Montréal, Columbus, July/August 1986

This time it is I going away and the others staying. But to start at the break in routine two days ago: an overnight boat trip. Sheila's morning squash match with Rachel fell through, so we went first to Summertown for picnic stuff. (In fact, we saw Rachel in Summertown. She said that Hester's violin examiner had been very hard, as well as ill-mannered, giving only one distinction and four merits in a whole day of examining (ten minutes a time), so Hester's missing a merit by just four marks is a very good achievement under the circumstances.)

Back home to load the car, then down to the boat. We loaded everything on, got ready to start, Sheila went off to take the car home -- then nothing. Total non-response from the electrical system. Not just a flat battery, but nothing at all. Sheila, who hadn't quite gone, went to get help. While she was bringing Phil back, the circuit suddenly came to life, and the engine started perfectly. Phil came, decided it was a bad connection, and cleaned the terminals. Off we went. Sheila left, we reversed into the pool; and, as I changed gear, the engine died once again. Fortunately, after we'd drifted for a while, Paul saw our plight, took a line and brought us in, and fetched Phil again. This time he located the trouble in the battery master switch, so took it off and put in an unswitched terminal instead. Then he noticed how badly the engine was idling, so turned the rate up a bit. It was nearly 1.00 when we finally set off.

Sheila was halfway down the towpath looking for us. We raced her back to Godstow. Since we were all hungry by then, we stopped for lunch on the south bank in the sheep meadow just past the ring road. Afterwards the kids played badminton while I emptied out the water under the floorboards, and we set off again, in blazing sunshine, up past Wytham hill.

Just before Swinford lock, Hester wanted the toilet. I'd intended to put chemicals in it but forgotten, so we pulled up by a grassy slope under a hill. The kids loved the place so we stopped for a while. More badminton; Hester and Neill are getting quite proficient. Then on through Swinford lock. The kids wanted a village to buy comics, so we stopped just past the lock and walked into Eynsham. Comics, drinks, and a walk round the pretty town centre. By then it was too late to make for Bablockhythe, our intended destination; we decided we liked Hester's grassy spot so well that we went back there and tied up for the night, then walked back to the Talbot for supper.

It wasn't very good -- underdone potatoes with textured fillings -- but it did satisfy hunger to some degree. We walked back to the boat, got the beds ready, and it was a late bedtime for the kids and quite an early one for us, tired. I had got a bit sunburnt, and didn't have a totally restful night; my unzipped sleeping bag came out from underneath, so that I was lying on the rough cushion.

Just before dawn it rained very hard, and we awoke to drips coming in through assorted leaks. We breakfasted off bread rolls, packed, played more badminton, and set off at 8.30 to potter gently home again. By now the rain had stopped and the day was broken cloud with patchy sunshine.

The return journey was quick and uneventful, getting us into Medley by 10, the children driving the last stretch from Godstow, after Sheila had left to get the car. We beat her to Medley this time, but I was last off, after more baling and cleaning, carting stuff out, filling the tank, and putting on the covers.

Back home. Instead of packing or thinking about this trip, I had to see Jacinta, Dave and Tracey. By this stage I had a slight temperature and felt quite bad. Dave had more than I could read in a short time, so he came round in the evening at 8. And by then I had finished reading it, despite also having cooked roast chicken and mowed the front lawn and stuck the Stanley Spencer postcards I bought in Eynsham into my scrapbook, and Sheila had been for a run and taken Neill to choir practice (because Matthew's family had gone on holiday without telling us). Dave stayed till 9.30. I saw for the first time his new theorem. The free group of countable rank has a permutation representation which is k-transitive, the stabiliser of k+1 points is trivial, and the free generators act as single cycles, and there is an invariant k-error-correcting code.

I still managed to get packed before bedtime, which left, for this morning, last-minute packing, watering the plants, washing the dishes, etc. Sheila drove down (with much grumbling from the boys about missing Transformers) and, after I'd packed up some papers in College, they all saw me onto the airport bus.


Contemporary British art ---
Rain-streaked concrete walls
Of the airport underpass.
So much for avoiding the queues at Terminal 4: it took us twenty-five minutes to get through airport security. The guards gave the Royal Wedding as reason enough, but there is no doubt that security was much tighter than usual. They opened my case, which they must have seen from the X-ray contained nothing but clothes; anything metal (keys, shaver) was in my shoulder bag.

Prior to that, everyone except me had disembarked at the central coach station, and the bus was nearly hijacked by a long queue of people whose bus to Oxford was late.


I am definitely not in a mood for extended writing. Boarding the plane was a complete shambles, and we took off more than half an hour late.

British Airways has replaced its live stewardesses used in the safety demonstration with a video.

On the way down on the bus, I gave some thought to my talk. I will speak on both themes of the conference, algebraic and extremal. I'll start off by indicating how the hypotheses and the problems with permutations closely parallel those for subsets. I got as far as my general bound for s-distance sets and Kiyota's Theorem. I must say something about geometric sets and groups. I also hope to drag in (at that point, perhaps) the vector transversals. This would perhaps make a good exit line.

The in-flight magazine has little but Royal Wedding guff; the entertainment is pretty thin. (I'm listening to the 60s program at the moment; it's loaded with Petula Clark, Dusty Springfield, etc., and odd bits of Hollies and Searchers; nothing hard, and only "Love Me Do" from the Beatles.) The film will be Clockwise. I'm in an aisle seat, so no chance of watching the clouds go by. Six and three quarter dead hours. I have books, though, and the possibility of doing some mathematics, so I suppose I'll survive.

0950 (Eastern)

I assume I have the time right. There is no information available. I'm hungry. I'd assumed that they would serve up lunch very soon after leaving; but we are over an hour and a half into the flight and there's absolutely no sign of it.

The Captain has just come on to tell us that we're going to be even later. We're going much further north than usual (we've just passed Benbecula) and are still being affected by a headwind. We expect to arrive in Mirabel at about 1510. We will pass over the southern tip of Greenland -- but even if I were in a window seat, it's practically guaranteed that they'll be showing the film at the time.

I've switched to the "Contemporary Music" channel. It's quite remarkable how recording quality has improved in twenty years, even if musical quality has plummeted.

They think to entertain us with in-flight videos: a combination of still more Royal Wedding, British Airways services, and trailers for films they're not going to show us. Me, I bought Andrew Harvey's A Journey in Ladakh, but I'm finding it too powerful a drug, too efficient at distracting attention from what's going on around me -- just the reverse, I think, of the effect that the experience underlying the book had on him.

Still no sign of lunch. The plane is almost completely full, from which I infer that their service just isn't equipped to cope with a full plane. I wonder if the same is true of the emergency service, in case of a non-fatal accident.


Lunch came twenty-five minutes ago. Such is modern travel, eating predigested food and drinking wine at what may or may not be half past ten in the morning. Lunch was egg mayonnaise, steak (eaten before the starter, and only lukewarm then), and fruit (oranges, grapes and melon balls in syrup -- I saved the brightly-coloured melon balls until last, but that was a mistake, they were totally lacking in flavour and texture by comparison with the other fruit).

When I had coffee, I caught myself dithering about whether or not to put milk in it. Fifteen years ago I dithered about sugar before giving it up altogether. (Not a perfect analogy -- you can have more or less sugar in your coffee, but milk is binary.) I suppose the next step would be to give up coffee. I don't quite know; I think alcohol might go first.

Andrew Harvey has reached Leh, and his mind is being expanded (except that he doesn't put it in those terms of the sixties) by the clarity of everything. The same that happened to Sheila in Mexico City, and to me in small doses in Toowoomba -- an effect of altitude? (Or, at least, partly so?) One could fantasize that mountains which had sheltered meditating Buddhists for twenty-three centuries might be more powerful than ordinary mountains. But I don't believe there is an ordinary mountain.

Glimpses of cloudbanks out the window have the look of a distant continent, one that exists but keeps itself aloof from us. It seems that it doesn't matter to us what happens on that sun-drenched strand. Does it? (This is a kind of koan. Perhaps it should be refined.)

The video is showing one of the Tom and Jerry cartoons we put on at the Bishop Kirk fete. Talking of fetes, two French-Canadians in a party on the plane have birthdays today. One is 15.

In the cloud-continent
Glimpsed through the plane window
Life turns, laws hold,
But they are of no moment for us.
Are they?
I haven't really captured it. The point of the riddle, the reason it has anything in common with a koan, is that it isn't even a question. But already there, leading off at a tangent, is an instance of the dilemma. Should I worry out a pointed formulation of that thought, or should I just record everything uncritically? In any case, the cloud landscape is past, and only a slightly lumpy haze is now visible.

To mathematics. The question on which I've slept for many nights now is: How many sumfree sets are there? The Hausdorff--Besicovich dimension of the set of sumfree sets is log2(limsup(fn)(1/n)), where fn is the number of sumfree subsets of 1,...,n. Why limsup, and how does liminf come into the picture? But, of course, I am convinced that the limit exists, and that its value is 1/2. But the question did lead me onto something else. What conditions on a graph ensure that the number of cocliques is bounded by cn, for some c<2? The particular answer which I hope will ensure that is that the minimum and maximum valency differ by a constant factor and are prescribed (and not too small). So, let c(d1,d2)=limsup(kn)(1/n), where kn is the number of cocliques in an n-vertex graph with minimum valency at least d1 and maximum valency at most d2. Then, for example, c(1,1)=sqrt(2), c(1,2)=(1/2)(1+sqrt(5)). I think it would suffice for me to know that lim c(d,2d)=sqrt{2}. I'd like to try this out on people at the conference. But it is obviously not enough to specify maximum valency only, and neither is it enough to specify minimum valency (a star has 2(n-1)+1 cocliques).

The point is that, if a sumfree set contains d of the first n numbers, then the patterns of exclusions in any interval of length greater than 2n is a graph with minimum valency d and maximum valency 2d (and, indeed, almost all vertices having valency 2d, if the interval is long).

No, that isn't quite right, we must assume that almost all vertices have valency d.

It's probably enough to consider regular graphs anyway. For the interval can be bent into a circle, and the resulting graph made regular, without changing very much the number of cocliques. In fact, more is true. The graph obtained in this way is cyclic, hence vertex-transitive, so it has 1-factors (if the number of vertices is even, which we may assume to be the case).


