I'd planned to walk the couple of miles to the Huntington Sheraton to wait for the bus, but it was raining so hard that I took up Sheila's offer of a lift. (She's so much more confident with that big car after a few months.) I bought my ticket and sat outside to wait for the bus while a wedding party came in; a very small wedding party, bride and groom and four or five others, and a photographer was waiting for them to arrive. She made one of the major-domos or bellhops or whatever hold the flashgun for her while she took the picture.
The bus came in, right on time, and set off with about seven passengers on board. In the rain I couldn't really follow the route it took to the airport after leaving the freeway; there was a lot of stopping and starting, and a lot of turning from one busy road to another. But we arrived at the terminal right on time, or a bit early. The driver had remembered everyone's airline and put us all off at the appropriate place; United was last. After checking in at a slow-moving queue, I went to a souvenir shop, bought a postcard, wrote it, and sent it off to Sheila and the kids. I bought it because it showed the downtown buildings with snow-capped mountains clear in the background. We had a nasty shock yesterday as we cam over the Santa Susana pass, after a day in Ventura with a sky of desert clarity, and saw the sky split in half, clear on one side, and with the smog, more formidable and threatening than a storm cloud and totally blocking visibility, on the other. I only hope for her sake that this rain clears the air; smog troubles her far more than it does me, she's always been very sensitive to nuances of the air, and it affects her mood directly, not just through her physical well-being.
I read some more of Kawabata's "House of the Sleeping Beauties", then, needing a snack, returned to the gift shop; it seemed to be the only shop in the terminal. It had candy bars; I found a beef jerky (far inferior to the ones we got at the roadside stall in Santa Paula on Thursday; they were real, not re-processed with all kinds of additives, like this one), and a granola bar (two signs that I am becoming quite Americanized). Then I went upstairs to look for the gate. I found that, while still not a shadow of a pimple on Toronto airport for shopping, the terminal did at least have a slightly better gift shop and a cafeteria the other side of the security check. The check, by the way, was perfunctory, nobody even checked that I had a ticket. (Maybe it is possible to buy one at the gate!)
Above the clouds, an amazing sight awaited us. We were in a band of clear air under another layer of yet higher (but thin) clouds, and the twilight in the west was vivid orange in the narrow rift. But in this rift, like alien cattle grazing the cloudfields, were some fine, intensely black, weirdly sculpted, wisps of clour. Sheila and I had been wondering this morning why clouds are so sharp-edged; this was a very dramatic example.
The cloud layer below stayed with us, but that above faded away out to sea. As the still-intense orange light shaded upwards into pale green, darker blue, and finally indigo, it held a sliver of new moon and two planets.
The plane had been delayed leaving the ground by the fact, according to the pilot, that the air traffic control computer had "broken down". An interesting thought.
In five hours I'll be in Eugene. In the dead of night, it will seem just like anywhere else, but in the morning I will notice the difference. Some lights are visible below. The cloud cover can't be completely solid. The retreating cloud above is textured by the pink skyglow on its charcoal grey. At one point, its horizontal layer blocks out the pale colours and appears to separate hard orange from hard blue.
It's quite dark outside now. The lights of the Bay Area are faint smudges below; the moon has nearly set, and no stars have appeared, only the same two planets following the moon down into the sea. The main source of light is of course our own, burning steadily out on the wingtip.
The plane is half empty. I find it a bit difficult to believe what Andrea in Thomas Cook's told me, that all other flights to Eugene today and tomorrow were fully booked. The fact that the train was also fully booked is even more startling. Of course there is only one train a day, but we saw it go by in Ventura, and it isn't just a couple of coaches stuck onto the back of a freight train. Perhaps, for all their apparent efficiency, the computer-dependent American travel agents have their drawbacks. Perhaps I was just over-impressed by Andrea's pleasantness...
With my head low, the acoustics of the plane become very different. I can hear a woman talking about Eugene. When I raise my head, almost none of the conversation carries to me, but here I could listen in on any number, if I wanted to.
