spent alternate years at Ankara and at Yale. Their policy was to stimulate research by young Turkish students by inviting to METU a host of researchers from the international scientific community. During my 4-5 months at METU I had a small overlap with T.D. Lee and G.F. Chew, and a larger overlap with Louis Michel. (I presume that my own invitation to METU came about as a result of Gürsey consulting with Salam.) There was a very relaxed and welcoming atmosphere at METU, and I certainly had an interesting and happy stay there; I enjoyed my interactions with Gürsey and Inönü, and it was good to renew my acquaintance with Michel. I will say more about Inönü, Gürsey and Michel below. 4.6. Some
non-mathematical
memories of Turkey
[I am not a real male; dog --- bite and Chew; a severed head; police interrupt a research seminar in order to track me down; untrustworthy tourist literature; in Beirut it never rains in ... but it hails; a highland fling for a crying bride; foolhardy behaviour at a Turkish well; a submerged car, and a boiling gearbox; a most delicious Bursa peach.] Partly because my lecturing duties at METU were light, on many weekends I had trips away from Ankara, visiting archaeological and historical sites, such as one where Hittite reliefs were carved out of the rock. On one long weekend some friends took me on a trip to a small town on the Black Sea. I was amused to discover there that, since I was an Englishman, Turks did not consider me to be a real male. For, on a wet afternoon, I was allowed into the local cinema even though the performance was advertised for females only. One weekend I remained in Ankara but decided to explore on foot the outskirts/arid countryside beyond Çankaya. At one point there was an apparently peaceful scene, with a calf, along with a small dog, lying down outside a small rundown dwelling. But as I walked away the dog came behind me and bit my Achilles tendon. Consequently I had to start a course of daily anti-rabies injections in my stomach. My Turkish friends managed to get the dog captured and put under observation to see if it had, or developed, rabies. A man, a Maronite Christian from the Lebanon, whose apartment was next to mine, told me that when a similar thing happened to him he went back to collect his family sword and then returned to sever the dog's head, which he presented to the hospital for rabies tests! (Actually what the Lebanese did was unwise, since the dog needs to be kept alive for some days to see if eventually rabies manifests itself.) By an extremely lucky chance a couple of days later I came across in an open-air stall the Proceedings of a very recent conference on Virology. I stood in the street a long time reading up about rabies, and learned that the stomach injections I was having originated from live viruses which had hopefully been killed by phenol, but that it was common for people to suffer a fever, and even catch rabies, as a result of the injections. I decided that my chance of succumbing to rabies was slight: I had been bitten in a part of my body far from the brain, moreover through clothing --- and the dog had still not become rabid. So I decided to stop taking the injections. Next day I travelled to Istanbul to collect a Ford motor car that I had ordered, since at the end of my stay I wanted to travel more widely in Turkey, and then to drive back to England. I was annoyed that the car was a left-hand drive model, even though I had specified right-hand drive. I almost had a (rather unusual) accident within the first two minutes of my journey back to Ankara: going round a corner of a narrow Istanbul street, I narrowly missed colliding with a dancing bear. On my arrival back in Ankara I went to METU and discovered a message from the secretary. This was scrawled in lurid red and read: YOU MUST CONTINUE YOUR INJECTIONS OR YOU WILL DIE. The next day a research seminar that I was attending was interrupted by the police: they were looking for me, to tell me that I must continue with the injections. But this I did not do. I later learned that some details of my rabies saga had percolated back to Cambridge, and caused some amusement --- chiefly because the story of the dog-bite was narrated by ------ Chew! Around this time there was a week's break in the semester, and I decided to have a 6-day holiday in the Lebanon, hopefully spending 2-3 days skiing in The Cedars and 3 days in Beirut, including some bathing and a trip to Baalbek. This went more or less as planned, although I was a lone end-of-season skier. Prior to my departure I had read in the tourist literature that: a) it never rains in Beirut in the months May-September; b) the Lebanon is an example to the whole world of how different races and religions can live together in harmony. Concerning a), I have to report that during my first night I was awoken by a violent thunderstorm, and on looking out of my Beirut hotel window I saw 2-feet deep drifts of hailstones. Concerning b), I was not surprised by future happenings in the Lebanon, having heard from my Lebanese neighbour some very shocking tales of slaughter in the past. Towards the end of my stay in Ankara I went in my newly acquired car on a trip to visit the rock churches of Cappadocia, taking with me two research students from METU. We stayed at Göreme, where a village wedding had been going on for some days and nights, but apparently the bride-to-be had still not met her future husband. One