I started to watch the film. But even the new design of British Airways headphones, though more comfortable than the old ones and far more stylish, couldn't really make the ordeal worth enduring; and, from so far back, the only way I can see the screen is to lean out into the aisle and get decapitated by every passing passenger. I think John Cleese will just have to do his thing without me. Music is tolerable, because it doesn't matter if you miss bits.

To put it another way, it's easier in the mountains, the facts of physiology are on your side. People have, it seems, always regarded mountains as holy, but those who really care haven't been afraid to go to them. There are examples, though, of the mountain which has never been scaled, perhaps guarded by a vengeful god.

"When there is a rainbow in the mountains," Wangchuk said, "we say the mountains are dancing. The rainbows are the scarves they wind round their wrists." (p. 40).
Or do the mountains always dance, but sometimes the dance does not require scarves wrapped round the wrists?


I gave John Cleese another chance, but there is more to it than the discomfort or the difficulty; I'm just not hooked by it. The point of a cinema is that there is absolutely no distraction, no possibility of doing anything else. But why then is television so addictive?


My first poem, this morning, was a small poem, a poem of bitterness and anger. This thought was clear when I read the Rinpoche's answer to Andrew Harvey.

His question isn't so different from mine. And if I can gloss the answer just slightly: there is a good way of working, of writing, of creating, a way which does no violence to the creator; but it is very unlikely that it will be recognized as valid artistic expression in our culture. Therefore it must be done, if need drives us, without expectations. That state of mind, perhaps more than the creation itself, is then of value.


On the ground in Montréal, sitting in the airport bus waiting for departure. The ride into "downtown" (in England it would be a literal translation of "centre-ville") should mean more to me now than on my first visit, seeing that I have a better idea of the geography of what I'll be seeing. It is a warm, humid, cloudy day, but the rain they promised us has held off, or stopped, and a weak sun is shining. In its openness, Canada looked brighter than England does.

The usual strange procedure for disembarking at Mirabel: little pigs come and attach themselves to nipples on the big pig, suck out the passengers, then run away and spew them out into the huge garbage bin. Getting through that was straightforward except that I'd forgotten to tick the noes on the Customs form, I suppose having assumed that not ticking the yeses would have the same effect.

Paying my bus fare reminds me that I'm in a country where the smallest banknote is worth about a tenth of the value of the smallest English note. My wallet will soon run out of space -- at least, I hope it stays that way. One thing I intended to bring but forgot was travellers' cheques. (Another was nail scissors, also handkerchiefs, vitamin C, and other such inessentials.)

The door is shut, but the bus is in no hurry to leave. There is no adjustable ventilation. What use is materialism without adjustable ventilation?

I felt I had left something behind me as I disembarked. I wasn't wearing a coat!


To the north, a rim to such featureless mountains that it is a level pale blue line above the dark blue of the forest. Nearer, dead ground between the highway lanes carries dry grass and yellow-flowering weed. The gentlest midsummer colours imaginable. But now, as we swing onto the main road, the trees are darker. Stands of forest alternate with meadows bordered with derelict hedges, spindly creeper-covered trees. Small patches of crops, far from ripe. The forests are very unkempt, too thick, and choked with undergrowth and dead trees. I think they might be boggy underfoot too: there are rushes by the roadside, and streams crossing. If this is their natural state, who would have dreamed them worth penetrating?

Little clusters of buildings flash past, European-style houses or farms with American barns, and now an out-of-town shopping center and an industrial estate, a huge factory. A football game with a few dozen spectators. Then bigger, greener fields and darker, taller, better-tended woods. One motorway crosses another, on three huge bridges, or that's what it seems from one side. But the bridges lead to a dead end.

Now across the Thousand Islands bridge onto the first island. More built-up now, but still not urban. Still very flat.

The second river, with its one big round island. We're on a fairly primitive, very bumpy, six-lane highway which yet has a new and unfinished look to it. And now a glimpse of the north slope of the mountain with its huge church and the city sprawling below.


I won't go on late. My eyelids are beginning to need propping up.

To continue: The bus carried on, parallel to the railway line under that odd, very straight bank, over a most amazing collection of concrete spaghetti, and to its stopping point down town. Out I tumbled and got my bearings, amid huge buildings whose reflecting windows distorted both shape and colour, so I could watch someone resembling me yet being wholly other walking along going the same way I was. But the most remarkable thing was that, at ten to five on a Saturday afternoon, the town was empty and all the shops were shut. I found my way to the Metro (harder than it should have been since the well-signposted way through the shopping arcade was closed), paid a dollar and went on, not forgetting my correspondence ticket. Suffice to say that, after a most straightforward trip, I arrived, signed on, got my room key, and found my room.

There was still plenty of time for a run, so I set of over Mt. Royal, a route I'd taken before. The long steady pull up was fine, I found myself running well. I decided to risk the plunge over the edge and, to my dismay, got totally and utterly lost. It was a blessing in disguise, because it gave me some most enjoyable rock scrambling on the cliff. I found my way down, and came back a quick way, having taken 46mins for the entire trip.

Then I showered, changed, phoned Ivo Rosenberg to tell him I'm here, and went out to eat, heading in the direction of the Kinh Do, which I duly found. The meal was very good, but especially the delicious soup, hot thin soup with finely chopped fresh vegetables and herbs thrown in.

Tired. Must stop.

Oh, before I forget, article by David Tritton in New Scientist on work of John Miles on a dynamical system exhibiting chaotic behaviour in response to a periodic applied force (a spherical pendulum, in fact). Relevant reference for the sumfree opus.

27/7/86, 0632

A good night's sleep -- the principle of tiring myself out worked well. The few times I awoke (usually because of people talking outside my window) I was able to get straight back to sleep again.

I didn't give any description of my surroundings. I suppose that, having seen it all before, it seems so familiar as not to need description. (Apart from missing the path on top of Mt. Royal yesterday, I've found everything exactly where I expected it to be.) The day was too hazy to yield really spectacular views from the top to the mountain, but the sea of haze in the west into which the sun sank was quite impressive. I noticed that, in spite of the unripe fields I'd seen earlier, the rowans on top of the mountain had very bright orange berries already. In a different climate, the balance of timing in the growth of plants must work differently.

I finished Andrew Harvey's book. My first reaction is that, despite the enormous amount of wisdom contained in it, it is really too glib. A foreigner on a brief visit can monopolise the time of two Rinpoches of great eminence and can master both the doctrine and the basic steps of meditation in a couple of weeks with no setbacks and no difficulties -- this seems less honest to me than Peter Matthieson's account in The Snow Leopard. It is also verging on being too clever. Many of the characters seem as if they are there to throw light on the author's development, rather than for any reasons of their own. In contradiction to the teaching he professes, the most enormous amount of ego fills the book. (So no doubt in any book, and that makes it readable to us in the West, fits it in with our own tradition; but more honesty would present this as a struggle.)

But of course, it must be said that I couldn't put the book down; I read it waiting to go through immigration, waiting for the bus at Laurier, etc.

The room was like an oven when I came, but after having the windows open all night it is much cooler. Only one mosquito came in; and now a gentle breeze occasionally stirs a tracery of leaves against a pale blue sky. Facilities are minimal -- towels but no soap; sheet and bedspread but no blanket. But plenty of cupboards (though no coathangers).

The number of grey squirrels is amazing. I just looked out the window and saw three of them playing in the courtyard between this building and the next. The park was full of them yesterday, and they seemed quite tame.


Breakfast time.

I worked for a while, finishing writing my talk, and then set out to look for the shops that Robert Woodrow and I found the day we went to Québec City. It's a glorious day, sunny and so clear; the view from in front of the main University building, over the ocean-like plain, was full of nearby tall buildings shining in the sharply-angled light, and varied greens of trees and bushes. Several things made me wish I'd brought my camera: shadows of maple leaves on a Children Crossing sign; a grey path where the paving was breaking up and revealing irregular patches of bright red foundation.

I think I mentioned in my travel diary last time I was here how slowly the joggers go. It's certainly true. They run a little faster on top of the mountain, but in the streets they amble along. One runner passed me (only just!), going downhill, with a friend beside him on a bike on the road, carrying out a perfectly normal conversation.

A man leaned to one side, out into the road, and spat. He then stayed leaning there for half a minute, as if one leg was shorter than the other.

After a false start, I found the shops. There was one Patisserie Provencal, which seemed much too high-class for my immediate needs (elegantly packed chocolates, etc.); but further down there was a 24 hour grocer, which, though quite as bad as such places usually are, could supply me with juice (reconstituted), bread (North American, not French), peperoni, fruit, milk, buns, etc. At present I'm sitting on a bench just outside, putting it away steadily while I write this.

How wonderful, to sink my teeth into a juicy, chilled Granny Smith!


I walked back up the street and, on impulse, went into the so-called American Historical Museum. What a swizz! Each word of the title was true only of a minority of the exhibit; and I think nothing satisfied all three. More than half of the building was devoted to dioramas of the early Christian life and worship in the Catacombs, with supposedly accurate costumes (though I have my doubts -- why were two Bethlehem shepherds listening to an Apostle preach?), with reproductions of the pictures and engravings on the walls (presumably accurate, though tidied up a bit), set off very gloomy tunnels so that many of the exhibits were invisible anyway. The rest consisted of a few dioramas of French-Canadian history prior to 1760, the founder of the Oratory being sculpted by an artist (art within art, with a photo of the actual event to demonstrate the authenticity of the model!), two wildly inaccurate modern scenes (one of the Apollo 11 astronauts, who landed on the moon in 1969, being greeted by President Kennedy, who died in 1963 after predicting their achievement), and several scenes from the life of the Holy Family. The whole thing confirmed any prejudices against French Canadians I might have had far beyond what an outsider could possibly believe reasonable.

I left and walked up the mountain. On the way up, I checked on one jogger, who was actually running just under half as fast again as I was walking (and I was walking briskly but without the smallest strain).

I'm now sitting at a picnic table in a beautiful grove of silver birch and maple on top of an outlying point of the mountain. I had a little mid-morning snack of marshmallows and an apple. While I was eating, a squirrel came up exploring. I gave her (him?) some bits of apple, and she sat on the table eating them and scattering the peel all around. I put one piece just a few inches from my hand, to see how close she'd come. She came up and reached out tentatively for it, which had the effect of knocking it closer to me. She plucked up courage and tried again; same thing, same result. It was now too close for comfort, so I knocked it away a bit. She did a bunk, but came back for it. I gave her the core, which she left on the ground after nibbling it clean. Once she was convinced there was no more, she ran gracefully away across the grass.