The moon has gone now. More faint light smudges show towns, far out to the west. The pilot said a while ago that Sacramento was visible out the left, but it wasn't from my seat, right over the wing (as usual). When will airlines stop penalising non-smokers by giving them seats over the wings? But perhaps, should the plane crash, we'd be better off here -- I don't know.
This is a trip within a trip. I am very disappointed about what I haven't done in my time in Pasadena. Of course, I'm pleased with what I have done: mathematics (an advance on the sum-free sets; a breakthrough in my own understanding of Baire category, especially as regards graphs which are ubiquitous in the class with prescribed finite subgraphs; odd bits, like Frankl and Wilson's graphs in which no t vertices have precisely t-1 common neighbours for any t; Chris Rowley's statistics of the symmetric group, and so on), writing (instead of nothing at all, I now have at least half-baked notes for quite a wide swathe of the book, and I've tidied up or corrected quite big chunks too), photos (I have a good visual record of Pasadena and of the children's various activities while they've been here, that should keep their own memories clear, or perhaps mould them in not undesirable ways; and a few of the photos I consider really good); sightseeing (I've done a reasonable amount, both grotty tourist places -- Disneyland, Universal Studios, the Zoo -- and things really worth seeing -- Twin Peaks, the Pacific Asia Museum, the hills above Ventura, the Buddha in the Norton Simon), people, things done (Big Rock, an Australian movie, a couple of restaurants) ... but why continue like this? My best description of Australia after my time there in 1979 was the beverages I had drunk, particular beverages linked to particular locations. But as I was saying ... I meant to run once a day, but only managed once a week. More than once I dreamt that I was back into serious running again. Maybe that is a good path for me. Perhaps I over-committed myself with lectures, but I still should have been able to do better. There was something in Pasadena, though, that didn't help me run well. The first few times I had to stop with very high pulse rate. Even towards the end, I twice found myself at the top of Lake but unable to run up the trail to Echo Mountain. (But at other times I have kept going without discomfort for nearly three hours.) My high-pulse-rate moments have in some strange way drawn attention to the lump on my neck; not because it hurt, or even because I could feel blood pounding in it. Ostensibly, beccause I felt the pulse in my neck, and that brought my hand in contact with the lump; not a gesture I make often. What is that lump? What does it mean? Next, I haven't done more than a few sessions of yoga, and I haven't meditated at all. I know those things work for me; why can't I do them? It is a question of organisation, I know that, and my job for the rest of this year is to get myself organised. I can't survive without it, I mean that. Dark thoughts of death, not the kind that advises and sustains but that which drags down to destruction. And, a very small after-tone, I'm just a little disappointed that I haven't been able to infect anyone here with the virus of infinite permutation groups. But perhaps that's too much to expect.
Frankl and Füredi determined all the collections of triples for which every quadruple contains zero or two, quite independently of my result (and came up with the same answer, which is encouraging; it's also encouraging that Lachlan found the same ones). Peter Frankl suggested the description (implicitly Lachlan's also) -- the points are arranged on a circle, no two opposite; the triples are those which contain the centre. This immediately generalises to higher dimensions -- do these structures give infinite permutation groups too? If so, this would be the first infinite family of primitive groups with nk=nk+1 for arbitrary k>2.
I have a two-hour wait here for the next flight. My opinion of Thomas Cook is not rising. But it gives me time to think about those groups for a bit longer. Is this the reason why I bought a three-subject notebook? And if so, what is the third part for?
This terminal is plush but not at all busy. There hasn't been a single flight departure call yet since I've been here, but dozens of calls for people to join their "party". Clearly it's easy to get lost in these endless carpeted corridors. On the walls are huge blowups of high-rise Portland (on one side) and rocky Oregon coast (on the other), and just in front of me, quite incongrously, a new car.