of the research students, Saime, knew the family of the bride, and, on a beautiful star-lit night, she arranged for me to visit the all-female group that surrounded the bride in some kind of alcove. The groom's group was of course all male, and they were being entertained separately by a dancer (a young boy). I questioned Saime to check that it really was all right for me to intrude on the female gathering, and once again, being English, it seemed that my maleness did not seem to count! I found that the bride was crying. Apparently I needed to place a gift (silver coins?) in her hand, which were dyed red, and to my horror I was also told that a song was expected. Having no singing ability whatsoever, I decided to do a Scottish Highland fling, which fortunately seemed to go down quite well. The next day, after a long drive, we arrived back in Ankara at night-time. Just as I thought my day was over, a car, coming down a one-way street the wrong way, clipped my wing, and my Turkish companions urged me to give chase. It was against my inclinations, but I did so and at least discovered where the driver lived. My main worry was to get back to my apartment before a military curfew set in, which I only just did At the end of my stay in Ankara I went with an English friend on a longer trip, ending up at Alanya and Antalya on the south coast. Often in Turkey I found myself throwing off my usual cautious approach to life, with worries over future plans, but instead living to the full in the exciting present moment. At times however I was foolhardy. On the way south I recall succumbing to a whim to ascend as far as possible up an extinct volcano. The track up was steep and ridiculously rough. Often we had to get out of the car to fill in deep potholes ourselves before we could proceed further. In the hot sunshine both of us, and the car's radiator, were soon in need of water. At this point we came across a deep well, and I drank copious drafts of deliciously cool well-water. A few moments later a nomad passed by, shaking his head, and then I was dismayed to hear the sound of sizeable creatures swimming down below in the well! On our last full day we managed to ford a wide river and bathe in some lovely places beneath the Taurus Mountains west of Antalya. On returning towards evening, the sun was no longer overhead, and as a result I failed to find my shallow route across the river, and we ended up half submerged with water covering our thighs. We managed to get towed out, and next morning set out back to Ankara. I had a disagreeable experience while ascending the Taurus range: I suffered a (slight) scald on my foot --- produced by the (water in the) gearbox boiling! Nevertheless the car did later succeed in getting me back to England without further mishap. My route to England was not the obvious one, since I decided to ferry the car from Izmir to Athens, and then from Patrai to Brindisi. On my way to Izmir I had an interesting stopover at Bursa. Helped by a ski-lift, I did some mountain-walking on Uluda? (the classical Mt. Olympus) and at one point cooled a really huge Bursa peach in the remains (it was now high summer) of a snowdrift. When I came to eat it, it was the most delicious peach of my life. 4.7. Erdal
Inönü
Erdal Inönü, my head of department at METU, was the son of Ismet Inönü (1884-1973), who in 1923 became the first Prime Minister of Turkey. Ismet Inönü was comrade-in-arms of Mustafa Kemal Ataturk, and after the latter's death in 1938 he succeeded Ataturk and became, until 1950, the second President of the Turkish Republic. Later he was elected Prime Minister again for a total of 10 times, and I believe he was Prime Minister during my stay at METU. Both Inönü and Gürsey were highly cultured people and at times I felt a little gauche in their presence. One evening, shortly after my arrival, I exhibited gaucheness in a definitely non-cultural setting. Erdal took me out for a welcoming meal, and towards the end of the meal an erotic dancer began to perform. At one point she handed me a box of matches, and Erdal was amused when I had not the slightest idea what to do with the matches. In theoretical physics circles Inönü is best known for a 1953 paper, written jointly with Wigner, which introduced group contractions to physics. Since I always thought of Erdal as a most charming, diffident, quietly-spoken, extremely civilized person, I was therefore shocked by an incident which occurred almost 20 years later. In 1982 I was attending the 11th ICGTMP in Istanbul, and at one point I was in a taxi, going up a hill from the Bosphorus to central Istanbul, when I noticed on my right the Inönü Stadium. So I told the taxi-driver that I used to know Inönü's son. The taxi driver immediately said "Yes, Erdal Inönü --- a very wicked man!". Now I was aware that at one stage, after my stay in Ankara, Erdal had become President of METU, and that there had been a lot of hostilities breaking out between left-wing and right-wing factions amongst the students. I can quite imagine that Erdal in his civilized way would have tried to see the point of view of both sides. But it seemed that he therefore did not clamp down on the unrest with sufficient authority and as a result displeased the military --- for I read that Erdal had at one stage been placed under house arrest. No doubt the taxi-driver's sympathies were with the (presumably, right-wing) military. Erdal did make a couple of appearances at the 1982 conference. When I talked to him I was somewhat surprised when he steered the conversation round from mathematical physics to politics. For some reason he seemed keen to hear my views on whether Scotland should be allowed some degree of independence. Light dawned some years later, when I read that from 1983 he had become involved in active politics, and soon became leader of the Social Democratic party. Then, from 1991 to 1995, he served as the Minister of State, vice Prime Minister and the Minister of Foreign Affairs, consecutively. However I can hardly believe that my fatuous attempts to say something sensible about Scottish separatism had any effect whatever upon the Turkish treatment of Kurdish separatism! 4.8. Feza