I can't get my mind to focus on this insight I had on the plane, getting an upper bound for sumfree sets by counting cocliques in cyclic graphs. But I'm sure there is promise!


I wandered down, across through the forest to the pavilion, looked at the view, walked up the hill behind, read the first Q story, had lunch.

I must stop slagging off the French-Canadians quite so much for the gothic content of their literature. Q's story was just as bad, and of course the same occurs in Dickens. Though of course the Celts give it quite a different flavour from the Catholics.

Early lunch because I'm intending to run this afternoon before registration. Part of the purpose of this walk, indeed, is to find a way down the mountain that is feasible to run. And I have found the bottom of one; so I guess I'll have to go round the other way. I found it by coming down a chimney in the cliff, far more precipitous than the route I took yesterday. Steadying myself after a slip, I saw a small raspberry growing out of a crack in the rock by my hand.

Walking through the forest at the top of the cliff was marvellous. Behind the green leaves, never out of awareness, was the blue of the flat landscape. Filling the silence was the sound of a carillon blending with an Oriental chant. I followed the sound of the latter and, at the foot of the hill, came upon a Hare Krishna circus. It was such a mixture. Sixteen booths under brightly coloured awnings, sense and nonsense inextricably jumbled beneath. A careful account of how the teaching of Jesus ("My kingdom is not of this world") was changed and perverted; a good account of the reasons for vegetarianism (omitting, however, the need to balance amino acids); a statement that science has failed, and must be rejected in favour of mysticism, because (a) theories of the Big Bang (ten years old, but no mention of that) involve a naked singularity, and (b) scientists can't yet predict the entire developmental path of an organism from its genetic make-up; the claim that reincarnation must be seriously considered because it is America's fastest-growing belief about what happens after death (with a long list of Hollywood starlets and pop singers who subscribe to it); a long history of His Grace A. C. Bhaktivedanta Swami Prabhupada; a free food counter (I wish I'd known before I had lunch; I didn't enjoy the dry bread and peperoni at all). The whole thing calls itself the Festival of India, which is on a par with the American Historical Museum.

I forgot to mention. I had to laugh out loud at the name of Buddha up there with Krishna, Jehovah, Allah, et al. Nobody turned a hair.


But whatever else you want to say about them, it's certainly true that their food is delicious! Yes, glutton that I am, I lined up for some. It consisted of pasta and chickpeas in tomato and herbs, some sweetish brown sludge-cake, green salad, poppadums, and farfar (cooked by a man at the back of the tent while we watched -- he had a giant wok full of oil; he would throw in a packetful of farfar, which would sink, and then turn it once with a curved instrument, on which it would rise to the top like multicoloured flowers in a bowl of water), and a delicious cold pink squash. I sat on a chair on the front of the stage to eat it, in the blazing sun, while the speakers poured out
Hare Krishna Hare Krishna
Krishna Krishna Hare Hare
Hare Rama Hare Rama
Rama Rama Hare Hare
over and over again, in a variety of styles, and a volume set to carry to the top of the mountain. Sporadically, people around me would try to clap the complicated rhythms for a few bars.

I have to love these people better than the priests up on their mountain. (That museum, so close to the Oratory with its Quebec flag, had to be in the priests' control.) However inconsistent they are, it is a sign that they are throwing everything out to try to catch my attention. They don't love scientists, which isn't easy to forgive, but they love most people.

The recorded music gave way to live, two girls singing Hare Krishna with beautiful voices, then a little talk about the event -- it is a festival dedicated to Juggernaut, "one of the names of God" (like Buddha: it is true that Buddha is deified as an avatar of Vishnu in some parts of Hinduism) and will continue till about 8.30 -- then more music, more traditional Indian, with the opening- and closing-doored harmonium, side drum, and finger cymbals. The drummer is playing a wonderful intricate repeated figure.


While they were playing, I heard a noise behind me, and turned and saw a huge tower, dazzling in green and red, on a wagon on the grass. Suddenly the place came alive; monks who had been in procession with the tower moved among the people, singing, dancing, and clapping. The excitement lifted the performers to greater heights. But they didn't get the whole crowd to come alive, and when finally the music stopped, and we were given a repeat of the welcoming speech, I decided it was time to move on.

Back at the residence, who should I see walking down the road but Peter Frankl, with a stranger who turned out to be Michel's son.

I set off straight away for a run. It really was too soon after all that stuffing myself with food: I got a stitch after five minutes. It went off after a minute's rest, breathing and bending, but continued to niggle for a while.

The loosely metalled road I've crossed a couple of times in different places actually stretches all the way from the bottom of the mountain to the top, and is much used by cyclists and walkers. It goes at a gentle but scarcely varying slope for quite a long way. A fairly broad path turning off uphill at one point I decided to call the Way of Zen, because it makes it possible to reach the top in one lifetime. (Anyone who has run up the road on a day as hot as this would agree that it takes many lifetimes.) I ran up at a fair pace, faster than several cyclists. The loop round the top was really spectacular; I can see why people pay to do it by horse-drawn carriage. Coming down, I took a short cut down the steps, and stopped off for a cooling glass of the Hare Krishna people's iced orange squash. Both weakenings punished me by making my body seize up so that running afterwards was much harder. I ran for 63 minutes, not including the two stops.


After I'd showered and changed, I went out and sat on the step for a while, and thought about sumfree sets, somewhat inconclusively. Then I decided that if anyone were here, they would have entered the building containing the offices from the back. So I went up there. Sure enough, there was Michel, talking to Fuji-Hara, with more people arriving all the time. Soon they took us through into a separate room, where soft drinks, fruit and nibbles were provided free (they couldn't compete with the Krishna people, though!) and we talked there for a while. I watched a lovely orange sunset flecked with purple clouds while talking to Steve Smith and Rick Wilson. But I bowed out soon afterwards, feeling that familiar drooping so characteristic of jet-lag.

28/7/86, 0618

Woke at 5 after a good night's sleep, dreaming about Hottentotenpotentatentante. I lay in a near doze until just before 6 when I got up to do yoga. Problem: the floor is so hard that back rolls are impossible, hence I don't get loose and my attempt at a shoulder stand was total disaster.

But, for all that, I feel well, not too stiff. I'm making a fourth meal out of the $15 worth of food I bought yesterday morning (I was spending money too fast over the last two days!). Then some work before it's tome to go and talk.


Last night Michel took great delight in showing me a newspaper article about Australian men who put "objects" up their rectums, giving feeble excuses when they have to go to the doctor to have them taken out. He also spread consternation among Steve and Rick by saying something to them which demanded a reply but which neither of them could understand in the least.

Today is again gloriously sunny. I'm told that the rain stopped about half an hour before my plane touched down. I think, though, that these pages will fill much more slowly from now on. In less than an hour I will be giving the opening talk of the conference.


Some now, more later maybe.

My talk went well, though perhaps I went too deep too early with all that character theory. But it produced an answer. Someone (whose name I must find out) observed that, if 2s is the correct exponent in the case s=1, then it is in general. For let F1 be a set of c(n/s)2 permutations of {1,...,n/s} with one distance; then the direct sum of s copies of F1 is a set of (c(n/s)2)s permutations of {1,...,n} with s distances. The good thing is that in my bound the constant cs differs from this by a smaller order of magnitude than for the Ray-Chaudhuri or Wilson bound or its derivative (exponential, as opposed to factorial).

Rest of the talks nice, not stunning.

Had lunch in a grotty "chalet suisse", with Bannai, Ito and Terwilliger -- chicken and chips and beer. Afterwards, in the tea break, I got everything I need (I think): postcards, soap, dental floss, writing paper, vitamin C (got postcards at lunch).

Michel conjectured that if there exists a L-clique and an L-coclique in Sn with product of cardinalities n!, then L={0, 1,...,t-1} or L={t,t+1,...,n-2}. I have one counterexample: D8 and C3 in S4, with L={0,2} and {1}.

After the last talk (Peter's, on Erdös-Ko-Rado, and one of the best), I ran over the mountain with him. We ran right to the top, which took 64 minutes, longer than he had intended -- if we'd come straight down from the car park, it would have been about right. Slow by my standards: my breathing scarcely perceptible on the way up. We saw one pair of really serious runners near the top. (The higher you go, the faster they run!)


Back from the wine and cheese party, an excellent affair with much more than just wine and cheese: ham and salami, salads, lots of fruit, and coffee and some magnificent cheesecakes.

Ivanov invited me to visit Moscow; Michel wanted me to come to Columbus with him for a couple of days after the conference. Possibilities may be opening up around me. I really do like travelling! but, modulo the American visa problem, Columbus would mean the end of any dream of losing myself in the Canadian wilds for a few days -- but with Bannai, Singhi, etc., it's too good a chance to pass up. Ivo Rosenberg has asked one of his students to find out from the American consulate just how difficult it would be for me to get a visa.

An impression: a transmitter of some sort on Mt. Royal, painted grey and surrounded by grey railings, in dappled sunlight through the trees, seemed to be wreathed in smoke.

I was going to work tonight, but it may have to get left until the morning. I have to write my paper and read Steve Smith's. His talk, incidentally, was as good as he usually is, explaining very clearly the different aspects of the (Buekenhout) geometry/(simple) group interface.

Tomorrow I'm chairing the afternoon session. There is also a problem session, to which I must give some thought.

28/7/86, 0943

I woke at about six, feeling raring to go. So I got up and wrote my paper for the conference. In fact, I adjourned for breakfast at 8 having not quite finished, but polished it off soon afterwards. It is a kind of mirror image of my Leeuwenhorst paper, covering much the same ground but proving the things ignored there and vice versa, as seemed appropriate for the conference. I reversed the order in the section on bounds, giving the example first. The character theory bound reduces the ratio between upper and lower bounds from factorial to exponential, which perhaps makes it more worthwhile.

Last night, too, I had been through Steve Smith's paper, making some comments, either favourable or nit-picking.

Today is overcast: not heavy cloud, more like thick haze. Looking out of the window at breakfast time, one could believe one was in mist-shrouded mountains; the effect of the haze was amplified by the dirty windows.