Twice since being in California I've had toothache. The first time it went away by itself quite quickly. The second was much worse. Cloves didn't help, but constant massage of the gum seemed to do some good. Now there's no pain but it's too tender for chewing any but the softest food. Unfortunately, there is an exposed nerve on the other side which won't take anything cold, while the sore gum seems to be soothed by cold. It takes careful juggling! The upshot is that I travel with dental floss always to hand; useful, just now, in removing an irritating bit of food, probably apple peel from lunch.
What are the economics of airports? These huge buildings must cost quite a lot to build, and they exist even in quite small cities. People walk around emptying ashtrays over and over; indeed, cleaners often seem to outnumber passengers. Do the airlines pay for them, directly or indirectly? The fashion here seems to be for airlines to have their own terminals. Of course, the passengers usually pay the travel agents for their tickets using their credit cards, so the route of the money from passenger to airport authority is somewhat indirect. And yet there doesn't seem to be an obvious way for a determined entrepreneur to bypass all the middle-men. Too many of the hands in the pie are official ones, and the sums involved are too large.
In front of me is an orange patterned carpet with strangely luminous blue spots which echo the blue of another carpet further over. At my back, a panel with stylised "oregon wildflowers trillium" in blue and fawn (but a darker blue).
I explored these shops after disposing of Peter Frankl's construction -- it doesn't work. To put it simply, in the case of tetrahedra on the sphere, a 4-coclique consists of four points in a hemisphere, which may or may not form a convex set, and are distinguished by the possible non-null 5-sets containing them: abce&bcde, bcde&cdae, cdae&dabe, dabe&abce in one case, abce&abde, abce&acde, abce&bcde in the other. So hypergraphs big enough to force the 4-cocliques to be different can't be amalgamated along their 4-cocliques.
The silent display on the wall thinks the time is 1.27am. It also thinks my destination is spelt "Eugegne", and it changes words so painfully slowly that it's very hard to watch. Fortunately it has only two messages, its crazy version of the time and an advertisement for a new Cascade Airways service.
I trekked off to Gate 37, a very long, mostly downhill, walk. The only person there was a cleaner! A bit later, a plane came in and a handful of passengers disembarked. Then a few more people arrived and finally it was time to board. Apart from walking out and up the stairs in the rain, there were a couple of features not found on larger planes: my bag had to be stored in the nose of the plane; and, once on, the passengers had to be rearranged to get the centre of gravity further back. So I've wound up in the next-to-last row, behind the wing this time, though there's nothing at all to see. (Except that a light on the wing just flashed on and then off again.) The plane is bumpy, as I said, and also very noisy. There are six passengers, as far as I can tell. Flying in small planes has its charms in the daylight, but at night, when you're tired, it isn't quite so marvellous.
It was marvellous to see Bill and Phyllis again, really. I feel so much happier going to bed tonight. I am really going to survive!
Yet actually there isn't much to tell. We talked about mathematics and mathematicians, mainly; I skimmed Bill's and Francis Buekenhout's Como papers, and Bill's and Gorenstein's Anaheim talks; too much detail there, especially on GABs, for me to take it all in, but amazingly I read for hours without fatigue.
I saw the other miracle that has come to pass in their house. The loom is connected to the computer. This is an interesting reversal of history, taking the computer back to its roots. The computer simply raises and lowers the harnesses appropriately, displaying on the screen a copy of the pattern it thinks it's making. Phyllis still throws the shuttle, choosing the colours and then beating, and adds any hand-done threads before the next throw.
The most picturesque event was a run up Spencer Butte, the oddly-shaped mountain south of Eugene that has haunted my dreams since my visit here eleven years ago. I started off along the road on the east side, and turned up along the Ridge Path. At a signpost, I found I was 1.3 miles from the summit. Until then it had been muddy underfoot and requiring care, with undergrowth of ferns and a definite rain-forest look, despite the pines. But up a bit, I came suddenly and dramatically into a forest of tall straight pines without undergrowth, blanketed by light cloud through which the sun shone from the west, ahead of me, making a magical picture. From a small clearing I came above the cloud, to a rocky scramble to the summit. There was a man, one Ben Ross, up there; we chatted for a minute, then he took me down the much steeper western face at a great rate, to the car park. On the way home I stretched out, except on the steepest downhills, even sprinting the last few uphills, but was a bit demoralised to find about six runners coming the other way, two of them female, in expensive-looking tracksuits, and going very fast but hardly raising a sweat. On the way back, I ran on the road I had taken eleven years ago; old memories came back, especially passing the memorial gardens.