Gürsey
(1921-1992)
Feza Gürsey went permanently to Yale in 1968, and occupied the Gibbs chair there from 1977. He was awarded the Wigner medal in 1986. He is perhaps best known for his work in SU(6), chiral Lagrangians and exceptional Lie groups in theoretical physics. In the late 1980's I myself became interested in the mathematics surrounding the octonions, and had obtained some results that I began to think may have some relevance to physics. In 1992 I was therefore looking forward to linking up again with Feza, the world's expert on octonions in physics, at the XIXth ICGTMP, to be held in Salamanca. On arrival at Salamanca, I was therefore especially shocked to hear that Feza had died a few months earlier. (On a personal note, I was particularly devastated to learn that Feza had died of prostate cancer, since on my first morning back in Hull I myself was scheduled to have a prostate investigation. I later jokingly told people that, since octonions were clearly bad for the prostate, I resolved in 1992 to forsake octonions, and that as a result I have had (to date) no further prostate worries.) Apparently there is now a Feza Gürsey Institute in Istanbul: see 4.9.
Louis Michel
(1923-99)
Louis Michel was an eminent French theoretical physicist who became the first permanent professor of physics at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette. His name is attached to the Michel parameter in muon decay, and to the Bargmann-Michel-Telegdi equation describing relativistic spin precession in an electromagnetic field. He was renowned for his use of modern mathematics in physics, and he made many valuable contributions to studies of internal symmetries and spontaneous symmetry breaking. (Incidentally the example of a spontaneously broken symmetry which I gave in my 1991 inaugural lecture was lifted from a Michel paper.) I certainly think that my attendance at Michel's 1962 Istanbul lectures and his 1963 seminars at Ankara, and also occasional interactions with him at the ICGTMP series of conferences, influenced the direction and style of my own future researches. Louis Michel died on December 30, 1999, and in January 2001 a conference "Particles, symmetries and structures: the legacy of Louis Michel" was held in Paris. Talks included one by Arthur Wightman entitled "The Scientific Work of Louis Michel; Some Historical Remarks" and one by Alain Connes entitled "Symmetries, from Galois to the quantum world". See: http://www.ihes.fr/EVENTS/Conference/michelA.html . 4.10. My
2-volume
work "Linear Algebra and Group Representations"
I devoted most of the years 1970-82 to work that (in part) resulted in the writing of the two volumes: Linear Algebra and Group Representations, Vol I, Linear Algebra and Introduction to Group
Representations, (269pp.), Vol II, Multilinear Algebra and Group Representations,
(309pp.), It did not start out like this! My original plan was to write a single book "The Mathematics of Special Relativity". This seemed a very modest aim --- especially in view of the existence of books that dealt with the VERY much tougher area of the rigorous mathematical treatment of quantum theory. Originally my book started out with one chapter dealing with the linear and multilinear algebra required for the rest of the book. Then this one chapter became three chapters: Ch.I. Linear Algebra and Group Representations; Ch.II. Multilinear Algebra and Group Representations; Ch.III. Antilinear Algebra and Group Representations. (The latter was needed, for example, for my treatment of Lorentz (j,j')-spinors, cf. Section 11.4 of Vol.II. As another instance, a canonical form for the Minkowski space Ricci tensor can be derived via a canonical form for antilinear mappings of complex Lorentz 3-vectors.) In the event these three chapters became enormously long, and I realized that perhaps I ought to write three BOOKS, namely Books I. and II., see Vol.I. and Vol.II. above; Book III. "Antilinear algebra, quaternionic spaces and group representations". (For a slight hint of the latter, see my 1986 paper "The ten classical types of group representations". ) After this I could then get back to my original "Mathematics of Special Relativity" book. Indeed, I did at one stage have contracts with Academic Press, not only for Books I, II, III and but also for a fourth book: Book IV. "The Mathematics of Minkowski Space: the linear, multilinear and antilinear algebra of Minkowski space and the Lorentz group, and of associated spaces and groups". In the event I completed the writing only of Books I and II, since I became pre-occupied with other things. So Books III and IV, although 80-90% finished, will I fear forever languish in the drawers of my study. 4.11.