I feel my duty is discharged now. I've had my plane ticket copied, given Michel my paper and Steve's with comments, and can now listen to Füredi's lecture (mainly an enormous list of results on Erdös' ex(n,H) function) with no guilt at writing in my diary.

Actually, he distracted me momentarily then (I use the word in the English sense) by talking about the "sunflower method", which I recognize from Peter Frankl's lectures at Caltech but which still seems to me to be very powerful and waiting for applications.


At lunch I ran over the mountain again, skipping the loop to the top, in 42 minutes. I had to be back to chair the afternoon session, and indeed I only just made it even with the shorter run. Also, with more than 30 miles in four days, I'm well above recent average. But these were really rationalisations of my tiredness when I hit the ridge; I'd run fast up the hill, much faster than with Peter yesterday, and it was a hot and humid day. But I'm pleased with the run.


I chaired the afternoon session without mishap. Perhaps the most interesting thing to emerge was that Ceccherini and Tallini have constructed projective space analogues of Steiner systems with all reasonable parameters over any countable field. (An obvious necessary condition is 2k<=v+t-1, since two blocks must meet in at most a (t-1)-space.) The construction is simple. Enumerate t-spaces and k-spaces. At the nth stage, if the nth t-space is contained in a block, do nothing; otherwise, choose the first k-space which contains it but none of the (finitely many) earlier t-spaces, and add it as a block.

Visa: it seems that it is possible, given a letter from the University (which Eiichi has agreed to provide, though of course it can't be on Ohio State letterhead) and several hours waiting around at the US Consulate, which is at the Place d'Armes metro station. And a photograph is required, but I'm assured I can get it taken there.

In the problem session, after dithering round considering various things, I posed the question about asymmetric elements in complete sumfree sets mod n, an old favourite.

Notes from running. Both yesterday and today, I saw a chipmunk on the path. They are much rarer than squirrels, as perhaps befits their better looks. Another dig at the French: How French it is to have a Musée de la chasse et de la nature!

Tonight we went to the Chinese buffet, all paid for except for taxis and drinks. So the evening cost $8, whereas the meal price was a very cheap $7 and some. I haven't the capacity I once had: two large platefuls rendered me uncomfortable, despite having missed lunch. It was quite good plain Chinese, no particular subtlety. I thought of Jacinta when Richard Anstee mentioned a dish he'd had in Florida called "Death by Chocolate" (apropos of the chocolate ice cream they had at the buffet). One little bit of excitement on the way back, when the taxi driver took us to entirely the wrong place.

Kiyota is weighing on my mind. I must do something about writing that paper. If the worst comes to the worst I must take it on holiday with me and do some there.

30/7/86, 0645

The toilet paper here, and the paper towels, are perhaps the thinnest I have ever come across. It seems counterproductive; one uses an enormously greater length of paper when it's so fine.

Incidentally, I seem to be revealed as a "big chooser" in Knuth's sense, though this may be an accident of the particular cubicle I use (the first one), where the larger roll is nearer to hand and turns more easily.

A nice simple problem from Kalai. In projective 3-space over a commutative field, if l1,...,ls, m1,...,ms are lines with li cap mj = emptyset if and only if i=j, we must have s<=6. The problem is to prove this over non-commutative fields (even the quaternions would make him happy).


Today it's raining for the first time (though it tried hard enough while we were waiting for the taxi last night). We started early, to fit in Clement Lam's lecture (to avoid having him talk on Saturday when everyone will have gone).

A nice argument told me by Noga Alon: Suppose you want to cover the unit sphere by strips of finite width. Then the sum of the widths must be at least 1. Proof: Project up onto a hemisphere and note that, by magic, the area depends only (and linearly) on the width of the strip!

Eiichi's new concept is that of a rigid spherical t-design. X fails to be rigid if, for every positive epsilon, there is a perturbation X' of X, its points distant less than epsilon from the points of X, which is a t-design but not a rotation of X. Any tight design is rigid but not conversely; however, it seems that rigid designs are fairly restricted (e.g., bounded in size or number of isomorphism types given t and the dimension).

On present appearances it looks as if I might have to type my own invitation to Columbus on one of the typewriters in the department! The secretaries are all on holiday.

This room (Salle Ernest Counier) is painted lilac-grey, with the arches picked out in maroon and the beams in puce. Daylight-coloured lights alternate with yellowish ones. It is illegal to have more than 180 people in the room, but we're quite OK.

Light on O'Nan's Lemma from Tatsuro's talk. If X is a transitive permutation group and Y a subset of X, then Y is uniformly transitive if and only if Y is a T-design in the conjugacy-class association scheme on X, where T is the set of all non-trivial irreducible constituents of the permutation character. O'Nan's Lemma is immediate (though the proof of the above assertion is much harder than the direct proof of O'Nan's Lemma!). But also you see that t-homogeneous groups are particular kinds of t-designs.


Clement Lam estimates 2300 days (six and a half years) of CPU time on a VAX-11/780 to complete the search for a projective plane of order 10. He is now writing a program to put it on a Cray.


Bilingual signs: In the shower, both taps are labelled "C". Presumably "cold" and "chaud".

After Clement's talk, I asked a few people what they'd do if offered a month on a Cray. Michel had an immediate answer: he'd look for his perfect matroid design on 183 points, with lines and planes of size 3 and 13 respectively.

Michel also said at lunch that A. V. Ivanov's presence (an accident: the other Ivanov was asked) and Clement Lam's talk (another accident) might be a happy conjunction -- if the Moscow group get to work on the plane of order 10, they might halve the estimated completion date (which Clement puts at two years now).

I used a chunk of the free afternoon running, up the hill again. I was once again wooden at the start, something that hasn't happened while I've been in Montréal. But I was only slightly slower at the start, and soon had speeded up to the same pace as yesterday. I wonder whether it was the weather (cold and windy, though no longer raining), starting from a desk instead of changing after hurrying back to my room, or the very tight constricting socks. Anyway, I suppose the cool helped after a while. I sprinted to the top of the mountain. On the way down, I got a nasty feeling in my right thigh, but it held off and I charged flat out along Edouard Montpetit to the residences, to finish as fresh as anything, pulse scarcely 95 after a minute. I didn't bother looking at the view at all, but I passed everything in sight. The time was 10 minutes faster than when I did it with Peter.

The other thing on the afternoon's programme was writing comments on the new triple paper. I still don't like it. In fact, what I had to do was to extract comments from my letters of December and January to Michel and Peter, and put them into suitable form for a paper.

Tonight we have an extra session on association schemes. I went and bought some lunch and am eating it in the common room. There's more than I can eat but some will do for breakfast tomorrow, to get me off to an early start in the direction of the American consulate. Fingers crossed. It just occurred to me, I have to come back into Canada too! I hate crossing frontiers!


A long, long evening. We went on for getting on for 2 1/2 hours altogether.

Walking back, I saw the searchlight flashing in the sky. It's the first time I've noticed it on this side of the mountain -- probably because of the clouds giving it something to shine on.

31/7/86, 0810

Last night Michel was getting into a bit of trouble translating Russian and coming up with words of assorted languages.

I woke this morning at about 6.20 feeling more tired than usual, the one morning I have to be up early. I dressed and set off, eating my breakfast on the way to the Mont-Royal metro station. By traversing the start of my route up the mountain, I established that I run 2.3 times as fast as I walk, though one jogger who passed me on the way was running only 1.4 times as fast. Then down through a fairly seedy part of town (the Rue du Bullion suggests interesting scenarios) to the metro for Place d'Armes. Then I was faced with an incredible labyrinth both above and under the ground, up and down escalators, through doors, past shops; but, strangely enough, I didn't have to ask the way until I was within 30 paces of the place I was looking for. (Its presence was advertised on a big sign which I'd missed, but not on the directory of the tower block of which it is on the bottom.) I wouldn't have done so well, or been prepared to stray so far from the metro station, but for having looked it up in the phone book this morning and finding the keyword Desjardins. (This had worried me at the time; I couldn't find a street of that name anywhere near the Place d'Armes on the map -- but it is one of these huge underground shopping complexes that proliferate in Montréal. The office opens at 8.30, and there are just five people ahead of me in the queue. (It was six, but one was persuaded that he needed a photo, and went off. The person immediately in front of me had, like me, sought out a photo machine before joining the queue; but he had mistakenly got one portrait photo for his two dollars and had to sit again. I had let him get back in front of me by stopping for orange juice.)

I'm about halfway through the Q stories now. I can begin to see why he appeals to Helene Hanff. There is no mucking about with description or poetry or anything like that. Still there is an air of the supernatural about most of the stories, be it just the colour of the sky or the effect of two bottles of wine.

But what a sign of the times, to be appointed by the Government to a chair at Cambridge! There must be an even more interesting story concealed by those few words. The cover blurb goes on to say how he became a very popular lecturer, hinting that it was to the surprise, even against the opposition, of the entrenched dons.


Outside Laurier metro.

When I arrived on Saturday, the bank across the road gave the temperature as circle. I failed to understand. Now the temperature is circle, but all is revealed: the time is : .

And of course this means that my wait in the US consulate is over. I was too inhibited to write while I was there; the atmosphere of such a place, reeking of enforced nationalism, upsets me quite profoundly. The consular official who interviewed me was as nice as could be, asking me about Cambridge winning the Boat Race and suggesting that I teach his wife mathematics. But with the next person after me, a Korean girl whose English was less than perfect, and who had entered the US on a tourist visa only a few days before she decided to take a language course, he was fierce and rude. After this I assumed that his front was just a front and my application would be turned down, but they've given me three months multiple entry; he was apologetic that it couldn't be longer, saying that I should apply in London.

Out of that place, I felt the need to celebrate in the most disgusting, debauched way possible -- chocolate and root beer.

Curiously, while I was waiting in the consulate, I didn't notice that it is on street level; one could see out of the curtained windows when the grey light outside was bright enough. I just hadn't taken that in before.


I missed Rick Wilson's talk but am back in time for Noga Alon's, fortunately. I've just walked into the lecture room to find a board of Rick's still up, as legible and clear as a textbook, in beautiful handwriting.