An excellent day.
Monday: up not too early, in to the department. Talked to Bob Liebler for a while. Nobody was running, so instead I had lunch with Gary Seitz. He wants the answer to the following question: Find all triples (m,n,V) where m<=n, Am is embedded in An arbitrarily, and V is an irreducible An-module whose restriction to Am is still irreducible.
After lunch I found the university bookstore and bought "The Narrow Road to the Deep North" for Bill and Phyllis, in anticipation of their trip to Japan (but in a different translation, called "The Narrow Road to a Far Province"). I wonder if they'll read it. They have borrowed lots of library books on Japan, mostly Japanese woodblock prints it seemed, but Phyllis was most involved in a book by Pearl Buck, who looked at Japan somewhat differently from Basho.
My talk went at least as well as I've ever done it. Many enthusiastic comments. Afterwards we went to the Spring Garden Chinese Restaurant in Springfield. Gary had been there before and knew that the right thing to do was to get the "old lady" to come and advise us. She was marvellous value. She was a hard-liner, virtually refusing to let us order any vegetable dish without meat, and killing off a minority move to order curry by replying, when asked how the curry was, "I don't know." She made us eat the garlic prawns furst, and gave us a pitch for the Chinese new year celebration (parties of ten, $9 a head with a dollar discount for couples married within the last year.) The food was very good, though I thought we could have ordered a little more; but to feed ten people so well for $57 was quite something. The old lady was categorised by the company as a "Chinese Jewish mother".
After that, back to the Kantors' for cake and wine, and quite a late evening.
The next day, a run had been set up, so I had two fried eggs for breakfast. The weather was bad, and got worse. I arrived just in time for Bill to give me a hurried account of his many GABs which are locally like a (4,4) geometry with residues POmega-(6,3) and PSU(4,3) (or is it the other way about?) example in his old EJC paper. Then to run, with Charlie Wright and Harvey Schmidt.
We walked under umbrellas to the gym. There I had to have my hand stamped and pay $2 for a ticket which entitled me to a lock. At another counter I got the lock (in exchange for Charlie's driver's licence) and he persuaded the girl to lend me a sweatshirt from their vast stock (strictly illegal). We changed and set off. The wind was in our faces, the rain (quite heavy) turned icy, and the path was underwater in many places, so that soon I was carrying pounds of water in my sweatshirt and more in my shoes; and still, it was a very good experience. I think I could have outdistanced the two of them quite easily had I wanted to, but felt very little competitive urge. We ran up beside the river for a few miles, through park that would have been rather pretty in better weather, then across a bridge and back on the other side, past supermarkets, freeway, etc., back over a bridge (both bridges pedestrian-only and quite new), and back to the gym. Seven miles, they said. A feature of the path was stencilled outlines of people with the stark label "holocaust".
A shower, with masses of hot water, hot air, and soap, all free, made me feel terrific. Then I went to lunch at the canteen with Charlie and Harvey, and found the mathematicians' table.
Afterwards, I talked a while with Bob about representation theory of GABs. The Hecke algebra of a locally finite building "is" exactly the generic ring with the parameters substituted for the indeterminates. His question was: How do you recognise which of its representations actually occur in the Hecke algebra of some quotient? My half-baked suggestion was to get the homotopy group associated with the covering to act on the irreducibles, and show that precisely the fixed ones occur; as he said it, enlarge the Hecke algebra of the building by adjoining the group to it, and consider Hom from the resulting ring to each irreducible representation space.
I talked to the algebra seminar on transvection groups, but I didn't feel I did as well as the previous day. The audience was huge, in a seminar room -- maybe nearly 50 algebraists. I kept feeling that I had pitched the level wrong, and then I got muddled and left out hypotheses of theorems; the whole thing got a bit out of focus. But it wasn't a disaster.