Reception
of my books
I was gratified that that my two volumes received some quite good reviews. I was particularly pleased with Gian-Carlo Rota's one-sentence review in Advances in Mathematics 57 (1985) 91, which read: "Important notice: This is probably the first comprehensive and informative non-doctrinaire presentation of multilinear algebra ever written." For a much longer review, see G.J. Janusz in Amer. Math. Monthly 92 (1985) 517-519. See also Mathematical Reviews 84m 15002 and 84m 15003. I was also delighted to receive several unsolicited letters from people that I was not previously acquainted with. A letter from H. Urbantke, Institut fur Theoretische Physik, Vienna, commenced "Last year I became aware of your magnificent textbooks: let me first express my enthusiasm about them!" A letter from A.G. Coleman, Queen's University, Kingston, Canada, commenced "Just a brief note to congratulate you on the lovely book you have produced about Linear Algebra and Group Representations." 4.12. Gradual change in research interests Possibly what I record under this heading may lend support to those who argue against the separation of teaching from research. In the 1980's I was giving courses on linear algebra, or on 3-dimensional vector algebra, to fairly large (~100) classes of first year undergraduates. In my first couple of years I told my students that, in contrast to the dot product a.b, the cross product a x b did not generalize, and was special to dimension 3. (I was aware that a ternary product a x b x c existed in dimension 4, and an (n - 1)-ary product in dimension n, but I was convinced that, for any sensible definition of a cross product of just two vectors, you had to be in dimension 3.) How wrong I was! I learned that it had been known for some considerable time that a cross product a x b existed in dimension n > 3 for precisely one value of n, namely n = 7, this exceptional property of dimension n = 7 being related to the existence of the 8-dimensional nonassociative algebra of the octonions. And so at this stage I became very interested in the octonions, and especially in the related 8-dimensional ternary composition algebras. I also at this time developed an interest in the real Clifford algebras. I was particularly intrigued to discover that certain aspects of both the octonions and the real Clifford algebras involved geometry over the smallest finite field GF(2) = {0,1}. I had never previously seriously thought about finite geometry, but I very soon became entranced by that area. The titles of my research papers for the period 1987-1993 give a good indication of how my research interests were changing. See also my 1991 inaugural lecture "Symmetry", where the contrast between the "vumula" mathematics of Spin(7) and the "kateka" mathematics of PG(n,2) is touched upon. It was somewhat ironic that I was awarded a Personal Chair in Mathematical Physics at just the time (aged ~60) when I was forsaking Mathematical Physics for a new love: Finite Geometry! 5. The years
1990-present
Originally my research dealt with the role played by symmetry in fundamental particle physics, but, roughly since my 60th birthday in 1989, my research was almost entirely in the area of finite geometry. For a few years after my 60th birthday I did maintain a foot in both "camps". Thus I attended the 2nd (Montpellier, 1989) and 3rd (Deinze, Belgium, 1993) conferences on Clifford Algebras and their Applications in Mathematical Physics, and I also attended the XVIII ICGTMP (Moscow, 1990) and the XIX ICGTMP (Salamanca, 1992). (ICGTMP = International Colloquium on Group Theoretical Methods in Physics.) At Salamanca I met for the first time Steven Weinberg, who was awarded the Wigner medal at the conference. There was certainly a strong finite geometry ingredient in my talks at all four of these conferences: Montpellier, 1989: Finite geometries and Clifford algebras III; Moscow, 1990: Clifford algebras, spinors and finite geometries; Salamanca, 1992: Composition algebras, PG(m,2) and non-split group extensions; Deinze, Belgium, 1993: Finite geometry and the table of real Clifford algebras. The next time I gave a talk at Deinze was in 1997, at a Finite Geometry and Combinatorics conference. At the 1997 conference I expressed mock surprise that the two audiences at the 1993 and 1997 conferences were disjoint, and jokingly claimed that my mission in life was to unite the two halves (Clifford Analysis, Finite Geometry) of the Department at Ghent. 5.2.