Sabidussi's problem about focal graphs was this. A graph is focal if, given any edge e, there is a unique vertex v not in e such that any automorphism fixing e fixes v. (This is a little different from what Dominic said, which was that any non-trivial automorphism fixing e fixes a unique vertex v not in e.) Show that deciding whether a graph is focal is equivalent to graph isomorphism. He pointed out that there are many focal graphs; e.g., take any bipartite graph, double the vertices on one side and triple those on the other.


Things have begun to slide again. That weird feeling of not recognizing my surroundings. The feeling is so instantly identifiable that it makes absolutely no difference whether I'm in an unfamiliar city. It came at lunch today and again now during Joe Hemmeter's talk, but it has receded again.

Last night we went to dinner. Michel was taking us to a Vietnamese restaurant. When we got there it turned out to be a Danish restaurant, to our amazement. Some, notably Richard Anstee, couldn't jut accept the miracle but had to go outside to check exactly what had happened. (It was simple enough: two doors, and we had taken the wrong one.)

Another association scheme session this afternoon, except for Labelle who is next up.


I forgot to mention two squirrels playing outside my room this morning, making the most wonderful shapes with their tails and bodies, in space and time, on the ground or on the bottoms of treetrunks. It may have been a courtship ritual, but it looked much more like an exuberant game of tig.


I ran up the mountain again, including the loop. Despite a slow start, it was the same as yesterday, or six seconds faster, to be unnecessarily precise. I was helped by enormous numbers of runners going up the same way. I passed all of them, none passed me. My empirical correlation between speed and altitude could have been given a real test today; I'm sure it would have passed the test. (Of course, there is a good reason why such a law should be true.) I was a little surprised that, with such huge numbers out running, still nobody passed me. Surely someone in this town runs faster than I do!

The grey metallic structure I mentioned earlier is actually the hideous illuminated cross that shines from the top of the mountain at night. Its orientation shows that my perception of the mountain top is slightly skewed.

Afterwards we had the party. I had started thinking about a tiny point from Ito's talk, that in any set of permutations X, the average of |fix(x-1y)| is at least 1 with equality if and only if the set is uniformly transitive. Michel immediately thought of some analogous Russian result for the cube, which got me thinking too. One can show that, in the cube, the Johnson scheme, and this case (the symmetric group), the average distance between points of the set is at least some simple function of the space, with equality if and only if X is the appropriate kind of 1-design. But for other spaces, one will have to replace "average distance" by "average of a suitable monotonic function of the distance"; in P- and Q-polynomial association schemes, this function should be 1-x.y (in the first eigenspace), equivalently the square of the Euclidean (not spherical) distance from x to y.

So we'll have something to discuss on Saturday.

1/8/86, 0717

What we need, in fact, is that, if a scheme is P- and Q-polynomial and we project into the first eigenspace, the inner products of the vectors (which depend only on the distances of the points in the scheme) are a monotone decreasing function of the distances. This should be true, though I can't see a proof. (At least, it can't be universally true because, in a strongly regular graph, both the graph and its complement are possible, and so the order isn't determined.)


I felt wonderful as I stepped out this morning; despite slight aches in various places, everything seemed in harmony. Why do I look so old in the photo yesterday?


I ran the same route again at lunchtime, after going to the bank to cash my cheque. It was better and faster than ever before -- a minute faster than yesterday, the gain fairly uniformly spread over all parts of the run. It was a warmer day than yesterday, but not too hot, and no wind; I flew up the last stretch of the hill and all the way down, despite an incipient stitch. Still nobody on the mountain to touch me.

It is not the squirrels or chipmunks in the park that attract people's attention, but the pigeons! Several times I've seen them being pursued by photographers; once, a home video buff was trying his hardest to make his daughter attractive to a pigeon. So far as squirrels go, I saw a man lying on the grass putting food on his stomach, trying to encourage a squirrel to come on board and get it (without success, at least while I ran past).


I've missed exactly two talks so far. One was Rick Wilson, while at the consulate; the other, Avis on repeated distances between sets of points in Euclidean space, while I was running today. There are a few other talks that I wouldn't have minded missing, for example, the present one, on some rather unmotivated stuff on "chromatic difference sequences" for graphs.

The answer to the question about P- and Q-polynomial schemes is no in general. Take such a scheme with more than one P-polynomial structure, e.g. J(k, 2k+1). The inner products can't be monotone functions of the distances in both orderings! It seems, though, that it is true in many cases.

It looks as if I will have to do my present shopping tomorrow morning downtown. We fly out of Mirabel tomorrow afternoon; I get back on Thursday at lunchtime and would only have an afternoon at best (barring accidents). If it is done, I can sit in the airport and work all afternoon.

It looks, too, as if I'll miss a couple of days' running. Tomorrow, I think I've earned a holiday. Thursday will also be quite out of the question. But this week has been so good that I can't complain in advance about next.

As to stereotype Chinese who confuse l and r sounds, Zhou actually says "colorraly", which is interesting in a talk about colourings!


Conference over. It's pouring with rain, so going to dinner will have to be deferred for a while. I was so tired that I very nearly fell asleep during Michel's lecture; the first time that had happened during the conference, and I don't really understand why.

Having missed lunch I am quite hungry, and have a mind to try the restaurant that Robert and I went to after our eventful trip to Mt. Tremblant last time I was here. This is failing the accident of running into someone else on my way out.

It's fascinating to sit here watching the random flutterings of leaves hit by raindrops.


So I put on a tie and went out to the restaurant, the Coin des Côtes. Though the service there is almost too quick, the food is excellent, much better than the menu might lead you to expect. I had aubergine in parmesan, sirloin steak, and blueberry pie, with two bottles of Bras d'Or, and coffee, for twenty dollars, and had no shadow of complaint about anything I ate. And the magnificent feeling, sitting there, summoning up good things, sipping the malt liquor ...

The Hare Krishna people's argument, that most people look half dead because they eat corpses, is beneath dismissal as an intellectual point, though it is a useful reminder. But when on unfamiliar territory, it must be extremely difficult for a committed vegetarian to maintain a balanced diet, especially one capable of sustaining him for fifty-five miles' mountain running in a week. So I have no regrets at all about my steak, only gratitude and obligation to the animal concerned. I think that what upset me more was their dismissal of the point about killing vegetables. They maintain that eating fruits and seeds does not involve killing. I would say that the latter, at least, is no less than murdering the unborn. Of course, I don't stop eating for that reason.

One thing I like about this campus is the extensive use of natural rock for decoration. It is far prettier than the appalling buildings that sprout all over the place. I guess that the very large amount of rock isn't making things any easier for the diggers of the new metro line, which seems little further advanced than it was last time I was here. I see that at Côte des Neiges station they have deleted the estimated completion date on the signboard and written in 1987.

On the other hand, what makes the place less satisfactory as a conference venue is the absence of bar and common room facilities. Sitting alone in one's room from the time lectures end doesn't make for the sort of atmosphere one expects, where a new theorem is being hammered out in one corner while less creative effects of alcohol are being felt in another.

I wonder how much mathematics I might have written or done if I hadn't written so much in this diary. In fact this must be one of the best documented of all my trips, even though it hasn't been the most eventful. I was thinking, at dinner, of the events of the day which brought Robert and me to that same restaurant two years ago.


The rain eased off as I was walking back from the restaurant, and the broken high clouds gave promise of a change in the weather. But it has now set in more steadily. Lights outside reflect in the wet path, and the leaves are too sodden and heavy to play with the raindrops any more.

I've been through my conference paper, and decided to add some bits and pieces: the clique-coclique bound and the Deza--Frankl conjecture, sharply edge-transitive groups, d-type, and Delsarte's theorem motivating the vector transversal designs, which Peter showed me at lunch yesterday.

2/8/86, 0800

The cafeteria is closed at weekends, and I failed to discover the existence of anywhere else where you can get anything to eat. I guess I'll have to look for food down town.

I slept badly last night -- or rather, well enough when I got to sleep, which wasn't until after midnight. So I'd got up at various times, to put on the light to read, jot down a line of music, go to the toilet. I've now finished the Q stories, and started in on the (more fantastic, but less supernatural) Peter Carey stories. He writes very well.


I'm now sitting on a very upright wooden park bench in front of fountains of the most secretive kind: solid featureless blocks, through gaps in which you can glimpse the water playing on rough-hewn granite-like surfaces inside. Behind them is a monstrous public building in mock-château style, remarkable chiefly for the brilliance of the verdegris on towers, flashings and drainpipes. To left and right are huge modern buildings. I haven't yet found either breakfast or a likely place for present shopping.

On the street, I saw a huge brown stain where someone had dropped a bottle of chocolate sauce.


Last time I went to vieux Montréal I found a full-scale tourist-exploiting industry. This time, I have found only urban decay: a forlorn yard full of fountain statuary (not at all like the one I just described); a mural with the plaster peeling off, creating a torn-paper effect with old brickwork underneath, more beautiful than the original; two butterflies dancing in an empty parking lot; a man trying to knock down a signpost with a broom (or was he just trying to break the broom?); huge walls of glass glimpsed at the end of filthy streets; eroded arches high above street level crumbling away. But just now I seem to have come to the tourist trap. This square has some discreet new brickwork, the flowers are well tended (and shine dazzlingly in the pale sunshine), and the two vehicles to pass have been a horse-drawn cart and a police car.

The one bright spot in my travels earlier was Chinatown, with triumphal arches and brass plates in the pedestrian street and well painted cafes. But nothing was open.

I foresee failure.


Yes, it wasn't at all touristy. Not a sign of a shop of any description. The buildings were in slightly better repair, and there were tantalizing glimpses through grilles or down alleyways into little courtyards, but it certainly doesn't serve the function it did last time. It bothers me, too, that things are so shut. Don't people here go to town on a Saturday? I've actually, at last, found a cafe open for breakfast, and am filling one need. After that, I have no choice but to make for one of the big underground shopping centres to see if something is open there.

Another surprising thing is the absence of maps, guides, or other public information. The only two such plaques I recall seeing are the one at the cafe on Mt. Royal describing Cartier's early visit, and one on the railings of a cemetery commemorating the first historian of Canada in the French language. (And that one didn't even say why it was where it was, but somehow gave the impression that that was irrelevant.)


Just squeezing in a report of the rest of today, nominally! It hasn't been a good day.