We had dinner -- all five sitting at table together, for the first and only time of my stay -- then went out to a concert. The Beethoven Quartet of Rome played Schubert's Triosatz, Mozart's Piano and String Quartet in Eb, and Chausson's Quartet in A.
The playing in the first half was very polished, but the music left me cold; even the playing seemed a bit wooden, and there was no clarity -- I think the acoustics didn't help. I kept drifting off into mathematics. I was sitting next to a visiting mathematician, who himself was in the seat of the ill wife of another mathematician (who was in the next seat), and during the interval it was clear that huge numbers of mathematicians were there.
Then the second half. From the moment the quartet attacked the opening chord, it was clear that we were in for something different, something into which they could put their hearts and their backbones. The music was full of beautiful patterns and themes (the first pentatonic theme C'GAEGACDE the most memorable but not the most haunting). I was spellbound. I would have given a lot for a bit more clarity; there was a tendency for the music to smudge over when they got very enthusiastic. Afterwards, I clapped. The audience clapped for an encore. That was a pity. It was so obviously a game or ritual that had to be performed. I didn't know the piece they played; bouncy, but boring -- they used their strings like synthesizers -- and such a let-down after what had preceded it.
Back home, nobody mentioned the music. I don't think it was because their experience had been like mine. We had more cake and wine, and talked more about mathematics and mathematicians.
One of Bill's concerns at present, about which we often spoke, was a long section of Gorenstein's manuscript for the AMS Anaheim meeting, in which he cakes care answering comments made by Michael Atiyah in an interview in the Mathematical Intelligencer. Atiyah's comments were predictable. In an interview in which he stressed often that he was expressing a personal opinion, he described theorems as staging posts, and proofs as merely setting a final seal on our understanding, this understanding being the important task of a mathematician. So, when asked about the classification of finite simple groups, he made two points. First, the understanding inherent in the classification is limited, because the proof is so complicated. About that I think he's quite right. Second, a classification isn't so very important; for example, in the analogous Lie group situation, all one ever meets in practice are the classical groups, and only knowledge of them is necessary. Gorenstein spends a long time on the first point, arguing that the proof of the classification is inherently very difficult. The second point he fails to answer explicitly, but part of the answer is implicit in the array of applications he quotes; one needs to know that most groups are classical in order to be able to use this information in a potentially non-classical situation. Anyway, Gorenstein's fear (which Bill shares in milder form) is the political consequences of such statements. For example, an erstwhile classifier is up for tenure, and the committee chair has glanced at Atiyah's comments and concluded that this mathematics is a waste of time.
I think Bill's real interest is as an ardent collector of applications. This is his subject in Anaheim.
Anyway, this morning, breakfast, a little more mathematics (including a potential application to Abelian varieties by Dodson), then to the train station, where we said our goodbyes, regrets that Sheila couldn't come, hopes that soon, etc., and on my side heartfelt thanks.
Outside the station, pools of water lay on the platform, and passengers sat on a wagon with blue painted sides and red wheels. About fifteen minutes late ("on time" according to the official announcement) came the now-familiar sixth, or minor seventh (depending on your point of view), and the train arrived.
This was very different from the other Amtrak trains I'd been on. Three locomotives; dining car, buffet car, baggage car, sleeping cars, and just three coaches (double-decker), with an attendant on each to direct you to on and then come and explain Amtrak travel to you (the dining car, the bathrooms, etc.) A ticket collector and a man to record destinations came round soon afterwards.
The train made good speed, except for a crawl past a long freight train, just out of Eugene. Under a very heavy sky, with bursts of rain, no mountains were visible; just lush green fields (with water lying in many) and green hills sloping up into the cloud cover. Further on there were fewer traces of hills, more rivers and trees. A sad waste of potentially lovely viewing.