Salamanca
1992, two Mexicans, Lochlainn O'Raifeartaigh (1933-2000)
As mentioned earlier, this conference was overshadowed for me by the death a few months earlier of Feza Gürsey. However I did come away from Salamanca with two happy memories. One morning, at the hotel at which I was staying, I had to share my breakfast table with two other people. I did not at first realize that they were at the ICGTMP, but when we discovered this, we of course introduced ourselves. They turned out to be Mexicans, from an obscure (to me) city Guanajuato. When they heard my name I was astonished, and thrilled, to hear them excitedly enquire "Are you the author of the books on linear algebra and group representations?". Apparently they were using them for a course of lectures to their students. I told them that I was therefore impressed by the calibre of their students, as I had always judged my volumes to be a bit too advanced for my own students. The second memory also came about from a chance interaction. At the conference banquet I happened to be placed next to Lochlainn O'Raifeartaigh, of the Dublin Institute for Advanced Studies. He told me that he had plans to write a book on the origins of gauge theories, and throughout the meal he kept on asking questions about the contribution in my Ph.D. dissertation. After the conference I was in correspondence with Lochlainn, and the upshot was that in his 1997 book "The Dawning of Gauge Theory" he reproduced the relevant part of my dissertation in his Chapter 9: "Shaw's SO(2) Approach". During the course of my correspondence with O'Raifeartaigh I remembered that I had filed away a transcript of a 1984 BBC Radio 3 programme Science Sublime, in which Lewis Wolpert questions Abdus Salam, and so I sent him a copy. I came across this programme by a remarkable coincidence. I was flicking through radio wavelengths at the time, in search of some suitable background music, and during the few seconds that I passed through one wavelength I heard the one phrase "... my pupil a man called Shaw ...". On returning to the wavelength I recognised Salam's voice, and so wrote off for the transcript. I recall that I had some minor difficulties in deciphering the transcript: Hugman Whyle = Hermann Weyl, Gowse = Gauss, etc. In 2000 O'Raifeartaigh was awarded the Wigner Medal, but sadly later that same year he died. For an obituary, see: 5.3. A new
finite
geometry configuration, and an old property of the icosahedron
Around 1994-95 I discovered one advantage of switching research fields from Mathematical Physics to Finite Geometry, namely that it is much easier to discover a new configuration in Finite Geometry than it is to invent a viable new particle in Mathematical Physics! I called my new geometrical configuration a double-five of planes. It occupies 35 points out of the 63 points of the finite projective space PG(5,2). The 35 points admit a decomposition into five mutually skew 7-point planes a1, a2, ... , a5, and also a second decomposition into five mutually skew 7-point planes b1, b2, ... , b5, such that ai meets bj in a single point for i not= j, and ai meets bj in a line if i = j. In some respects the 35 points of a double-five mimic the 35 points which lie on the Klein quadric in PG(5,2): for both sets of points are intersected by any hyperplane in either 15 or 19 points. With some difficulty, and perhaps clumsiness, I showed that a double-five exists and that its symmetry group is isomorphic to Alt(5) x Z_2 , which happens to be the symmetry group of the regular icosahedron. Then gradually it began to dawn on me that this last was perhaps no coincidence! Eventually I saw how I could start out from an icosahedron in Euclidean space and construct my double-five in finite projective space. At one stage I even used my knowledge of the double-five to deduce that, using 5 colours permuted by the rotational symmetries of the icosahedron via the alternating group Alt(5), the faces of the icosahedron ought to be colourable in two enantiomorphic ways. In this perverse way I "discovered" properties of the icosahedron which everyone else but me already knew! Of course, in writing up, for Europ. J. Combinatorics, my 1997 paper "Icosahedral sets in PG(5,2)", my logic proceeds in the natural direction, from the known (now even to me) properties of the icosahedron to the previously unknown double-five configuration. 5.4.