I went to Place Bonaventure and found the shops open, even uncrowded -- virtually deserted, in fact -- but wasn't able to find much in the way of presents for the kids there. Apart from low-grade fashion shops, and one eskimo carving shop with pieces a bit out of my range, there was really very little, so I gave up in disgust and went back to the University.

I found Michel there waiting. We agreed to set off for the airport straight away, and I let him talk me into taking a taxi. Once there, he insisted in going round every shop in the terminal. I would have preferred saving them up for the way back. After my experience this morning, I've no great desire to spend my last few hours on Canadian soil in downtown Montréal. We had a quick lunch and then went through security check, to find ... yes ... yet more shops.

But also to find an hour and a quarter's delay to our flight. And that was as nothing. People Express has a lot to do to convince me they are not a bunch of hopeless incompetents. At Newark, we had to change terminals, and they promised us a bus in five minutes. It took them half an hour to figure out that they'd lost their bus, and then some time to commandeer one from the New Jersey Transit Authority. We found the other terminal jammed with people, so that it was almost impossible to get to our gate. When we got there, they tried for another hour and a quarter to persuade us that our flight was about to board. The problem, it transpired, was that the crew who were supposed to fly us to Columbus were somewhere over our heads in a holding pattern. We've just taken off, more than an hour and a half late, and I dread to think what will await us in Columbus at something after 1am in the morning.

What bothered me far more than the inefficient organisation was the fact that nobody knew what was happening, and in the absence of confirmation, they were quite ready with lies to stop us from stampeding. There were four flights supposed to be leaving from our gate, which was not a gate in the usual airport sense, much more like a gate in a bus station. (Indeed the whole enterprise was reminiscent of a seedy bus company.) The first of these was due out to Buffalo at 8pm. One very angry man had been waiting for this flight since then (we arrived about 10pm) and had just discovered that another flight to Buffalo had left, unannounced, from a different gate at 9.30pm. The ground staff refused even to offer him an apology, and the other passengers soon turned against him, apparently believing mistakenly that if he'd shut up and go away the ground staff could expedite their own departures.

In our case, too, there was the oft-repeated bland assurance that boarding would be "in ten to fifteen minutes", or "momentarily".

The other thing that disturbed me was a smiling photo of Ronald Reagan, with some bland saccharin message, hanging over the entrance to immigration control for non-U.S. citizens in Newark. Presumably anyone reacting adversely to his portrait can be excluded from the country.

The one profitable thing from the journey was an example for Michel of a paratransversal design. Essentially, what he requires is a group with two families of subgroups with the properties that subgroups from different families are complementary, and the union of all the subgroups is G. This holds if G is D8 circ D8, one family consisting of the V4s not containing the centre, the other consisting of the Q8s. This generalizes a trivial example in D8, but the sequence stops there.

3/8/86, 0904

I don't think I mentioned that part of the reason for the chaos last night was a storm over Newark -- at least, that's all that People Express would admit to.

One more instance of their failings awaited us. On arrival in Columbus, we found Navin Singhi wasn't there. It turned out he'd been told by People Express that our flight hadn't left yet! Michel phoned him, and he came out promptly to pick us up. There is a bed and a sofa; we tossed, I got the sofa, and that's all I remember until now.

4/8/86, 0727

And that was the only free moment, apart from a run.

We got up, had tea (with hot milk), then later the most marvellous breakfast (spicy vegetable sandwiches, fruit and gulab jamen). Soon afterwards I ran, suspecting that no further time would be free. It was already a very hot day. I found on the map that there was a little river not too far away (I guessed 3 miles but I'm sure it was closer to 4) down a straight road, and set out to run there. The straight road, just an ordinary suburban crosstown street, had six lanes and about three times the traffic density of the Oxford ring road. (Where were they all going, on a Sunday morning? Not to church; I passed a church, totally deserted. Some were going shopping, in the huge shopping areas ribboning the road.) Also, it was without a footpath, so I had to run on the edge of the mayhem. With this, the heat (already blazing sun at 11.30), and the general unfamiliarity, to say nothing of tiredness in my legs from all the standing the day before, of course I didn't run terribly well. But it was 50 minutes, and not disastrous.

Came back dripping with sweat, and had a shower, which didn't really stop the dripping. Then Michel and Navin and I went out to spend the day in the local Wendy, pretending that it was a Paris or Tokyo coffeehouse. I must say it served the purpose remarkably well. We stayed there until about 8pm, during which time we drank huge quantities of tea and coffee, ate cookies and salads, and talked a lot of mathematics. We talked around various things, and finally settled on sharp sets and groups of type {0,2} on infinite sets. We decided that a free construction would almost certainly give such a thing -- Navin wanted to know the connection with free planes and free algebraic objects, I didn't care so much about this -- and, much to my surprise, showed that no groups could exist. (Perhaps not so surprising, as I'd been turning the problem over in my mind for a few days, lubricating the connections.) Then, after discussing religion for a while, we went home to a dinner even more delicious than breakfast: vegetable curry, spicy cauliflower, some non-spicy dishes (for Michel) including peas and carrots in a kind of custard, beautiful chapatis, chopped melon, yoghurt. After dinner, we looked at Michel and Navin's "ternary projective planes", {0,1,-1} matrices A satisfying AAT=nI+J. We started out trying to prove that they had a certain structure, and ended out with an enormously flexible construction for weighing matrices which gives many new ternary planes as a by-product. Michel says that weighing matrices are an Australian preoccupation -- I guess that Jenny Seberry has written a book, or something -- and that the construction will be food for generations of Australian research students to cut their milk teeth on!

We got to bed well after midnight, and that was the day. Except that many things are inevitably left out of such an account, such as playing with the twins, who are entirely different but enormously appealing -- the old experience of dividing myself between thinking about mathematics and attending to the demands of a child (among their limited English has appeared the word "uncle"); the singing of crickets in the trees across the parking lot; broad swathes of light reflected in the lake cut by mysterious shadows of tree trunks; the unexpected sight of Concorde flying over the town low, slow and noisy; failure to get even the slightest glimpse of the town centre, which I remember as a tiny cluster of tall buildings sprouting from dead flat plain without any foothills; the odd feeling of mixed cultures as we worked and the sunlight moved slanting across the cafe where people came for fast food to go and went (and I mean the clash between mathematicians and others -- we three, Russian, Indian and Australian, were quite in tune); and so on.


Another magnificent breakfast. Then a student drove us into town. Dijen Ray-Chaudhuri and Eiichi were there, and organized a letter from the Chairman giving me entree to the athletic facilities to shower after my run.

By this time it was early lunchtime; despite the late breakfast, I went to lunch with Dijen, Eiichi and Tatsuro. We met Tom Dowling there. I had just a salad. I seem to have become entirely vegetarian at the moment. I certainly feel well on it. I think I should cultivate listening to my body a little more. Slight problem with wind, which I associate rather with travelling than with diet.

After lunch I spent a while with Eiichi. He told me some interesting things. There is one known family of simple non-associative Moufang loops, parametrised by fields GF(q), order q3(q4-1)/(2,q) (he said -- I wonder if that denominator should be (2,q-1)). He claimed that its multiplication group is D4(q) on the cosets of C(sigma)<sigma>, where sigma is a graph automorphism of order 2. He also claimed that it has the same number of conjugacy classes as PSL(2,q3) (at least for q even), and conjectures the same character table. Remarkable: PSL(2,q3) does not even appear to be a subgroup of D4(q), let alone a regular subgroup of the permutation representation.

Also, a construction of distance-regular digraphs of girth 4 by Liebler and Mena (infinitely many, following an example by Enomoto and Mena). Also, a characterisation of the bilinear forms graph by its parameters and a weak form of the 4-vertex condition.

I went running after that. Things were not quite as smooth as planned. They let me into the gymnasium, but padlocks for lockers were a problem. There are some red ones kept for some reason by the attendants. They lent me one but made it clear that they wouldn't do this again tomorrow, with a strong suggestion (which turned out true in the end) that all else they would do was to come and take it off and remove it when I'd finished. This led to a problem about getting my towel and soap out to shower with after the run. In the end, I chanced leaving them on top of the lockers with clothes and valuables inside.

I lost my way going to West Campus, so ran by the river instead. I had to retrace my steps several times, because the path didn't go close to the river, or went somewhere else, or led to a busy road, or ceased to exist. But it was more enjoyable than yesterday. I ran for 43 minutes, counting time spent being lost. It was still very hot.

Then home, for another wonderful dinner, and out again to what I've come to call the Wendy House. This was less productive. Michel was pushing his ternary planes, which have gone as far as they are going. We did a little on infinite geometric groups, which is going somewhere, but only as far as the lemma that if G is 2-transitive and Gab is of finite even order then some involution in G fixes infinitely many points. This substantially shortens both cases in our earlier proof (as well as avoiding a point I'd been worried about), and will almost certainly do more, probably {0,l} for l congruent to 2 mod 4.

After a long day I'm ready for bed.

6/8/86, 0659

No time for the coffee house yesterday. I had intended to run in the morning, but was too lazy in the event, just sat eating breakfast instead. Then we went in to the University, and Michel propogandized me on the subject of paratransversal systems, with some effect -- I did the calculations for the example consisting of some lines of an affine plane. Also I found an 8×8 weighing design with a partition not arising from our earlier construction, so destroying our characterization of "ternary projective planes" of order 4.

We worked so hard that we forgot about lunch until after 1.30. We walked across the square looking for somewhere to eat but without success. There was no time for anything too elaborate because Michel was talking at 2.

It was interesting stuff, attempting to characterize embeddability of finite metric spaces in l^2 by "hypermetric" inequalities. Far too much material for an hour, though, and it didn't help that the projector didn't come until half-way through.

I followed on with sum-free sets. It went OK. Several group theorists came. I don't know whether they went away disappointed. But I enjoyed myself, even though it didn't have quite the impact it should have.

After that, some farewell words from Michel, then off to dinner with Bannai and Ito at a Japanese restaurant. Sushi on a large bed of rice, then shabu-shabu on another large bed of rice. Very good, enormously filling. Lovely calm relaxing surroundings, with the attention of a beautiful Japanese waitress to explain the technique of cooking the shabu-shabu; we talked little about mathematics, some about this and that, and I felt very calm.

We spun out our time at the restaurant until 8 with cups of tea and then went to Bannai's house, where he played with his baby and his wife served us with a delicious red bean confection and pieces of melon (which the baby loved).