Interconnections between coaches are all at the top level; and there is passenger accommodation above the baggage coach, and presumably elsewhere too. The scenery changed; we went through a town where the Willamette descended over a very big weir, a thundering mass of water; the town had industry and dockside warehouses, but also weatherboard houses in the American style (weather-resistant and so much less prone to rising damp and fungus infection than the English brick and plaster). We're now coming in to Portland, town of river bridges and highway spaghetti.
Over lunch I talked to the girl opposite, who is married to an Englishman and teaches in Kentucky. Our lunch was brightened by a very clear rainbow, a common occurrence here. (There was a rainbow in Eugene, bright over its entire arc, after we ran, as if to tell us we should have waited.)
We've just crossed the Columbia, its banks lined with ships, piers, factories, with expensive apartments sandwiched among them, and are even now pulling into the station, as a trainload of cars goes past. High above ground and with no vantage point except the moving train, the illusion of our movement is very strong.
Weather influences our perception of places to an absurd degree. I would not have recognised Portland, a dingy city of docks, railroads, and rain, as the crystal-clear town surrounded by symmetrical white peaks set in the blue of the nearer mountains and the blue (just as dark) of the sky, that impressed me last time. Thus Sheila in Sydney, I guess.
My lunch was a turkey sandwich, with gravy and mashed potato. Well, this is America.
We passed an old round barn with a cupola. The Sheldonian, translated and reflected.
Less than an hour and a half from Seattle now, and still hugging the line of the bay. We picked up time, presumably in Portland, and are now running pretty much to time. Here, it's not trees, but houses, presumably Tacoma. A house many miles back was pretty: sky-blue painted weatherboard with white eaves.
We just passed under a big suspension bridge. Not negligible, even here; Billy's mother, across the coach, pointed it out to him.
At the border, the Canadian immigration officer seemed to think I was likely to take a job in Canada, and got quite upset.
Approaching Vancouver, a strange pattern of lights in the sky, actually on some invisible mountainside, made a rapidly-evolving constellation. First a dog, then a bridge, then a jumbo jet taking off, and finally part of a maple leaf.
On the bus were five people: myself; two elderly Scottish tourists; a history and politics student from Toronto taking a year off to work; and a retired construction worker. The last two talked all the way. The old man, who had very accurate views about conservation, pacifism, etc., was much more enlightened than the student, a member of the Canadian reserves and quite reactionary in his views.
Our terminus in Vancouver was the Sandman Inn (not Sleepy Hollow as the travel agent had told me). The bus was 20 minutes early, so I had quite a wait for Dugald and Chris. Weather cold and drizzly.
Today I had trouble waking up. I went straight out to the bus stop and downtown to the US Consulate, stopping for a map, a passport photo, and breakfast. At the address given in the phone book was a bank. In the bank was a phone book giving a different address. At that address was a sign sending me back to the first address, with the extra information that I wanted the 21st floor. At that point the hassle was over. I was in and out of the consulate in ten minutes, an all-time record. They said I didn't really need to go there, but in any case gave me written authorization to show the officers at the border.
The weather was worse then -- fairly heavy rain -- so I abandoned sightseeing and decided to head back. On the way I found a tourist office. The man there must hold the world record for the number of free maps and pamphlets per second that he handed me. But he recommended Salmon Arm as a stop-off point. Present plan: leave Vancouver on Sunday night, stop off, arrive Calgary Tuesday night.
I found Chinatown on the way back. The fruit and vegetable markets were amazing. Unfortunately I had no use for fresh vegetables, but I bought a can of chrysanthemum juice and a packet of real dates. Then, while looking for the bus, I found a Buddhist vegetarian restaurant, so I had lunch there. I saw "The four hidden jewels" on the menu, so I had that, It was cashew nuts, four species of mushrooms (ordinary buttons, flat dark ones, squishy white (snow fungus?) and little ones like oysters), snow peas, and baby corn, with a couple of slivers of carrot. Delicious, though the button mushrooms tested my competence with the chopsticks; and lovely relaxing tea.