First Pythagorean
Conference (1996), John
Conway (click here for photo)
The First Pythagorean Conference took place in 1996 on the island of Spetses, Greece, and for me it was my first attendance at an overseas finite geometry conference. The conference commenced with a really delightful historical talk on Pythagoras given by Jean Doyen. Amongst other things, I was surprised to learn of the considerable numbers of females present in Pythagoras's school. I pricked up my ears when I heard Jean mention the fact that at the age of 60 Pythagoras married one of his students. For this gave me the opportunity of a good start for my later talk. Possibly as an excuse for any inadequacies in my talk, I wished to let people know that I was new to finite geometry. So I commenced by saying that, like Pythagoras, at the age of 60 I found a new love, in my case Finite Geometry. At this conference I met John Conway for the first time, and I recall spending an interesting (mathematical) half-morning with him, skiving off lectures --- see Picture Gallery (LINK?). In my talk I mentioned a particularly tightly packed configuration of projective flats which, with some hesitation, I ventured to call a "conclave". I was pleased that at the end of my talk Conway stated that he approved of my term. Actually I do not think that conclaves appeared in my contribution to the Spetses proceedings, but they were defined in my contribution to the proceedings of the Assisi conference held later the same year. Also in 1996, while I knew that a conclave of 8 planes existed in PG(5,2), I was not aware that such conclaves existed in one dimension lower, covering 28 out of the 31 points of PG(4,2). This realization came later, see the paper in J. Geom. 78 (2003) 168-180, which I wrote jointly with Johannes G. Maks. Apparently our conclave of planes in PG(4,2) is the (n,q) = (4,2) instance of what has been called a dual hyperoval. 5.5. An
unusual
conjunction
Q. At which conference, attended by the President of the London Mathematical Society, did the main speakers include the Regius Professor of English Literature, Cambridge, and the Rouse Ball Professor of Mathematics, Oxford? A. The one-day "A Celebration of Two Lives", organized by Roy and Monty Chisholm and held on 6 May 1995 at the University of Kent at Canterbury, two of the speakers being Gillian Beer and Sir Roger Penrose. The "Two Lives" were those of William Kingdon Clifford (1845-1879) and Lucy Clifford (1846-1929). I attended because of my fairly recently acquired interest in Clifford algebras, and because I had previously been impressed by accounts of Clifford's scientific achievements and of his full, but tragically short, life. (W.K. Clifford was a student at Trinity College, Cambridge, was second wrangler in the Tripos examination, and was elected to a Fellowship at Trinity in 1868. In 1871 he was appointed to the chair of Mathematics and Mechanics at University College, London, and in 1874 he became a FRS. Unfortunately his health soon declined and he died of tuberculosis when aged 33.) However, prior to the Canterbury meeting, I knew nothing of Clifford's wife Lucy, and was surprised to discover that she had quite a high reputation in literary circles. (Her obituary in The Times said she was a "distinguished novelist and for many years an honoured figure in Literary London, a link with the great writers and scientists of the Victorian age." Apparently in the 1910's and 1920's she frequently accompanied Henry James to west-end plays and films, and I believe she was the only non-family recipient of a bequest in James' will.) This unusual conjunction of Mathematics and English Literature resulted in the unlikely happening of my attendance at the same conference as my ex-wife Marion Shaw, Professor of English at Loughborough. Our daughter, Elizabeth Jane Shaw, also turned up, but only at one of the refreshment breaks. (Liz was a research student at the university but, despite the broad interests of the Cliffords, we thought that Liz's research area, Environmental Microbiology, did not quite fit in.) I also enjoyed the Canterbury event because I met for the first time Nigel Hitchin, who was then President of the LMS. 5.6.
Philip Larkin
and the President of the LMS
At the Canterbury "Two Lives" I had some interesting conversations with Nigel Hitchin. These started off mathematically, but at some stage I discovered that he shared with me a love of Philip Larkin's poetry. Apparently Nigel first came across Larkin through a jazz column that Larkin used to write. However I soon found that Nigel also knew Larkin's poetry, since when I struggled to recall some lines of Larkin that I was particularly fond of, Nigel completed them for me. Later that summer I was writing to Nigel Hitchin mathematically, but felt the need to remind him also about our Larkin conversation. I confess that I found it quite a struggle to find anything at all mathematical in Larkin's poetry! So I ended my letter (29 Aug 95) to Nigel (which included mention of icosahedral symmetry) as follows: "I had intended to end with an apt quotation from Philip Larkin, but I have been shocked to discover how little moved he appears to have been by the delights of icosahedral symmetry. Honesty compels me to point out that he appears also to have a lamentable lack of knowledge of Galois fields. In his poem "Counting", p. 108 in Collected Poems, he even appears to have difficulty with the smallest field GF(2): Thinking in terms of one is easily done --- One room, one bed, one chair, One person there, Makes perfect sense; one set Of wishes can be met, One coffin filled. But counting up to two Is harder to do; For one must be denied Before it's tried. (Incidentally, I see that the date of this poem is given as September ? 1955 --- which is one month prior to my first meeting Philip, when I took up my appointment at Hull.)" I do like the "One coffin filled" --- typical gloomily humorous Larkin. 5.7. A