At about 9.30 he brought me back here.

Nikhil and Nidhi were eagerly awaiting me and delighted to see me -- they had been asking whether uncle had got lost! They sat on my knee for Navin to take photos, and then we looked at their photo album where they delighted in identifying themselves, poppa and mummy in all the pictures, together with various other highlights of their life here such as the clown who gave away balloons at the shopping centre.

Then we worked for a while. We cracked geometric groups of type {0,m} for m even, and feel ourselves close to SETGs of this form. I wonder too whether type {0,m,2m}, when 3 divides m, is now within reach.

To bed about midnight -- I slept on the problem but without success.


Yes, we've killed {0,m,2m} with the same argument. Furthermore we checked the free construction and it seems to work.

I'm now sitting in Victor Klee's lecture to the summer school. Among other things, he says that Kenneth May searched the literature and refuted the assertion that "mediaeval cartographers knew that four colours suffice" as effectively as such an assertion can be refuted.

He has now turned to the chromatic number of the unit distance graph in the Euclidean plane, showing that 4<=c<=7. He also discusses the idea of a "safe" colouring, so it doesn't matter if you slop the plane a bit, and the 7-colouring of the plane is safe. (On the line there is a 2-colouring, but the best known safe colouring has three colours.)

This morning I woke up at about 6.30 but, after writing the above, dozed off again until 8.15. How noble: I got up and went out running. It wasn't very good. I almost turned straight round and went back again. My ankles were so rigid I could only shuffle along. It went a little bit better after a while, but it was clear that my metabolism was so slow that a long hard run wouldn't be a good idea. So when I came to the deaf school, I ran around the grounds (slowly, on the uneven grassy surface), and turned around and went back. A freight train went over the bridge, taking two minutes and 40 seconds to go by. It appeared to be doing a good 30 miles per hour, so over a mile long; a very small train, by the standards applicable here, I think! I got back in 37 minutes.

Although I had a shower, it was so humid already that I went on sweating for a long time and had to change after breakfast. Michel having gone, breakfast became spicier: chick peas with lots of chillis, orange juice, fruit and milky dessert.

After breakfast, and after Navin had taken the children to school, we came in to the University. We worked some more on the free constructions -- I keep having crises of conscience about whether it works at all, or whether it gives examples for {0,m,2m,...,(t-1)m}, {0,1,3,...,2d-1}, and {0,1,q,...,qd}. I never believe a free construction unless I can write it out carefully.

Victor Klee asks what is the maximum number of lines in Euclidean 3-space, any two at distance 1. Is it 7? What about other kinds of flats in other dimensions?


After the lecture I went back to Navin's room. We decided it was time to bite the bullet and get on with the business of developing an extension theory for association schemes. We worked on it for an hour, went home and ate lunch, went to the Wendy house, and worked on it again until 8pm, with only the slightest glimmer of a breakthrough. It is difficult to know what to do about this problem. But we do at least have a number to calculate.

Then we went home, played with the children, ate dinner -- a deliciously spiced dahl, cabbage, raita, rice and pickle -- then talked more mathematics. By this time we had finally managed to leave the question of extensions of schemes alone, but we ranged over many things, some inspired by my notes of the Montréal conference. I'm going to bed now; I've set my alarm for 5 o'clock in the morning.

Just before bedtime, Navin's wife came in with presents: a beautiful embroidered Indian dress from Lucknow for Sheila, and another dress for Hester.

7/8/86, 0550

The beginning of the way back, which means, I suppose, that I'll feel temporary for more than 24 hours, until I arrive back home.

I set my alarm for 5.00 but woke up at 4.55. (Would I have woken up but for Navin going to the bathroom at that moment?) I woke feeling tired and without conviction that now was the moment; I had woken once previously at about 12.50, and got up to go to the window where there was just enough light to see the time.

I dressed and packed my pyjamas while Navin made tea. I've done very little in the way of checking that nothing has been left behind; but I have my passport, wallet, BA ticket, and PE reservation number -- the essentials, I suppose.

I left the Inuit art calendar as a gift, then we set out for the airport. Who could fail to feel some excitement at starting a journey at that hour? We drove through near-deserted streets, past empty donut houses, and along a tiny country road, where the feeling suddenly overwhelmed me that the assumptions I'd made were wrong; I was travelling along an endless road in the middle of nowhere.

But we arrived at the airport, ten minutes before the PE desk was staffed.


Now through check-in and airport security and sitting in the PE lounge waiting, as I will be spending a lot of time doing. I checked my suitcase, a change of procedure for me; another change is having a window seat well forward of the wing and the chance to cloudwatch.

Impressions of America:

More important impressions are of Navin and his family. He's such a gentle person that I seem violent by comparison, and yet I feel quite at ease in his company. We can sit quietly without any strain, but it comes more naturally to us to talk about mathematics. And to find somebody with whom I work so well, think so well, is a rare experience. And the goodwill of all four of them overwhelmed me. Nidhi and Nikhil came out at bedtime to say, "Uncle, come back soon", and to wish me "Namas-te" in traditional manner. They'll have photos of me in their album, so perhaps they will remember me. How fortunate that I came to Columbus!

Navin and Michel were interesting to watch together. Apart from Marie, I haven't seen anyone so positively accepting and loving towards Michel; most people don't find him very easy to accept or love. But they are very important to each other, I think.

It rained last night while we were at Wendy's, though that hasn't eased the heat much. It is a little cooler this morning, but still humid, and the light cloud will be no protection against the sun. Light is stealing in; it seems earlier than the clocks allow, but we are very near the western edge of the time zone here; in Indiana it is an hour earlier. Against the touch of pink, the clouds are slate-coloured. I can't see our plane; either it is at the end of the building, or we are going to be late. I have an hour and three quarters for the connection at Newark.

Talking of Indiana: a glimpse of a signpost to Wheeling and Indianapolis (via the east-west highway crossing the north-south highway just north of Columbus) brought back memories of my cross-country trip in 1973. I stopped in Columbus then in the hours of darkness and saw only the inside of the bus station, as I recall.

A plane now exists. It has to be pulled up at some distance from the gate, but having to walk to the plane is the sort of thing I don't mind at all if it makes for cheaper fares.

This morning before we left, Navin tried out an argument which should extend our result on infinite geometric groups to SETGs. It's not quite right as it stands, but I have high hopes. He'll probably have done it before I get home. This piece alone has made the trip to Columbus profitable for me, but of course it's not the real reason!

This terminal is somewhere between an airport and a bus station (unlike Newark, which was nothing but a bus station). At the adjacent gate, almost all the passengers have bleached hair -- yet it's not a Florida Express flight. Our plane has now begun flashing lights; the noise level is rising; the barrier has been drawn back; we'll be boarding soon, I guess. Perhaps I'll sleep on the plane. But there may be a delay. The man behind the desk can't get his microphone to work, and he seems reluctant to raise his voice. Would they delay the flight for that reason?


Correction, it was Florida Express. And he did shout, or rather, raise his voice just slightly.

Light came in quickly and unobtrusively. The sun rose red and ineffectual behind the grey haze, but as it ascended through the haze it painted gold the torn clouds above, so that the light came in two layers, pink and yellow. Very soon it won't be dawn any more, but full day.

The thought crossed my mind of going for a run at Mirabel. But lack of changing facilities and safe keeping for my baggage deterred me, and in the end I packed away my running gear with my checked baggage.

Perhaps what this diary proves is that writing a book would not be too hard, if I could submit to the discipline of writing a little every day. (I think that writing at regular times would be too great a shock to my system.)


Still sitting on the tarmac. They can't start the engines. The engines have to be started by an air-operated machine hooked on by a big hose. Their machine wouldn't work, and they had to borrow another, which has just arrived; with what result, I don't yet know, though one of the men walking round outside without earmuffs had his fingers in his ears (a good sign).

No, this machine doesn't work either, so there will be a delay of "several minutes". They are opening the doors for fresh air because, without engines, there's no air-conditioning.


Moving at last. PE compensated us for the delay with free orange juice. Starting an hour late, I should hopefully still make my connection in Newark. But who knows what might still happen?


Well, this is it. One of the engines has stopped and has to be re-started. I can't help feeling just a little bit nervous. On the way up here, a shudder suddenly possessed me. And I've finished reading the in-flight magazine. Nothing to do but mathematics.

Weak sun spotlights the terminal buildings across wide expanses of flat space, in front of a rapidly advancing cloud bank from the west. There is an Eastern Airlines plane behind us; I wonder if perhaps it is not allowed, by regulation, to overtake. (According to the pilot, we are not allowed to use one engine to start the other. I wonder if that applies in flight.)

Now totally overcast, and the darkened terminal building has receded into the murk.


Eastern went past and took off. From this end-on angle, it seemed almost like a vertical take-off with rocket propulsion. After that, two small planes, American also, went past and away, and now US Air; but at least we have engines on. And, at that moment, fresh air, and a promise from the Captain of departure in five minutes.

Cloud is even heavier now, but small sun windows are coming over. One is passing us now, the light striking across the glasses of passengers across the aisle, making little glowing patches of bright colour.

The disorientation I've had lately has no relationship to the familiarity of my surroundings. I know the symptoms; I know without looking up that when I do look up things will be strange. The symptoms in the Math Department in Columbus are identical to those in Merton. It is a non-recognition which isn't intellectual, more a removal of the veil, a reminder of what is really out there.

US Air is away and, believe it or not, we're moving out onto the runway.


We were very quickly past the flag-adorned terminal and above the cloud layer, and now we're in dazzling sun between white and blue, with three tiny black shapes like ominous battleships on the horizon. Now we've just crossed a front onto a higher and more jumbled cloud layer; through a gap in the wall, the mysterious bronze colour of sunlight reflected from a lake below shows through.

The horizon is not the least remarkable thing. The smooth white of the lower layer darkens to grey, then there is a gold stripe across the grey, then a thick layer of even brighter white, surmounted by a yellowish-brown streak with the tiny black mounds in front of it. Then comes the blue sky, rapidly darkening above.


We're now ticketed and fed, and less than an hour out of Newark (I hope).