The bus back was a problem. It was nonstop. The driver was a bit puzzled and upset when I rang the bell but he let me off anyway.
When I got to the maths department, neither Dugald nor Chris was in. Brian Alspach took me to his office (he has a hi-fi radio on which he listens to an FM classical music station in Seattle), and also introduced me to the chairman; I saw Kathy Heinrich; then Dugald arrived. Soon it was colloquium time. The speaker talked on what you can tell about a topological space from first order properties of, for example, its lattice of closed sets of of zero sets or its ring of continuous functions. All the doughnuts, and even the bananas, in the seminar room were cut in half!
After that I lectured Dugald on spinors; we took the bus down the hill, bought wine, and went to Chris's for dinner, where we had excellent curry with real chapatis and banana raita, followed by Smarties from the jar with Tom Lehrer (the same songs as in the book Michael had been learning from in Eugene), and talked about why the lattice of subspaces of projective space has five generators. (We decided that five randomly chosen subspaces should generate with probability close to 1.) And so back to the motel.
My curtains were drawn, so I slept late, and woke dead tired. After I dragged myself out, I had to write my talk (this being the public unveiling of ubiquitous graphs) and have flour cakes and chrysanthemum juice for breakfast, and I didn't leave my room until after 9.15.
I walked up the mountain, managing to find a path through the woods that wasn't too wet altogether. There was snow on the ground at the top, where I didn't arrive until after 10.
Then, talking with Dugald took us up to lunchtime. By then the clouds were lifting and patches of sun appearing. So, when we sat down for lunch, we could see the forested hill opposite, and houses across the river. A little later, you could sense vast towering mountains just out of sight. Then you could see the bottom of the snowfields, and by the time we left, the top of a pass. It must be mind-blowing in clear weather.
After lunch, more talk, with Dugald and Chris -- what are the combinatorially 6-homogeneous (in terms of distance) graphs? And to my talk. It was far better attended than the department colloquium yesterday. I felt a bit bad that I wasn't giving my official colloquium talk, because although several people had heard it, many more hadn't. But perhaps either Dugald or Alan Mekler will make something of this stuff. I have a feeling that "absolutely ubiquitous" and "height less than omega squared" are not unconnected; maybe Dugald will get something there. He's certainly tackling the Pouzet stuff with almost as much energy as the Kantor--Liebeck stuff.
Dinner at Dugald's place, after adjourning to the university club, was chiefly distinguished by the fact that four-fifths of the people there were Australian, the remaining one British; not too different from last night. But at times there seemed two different non-overlapping groups there, those talking about Oxford, and the professional meat-eaters.
Today I saw the mountains for the first time. It was foggy again and, as we drove up Burnaby Mountain, shifting fog gave occasional glimpses of high snowy mountains illuminated by the early morning sun. Coming down there were tricks of light, fog, and buildings, the whole thing somewhat unworldly. But on the road, there were some more nice views behind us.
On the way, more fog, lying in river valleys with small farms on the Canadian side. The border guards let us straight across with only a cursory glance at my passport and all the complicated documents.
At lunch, after my talk, we sat in the cafeteria with a panoramic view over the bay; clouds shrouded the mountains but nearby visibility was good. But what was there to be seen? Dead calm ocean, with big ships at rest under factories with tall chimneys sending out wisps of white smoke.
Afterwards we looked at some of WWU's collection of endowed modern sculpture, none of which you could use for hide-and-seek (a cube of railroad ties was nice), and then up the hill, through a forest of ferns, for a little way before coming back for the talk.
Coming back in the dark, two impressions: fog beginning sharply the moment we crossed the Canadian border, and once again the huge expanse of Vancouver lights.
Before leaving this morning I booked train and motel in Salmon Arm. Dugald is going to take me up to Squamish for the day before time for my train.