few Philip
Larkin memories
In the autumn of 1955 I was walking along Cottingham Road with Philip as we left the University. I recall talking of motor cars, and we both agreed that cars were not for us. Philip had written his poem Church Going in the previous year, and an early well known image of Philip was with a bicycle visiting a country church. Nevertheless, two years later both of us had a car, with Philip's having a suitably spacious interior to accommodate his long frame. A terribly embarrassing lunch! At the beginning of one autumn term I recall having lunch in the Staff Refectory sitting opposite Philip. He immediately said what a God-awful time of the year it was, as academics returned from their long vacation away from Hull and insisted on telling you the boring things they had been doing. Despite this, I found myself going into considerable detail about my own long vacation! For some years around 1960 Philip lived in a university flat which occupied the second floor of 32 Pearson Park. I was a regular visitor to some friends who occupied the first floor flat, and so I occasionally bumped into Philip on the staircase. I particularly recall one bright sparkling frosty morning when I was in a joyful mood, and I was bounding along the path to the front door of 32. Philip emerged at this point, complaining gloomily about the awful weather! I recall being at various parties in the 1960's when Philip was present. At one quite lively party I remember Philip looking at his watch (it was ~11p.m.) and (?mock-)gloomily saying "Ah! at least the electric blanket has now switched on". One evening, around 1972 I think, we had Philip to dinner, the only other guest being Jean Hartley, a schoolfriend of my (then) wife. (When Jean and her husband had ran the Marvell Press they had published Philip's "The Less Deceived", which was his second collection of poems, but the first to receive critical acclaim.) We got round to talking about Philip's poem "Here". In it, "Swerving east" he arrives in Hull, whose denizens he describes in somewhat unflattering terms ("A cut-price crowd", "grim head-scarfed wives") and then proceeds further east into Holderness, and then beyond, to "unfenced existence". A line in the Holderness part of the poem reads: "Luminously-peopled air ascends;" and we asked Philip to what this line refers. I think he was surprised we did not know, but, for the record, his answer was: "Gnats". (The homepage of The Philip Larkin Society is: http://www.philiplarkin.com/ ) 5.8. Jaap
Seidel
(1919-2001)
I met J.J. Seidel for the first time at the Third International Conference at Deinze, Finite Geometry and Combinatorics, May 1997. On my arrival, I was greeted by Francis Buekenhout, who surprised me by immediately saying that Jaap was keen to meet me. I think that this was because Jaap had recently been using Vol 2 of my "Linear Algebra and Group Representations" as a reference in several recent papers of his, including "Spherical Designs and Tensors" and (joint with Calderbank, Cameron and Kantor) "Z_4-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets". Apparently like everyone else who has interacted with Jaap, I was greatly impressed by his mathematical enthusiasm and his friendly eagerness to communicate mathematics to others. I soon found myself being given interesting private tutorials by him. On the conference half-day break, the two of us we went into Ghent for the afternoon, and I received his guidance on the purchase of Belgian chocolates. In the weeks after the conference we exchanged papers, including some copious hand-written notes of his prepared especially for me. I am still regretful, because of being busy with other things (including my retirement), that I did not follow these notes up as I had intended. A June 1997 letter from Jaap started very generously: "Dear Ron, Your E-mail of 9 June 1997, and your sending of recent papers (such as your Research Report (1994) about Clifford and Dirac) appeal to me very much. It seems to me that we have been working along parallel lines, let it be in different languages!" I next met Jaap at the 2nd Conference on Geometric and Algebraic Combinatorics held in Oisterwijk, August 1999. This was on the occasion of his 80th birthday, and the proceedings came out as a special issue dedicated to him: Des. Codes Cryptogr. 21 (2000) no. 1-3. If you read the first paper in this issue, "Jaap Seidel 80", by N.G. De Bruijn, you will learn that Jaap only really started his research career seriously at the age of 47! What a prolific output thereafter! Sadly Jaap died less than 2 years after this conference. 5.9. Two
tough
classification problems