Those ominous battleships were just the distant signs of the most incredible cloud wall, made of twisted shapes tumbled and leaning at crazy angles, rising from the plateau so as almost to touch our underbelly, and to shake us with their turbulence. It was very difficult to see them as insubstantial; they were as solid as rocks on a mountainside. (The true lesson, though, is the other way.) They were mountains by the shore, the sea in this case being land, now visible through a light haze below.

According to the (very rough) map in the in-flight magazine, Columbus to Newark is slightly north of east, and we should be past Pittsburgh now, and crossing the Appalachians. And indeed the geometric farms gave way to forest just as another cloud plain screened the land from our view again and we hit another pocket of turbulence.


Now we could almost be over the sea. The clouds are wispy white calligraphy over a hazy blue, through which it is possible to discern details of the landscape if you look hard; it's like flying over Atlantis. The blue sky is quite a different colour, much harder and more aggressive. The main change, though, is an overhead cloud canopy. It's just overhead, almost possible to touch, and the sky is bounded by its edge. We are looking out from under a shelter.

We just passed a contrail, crossing our path, just above us and below the cloud canopy, like a streamer hung from the ceiling. As we crossed right underneath it I could look along its length and think that somebody went that way.


We approached Newark over an expanse of forest, with slow, meandering rivers. (The pine barrens? I don't know.) The suburban houses bordering the forest were set right in the green carpet; every other one had a turquoise swimming pool. (Deeper into the built-up area, there were even more pools.) A car wrecker's yard and a container terminal added bright colours as we came in.

On the approach run, I got an idea which should fix Navin's argument about SETGs. If I can recover it, I'll write it out on this leg and try to mail it to him from Mirabel.

At Newark, we came in, fortunately, almost opposite the gate from which the Montréal flight was due to leave. I boarded and found open seating with very few empty seats, so I took the first one free, an aisle seat in row 2. No cloudwatching on this leg. And indeed, the euphoria has evaporated in the face of customs declarations etc., mainly I guess just the thought of crossing frontiers.

We're now moving away, my first PE flight so far to leave within spitting distance of the advertised time. I just wonder if they got my suitcase on board! It's easy to invent any number of difficult problems (as Gauss is supposed to have said about Fermat's last theorem.)

On this flight they don't even wait for us to be airborne before commencing the ticketing procedure. Such optimism!


Hah! It was actually 10.45 by the time we left; there was a long queue for take-off. Still, that's better than their previous record by some way.

Also, my argument collapsed in ruins.


But, after all that, we came in to land within a minute of the advertised arrival time.

Then the unexpected. In the people-moving device to the terminal, I found myself sitting next to Jeff Kahn and Ayerung Moon, who happen to be married to each other, and who also just happened to be flying from Newark (their local airport) to Montréal. Jeff knew that I was in Montréal last week, having in the meantime talked to someone who'd been at the conference, but didn't expect to see me flying into Montréal. We talked about mathematics of course; he told me the following problem: Let G be a graph on n vertices with small valency k (such as c log n or nepsilon). Probabilistically one expects a small set of size o(n) to have induced subgraph of valency o(k). Problem: Find explicit examples! I told him a few things about the random graph. We had a very pleasant time until past immigration, when they disappeared to take the bus downtown.

I walked the length of the terminal building from arrivals to departures. The British Airways desk is next door to People Express at the very far end. Of course there was nobody in sight. So I walked back down the terminal building to the left-luggage office. It was closed. There was a number to call for service. I dialled it. The phone made three different sorts of noises at me, one the ringing tone, one very loud and unpleasant. I hung up and took my bags away. I went upstairs, found a bathroom, shaved and brushed my teeth, went to the cafeteria for lunch.

I'm now sitting at a table, finishing a donut. It's raining non-trivially outside, which gives me even less incentive to go downtown. It is interesting, though, being a tourist in this airport building. It isn't crowded; the facilities are pleasant enough; there are enough tourist traps, a bookshop, a post office (which might be open now). I'm sitting at the edge of the upper level, watching kids below me playing with the automatic doors and grey clouds raining on grey concrete behind the glass.

The man next to me on the last flight was full of suppressed anger. He filled in his wife's customs declaration because she was being too slow; and he became annoyed momentarily (in the English sense) when he discovered he'd spilt Coke on his trousers.

I'll go shopping now, then I'll take coffee and try to write out the free construction.


I got my suitcase checked in OK this time. I only hope there is someone there to take it out again when I leave!

I've bought a postcard and written to Navin, confessing failure to put right the gap. On the other side is a Canadian raccoon. I also bought a couple of books. But the better shopping is on the other side of airport security, to which I won't have access until British Airways begin checking-in procedures.


At least the free construction of sharp sets seems to work OK in all cases. I wrote Michel a postcard to tell him. I also weakened and did some shopping -- kids' junk and maple syrup.

The rain seems to have stopped, but fog is closing in. The road out of the airport, seen from the end of the terminal building just beyond the chapel, disappears completely from sight as it passes through a stand of trees. I overheard some people awaiting arrivals, discussing the lateness of the flights. I suppose it will be possible to leave more or less on time despite the fog (if, that is, the aeroplane isn't delayed on its way here from somewhere else).

I feel as if I'm re-entering life after a spell in a monastery. I felt guilty eating a corned beef sandwich and drinking a beer with my lunch.


A little French-Canadian Jesuit priest came and moved me on so he could celebrate Mass. He didn't mean to be offensive; he was very earnest and explained how the English treated him better than the French, and he'd been to Windsor Castle, and so on. It was just that the combination of his brusque manner and his first words to me had been a bit forbidding. So, after helping him to shift the seat I'd been sitting on, I decamped, and he lost the chance to save a soul (some chance!).

So I went and had coffee, then went up the other end and did a few exercises in a reasonably secluded spot, overlooking a wide tarmac. The view is green and grey and most shades between.

I bought an American running magazine. The standard of journalism is extremely high, though the style is definitely American and dogmatic. The pictures accompanying the article about Said Aouita, too, would grace a travel magazine. His career sounds quite incredible (and is peppered with guarded expressions of disbelief from the journalist and various quoted authorities, particularly concerning a training session of 5 x 800 with half-minute recoveries in an average time of 1.47, with the last being the fastest!)

On running: my heels are giving me some concern now. They feel OK until I start running, but then they just lack spring or bounce or even flexibility, so that I move in a very flat way until they've warmed up enough to bend. It's not really a question of pain, just this immobility.


Guess what? They've just announced that the People Express flight will be delayed, due to the late arrival of the incoming service.


Airports rattle me. In the duty free shop, they wouldn't sell me souvenirs at the main checkout, so I went to the souvenir center and got enmeshed in a big group of Italians who obviously thought Canadian dollars were going out of fashion. The woman at the checkout, hassled and unable to cope, told me to go to the main checkout. I declined.

Then, at the coffee bar, a man glared at me when I sat at his table. (It was the only table with fewer than two people at it.) Then a poisonous cloud of nicotine and tar came drifting over from a woman just behind me, so I moved. This was too much for him; he ordered me off the chair on which his wife had been sitting.

The airport, so dead during the day, comes alive in the evening, with all the European traffic. Naturally, there is a progression: the Russians leave first, the Yugoslavs next, and the British are the close of the evening's entertainment.

Still no luck with the SETGs. My brain is tiring. Of course, giving up and reading won't help.

An hour and a quarter until boarding.


The name of Tarthang Tulku has come up twice today. I saw his book in the bookshop (in French, so I didn't buy it), and then on page 15 of LaBerge's Lucid Dreaming is a quote from him: "Dreams are a reservoir of knowledge and experience, yet they are often overlooked as a vehicle for exploring reality", quoted from a book of his called Openness Mind.


I await a third encounter. In the case of Elisabeth Kubler-Ross, the third was finding her book in a shop, after being told about her by Liz Billington and then reading a magazine article.


Airborne. And with the minimum of fuss.

8/8/86, 0716 (BST)

And after a few hours' sleep and a drink of water, what more could I want?

Congratulations (almost) to BA for one of the best airline meals I've had, last night. After the sausage and cheese starter, there was a dish of chicken and kumquats on saffron rice, which was very tasty but spoilt by having been kept hot so long that it had dried out; then melon and strawberries, real fruit with real taste, served up on a real leaf of some sort. I was so tired I dozed off before it came and went very soundly asleep afterwards.

The film is exceptionally long, having been going since the end of dinner (which must be nearly three hours ago). I don't know what it is, though.


A terrible thought just occurred to me. What if I didn't really wake up at 5 yesterday but just dreamed that I did? If so, and if I'm still dreaming, then either I've missed my flight, or I'll have to go through all this again!


Full daylight, film over. Out of the window, blue sea with little pieces of cotton wool, pale horizon with grey band, and totally clear blue sky. Sunlight falling on the plane's wing, but I can't determine in which direction the shadow should lie.

For a moment I mis-cued and imagined us flying west. We must, in fact, be going some way south of the morning sun; the shadow is only some fifteen feet out on the wing. I guess the antisolar point is so far round behind us that it's impossible to see it from this window.

As I was dropping off to sleep, I was awakened several times by clamorous but unintelligible voices. This is what LaBerge calls the hypnagogic state.


The land crept in unobtrusively -- a cape, a round bay, a river -- and took the place of the sea. On the other side of Ireland, we had sea again, but now we've just crossed the coast of Wales. Despite cloud cover, thick in places, there have been many windows. The only problem is that I have to crane my neck so to get any glimpse of the view. It's easier to stick with the interior landscape of my book.

Breakfast has come and gone too, and we should be down on the ground in half an hour.


The approach to Heathrow is spectacular. It's clearer than I've ever seen it in England, with occasional little round clouds, and when the plane banks to turn (right, fortunately), the whole landscape of England looms over us. Following us on this merry-go-round is another plane: we are as if on the ends of an axis, though sometimes it seems less mechanical than that, a fish swimming through this aquatic world.

I suppose that this stacking will delay us. But I don't mind very much.


There seems to be a need to close the account.

After landing we taxied for about ten minutes and still had a long bus ride to the terminal. The in-flight magazine raved about T4 which, it said, has dispensed with corridors -- but I'd much rather walk down a corridor than that!

At the bus stop, some hike from the terminal, were a couple from Vermont going to Oxford where their son is graduating.

We're passing High Wycombe; I've finished the dream book; and this bus is too bumpy to write more.


One more thing, too good to omit. On a sign warning of roadworks: "Possible delays until December".