In the last few years most of my research energies have been expended upon the following two classification problems. A. To classify (under the action of GL(5,2)) all partial spreads of lines in PG(4,2). B. To classify (again under the action of GL(5,2)) all projective flats in PG(9,2) which are external to the Grassmannian G_{1,4,2}. I claim that both of these are pretty tough problems. Concerning problem A, this was solved jointly with Neil A. Gordon and Leonard H. Soicher, and we showed that there are 64 distinct GL(5,2)-orbits of partial spreads. Essentially the proof of this is computer-free, but at a partway stage there was invaluable guidance from MAGMA (Neil Gordon) and GRAPE (Leonard Soicher). I claim that our classification is currently the only complete classification of partial spreads in a projective space PG(4,q). Moreover I am of the strong belief that any future complete classification in PG(4,q) for any value of q>2 will only be achieved with much use of the computer and with much less theoretical understanding than in our q=2 case. Concerning problem B, this was solved jointly with Neil A. Gordon and Johannes G. Maks, and we proved that there are 10 distinct orbits of external flats. Again some use of the computer was of help at a partway stage, but our proof of the final classification was computer-free. An interesting feature was that as many as 7 of the orbits could be simply constructed, by means of the "even hyperplane construction" out of 7 of the orbits of partial spreads determined in the classification A. Again I would claim that the corresponding classification problem for a Grassmannian G_{1,4,q} for any value of q>2 will only ever be achieved with prodigious use of the computer. I confess that at my age I ask myself: should I not be spending my waning energies writing elegant brief papers rather than on solving these mammoth classification problems? (The proof in the case of classification A occupies more than 60 pages.) Surely these classification problems are more suited to a young and vigorous Ph.D. student? 5.10. The
quintic
Grassmannian G_{1,4,2}
In a 1994 paper, written jointly with Neil Gordon, it was showed that the 155 points x on the Grassmannian G_{1,4,2} (the Grassmann image of the 155 lines of PG(4,2)) are those points x in PG(9,2) which satisfy a certain quintic equation Q(x)=0. Now in the case of a quadratic Q, given a flat X one is used to looking at its polar X*; and in the case of PG(n,2) with n odd and Q nondegenerate we know that X** = X. Recently we have introduced a definition of the associate X# of a flat X in PG(9,2) with respect to the quintic Q of the Grassmannian G_{1,4,2}, and we have begun an investigation into its properties. I do not claim or expect that X# is going to be analogous to X*. WE ARE IN TERRA INCOGNITA! Exciting! However in the case when X is an external 4-flat we do have X## = X. At the present time we are confining most of our energies to looking at the even 4-flats, that is 4-flats in PG(4,2) which meet G_{1,4,2} in an even number of points, because we can prove the result: if X is an even 4-flat then X# is a disjoint 4-flat. Conjecture: if X is an even 4-flat then X# is an even 4-flat. On the assumption that this conjecture holds true, we can consider sequences of even 4-flats of the kind: X, X#, X##. X###, .... . It is usually an extremely formidable task to determine such sequences! At some point in the sequence we must have equality with an earlier member. So far we have examples of three different kinds of "first equality", namely (a) X## = X; (b) X### = X; (c) X### = X#. MUCH ARDUOUS (and exciting?) EXPLORATION NEEDS STILL TO BE
DONE! 5.11. Brief update (October 2008).
5.12. Brief update
(December 2009).
In "Trivectors and cubics: PG(5,2) aspects" I conjectured that, for any alternating trilinear form T on V(n+1,2), n>2, the set of T-singular lines in PG(n,2) is non-empty. This conjecture was taken up in a research seminar at Technische Universiteit Eindhoven led by Arjeh Cohen. Very quickly Cohen produced a simple proof that --- over ANY field F! --- if V(n+1,F) has even dimension then the conjecture holds. Moreover it holds in a powerful manner, in that through any projective point <x> there passes at least one singular line. The case of an odd-dimensional space V was taken up by Jan Draisma, a colleague of Cohen's, who, by use of a beautiful equivariant map from alternating trilinear form T on V(n+1,F) to polynomials of degree n/2 - 1, proved that the conjecture holds for any quasi-algebraically field F (and in particular for any finite field F). See the pre-print "Singular lines of trilinear forms" accessible from http://arxiv.org/abs/0909.5676. 6. Postcript:
near-assassination
of vice-chancellor-elect
As I read through the foregoing my conscience begins to prick me, since I have suppressed all mention of a certain rash act of mine in my early days at Hull. My behaviour was highly culpable, it could well have affected the whole future of the University, and certainly ended my career at Hull. On arrival at Hull I quickly found myself on the Staff Dance Committee. In those days this committee organized a Staff Dance, held on the last Friday of each of the three terms. For one dance we chose an Ancient Briton theme. I think this theme must have caused us difficulties, for one of the decorations that we loaned from a local museum was .... a boomerang! On the Saturday morning the Committee cleared up after the night before, and I emerged from the old Science/Library/Staff Refectory building with the boomerang. Now I had never held a boomerang before and I experienced an irresistible urge to throw it to see if it did what boomerangs are supposed to do. So I made quite a mighty throw over a lawn (where now stands the Middleton Hall), and was impressed when it actually did arc and start to return. Unfortunately I had overlooked the fact that someone was walking towards us from the Administration/Arts building, and it very narrowly missed his head. Brynmor Jones (poised to become vice-chancellor on the retirement of Lord Middleton) was understandably very angry. |
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