But I'm getting out of order. When I woke at 7:30, it was fantastically clear, with a heavy frost. In fact it was so clear that I changed my mind about not going for a run and went anyway. I ran down to the inlet, where smoke rose nearly straight from the mill and mist softened the reflected dawn light, and the high mountains on the other side were perfectly visible as they awaited the sun. By the time I turned back, their tops were already kindled. I took a different route -- a big road, which became a small road, which became a housing estate, which became a path through the forest, which became a stream bed piled up with litter. I had to crawl up the bank, tripping over a tree as I did so, and duck across somebody's lawn (luckily no savage dog there), and then navigate my way out of another posh housing estate and back to the motel. One fortunate thing -- it was so bitingly cold that, though I had (without intending to) run fast, I had hardly sweated at all. My running gear dried out sufficiently by the air conditioning while I showered and dressed.
I waited outside for Dugald; the sun was shining brightly and the day still clear. We drove up to SFU (with views of Mt Baker from the ring road) to look for my mislaid expenses form. (We didn't find it.) Then we stopped for breakfast at a Dutch omelette place which advertised "1001 varieties". I speculated that 1001 meant 14 choose 4, but in fact they had 26 different fillings and permitted plain or any combination, which makes 67108864 according to my calculations.
Then on to the mountain. A beautiful drive up beside Howe Sound, with its mountain-lined borders and precipitous islands, and on our side many streams, some with very nice waterfalls, and human habitation varying from ultra-modern marinas to frontier-type railway halts. I took a smudge in the distance to be mist at first, but it was much too dirty; it was pollution from the wood-pulp factory at Squamish. But it was across the sound from us, and quite localised.
We set off along the road to a small quarry and then up the trail (which was extremely well marked all the way). It climbed quickly up a rocky course, necessitating scrambling at several points. Before very long we came to the turn-off for the first peak, which we took. We climbed under a long featureless rock wall through increasing snow until the path doubled back in sight of the summit. Here we made our only mistake and continued round for some fifty yards before giving up, but we found the right path, up a gully, on the way back. The top was further than it seemed, and on the last stretch, up a treeless snow slope, we found three climbers ahead of us. We overtook them but stopped to talk at the top.
The view was marvellous. The sound, polluting pulp mill and all; the town and docks just below us; railway lines up the valley; huge mountains, including shapely Mt Garibaldi, covered in snow, with a more dappled mountainside rising well over our heads on the other side of the valley we'd ascended; and the second peak just a stone's throw away, across a vertical-sided chasm (it actually took us an hour to get there).
We didn't linger, but retraced our steps, and headed on up the path to the second summit. It climbed the gully we'd looked across, with its sheer rock sides. Then the trail markers pointedup a little spur leading up the right-hand face. It was quite a slippery climb without handholds, and eventually I headed for the outside, where at least there were trees to break a possible slide.
We came out on a narrow ledge, with steeply-sloping snow-covered rock face up to the summit. If that had been the way, we wouldn't have gone. But the path led us along the ledge and then up a narrow gully, out of which we had to scramble up the rock with only a few roots for handholds. Approaching the top, the snow was deeper; I saw the tracks of some light animal or bird over the snow, and wished to be like that, or like an elf, instead of like a snowplow. (In fact I found that if I could gain the top of the snow crust, I could often keep going for a while without sinking in too far, by keeping my legs moving fast in small strides; but it was, if anything, even more tiring.)
Most of the way up the second peak, we stopped for lunch, only to find that Dugald had carefully packed sandwiches without filling. Tank had taken the bread out of the freezer the night before, and Dugald assumed he'd made the sandwiches. But we ate them, in the gully out of the wind, with chocolate digestives and coffee.
After the second peak we pressed on for the third. We descended to a saddle, through which we had a fine view of the town far below through an even narrower gully. Then up again. Around a sharp bend, and over a ridge, we hit the waist-deep snow. We nearly despaired and turned back, but pressed on anyway. It was a case of trying to guess where the snow would be a bit lighter, not always successfully. The one in the lead would press forward a few yards, then stop exhausted and have to regain strength. Almost at the top, the snow really did become lighter, and we strode to the summit, finding a notice telling us not to leave litter.
Going down was much easier, at least to begin with. The snow was so deep that we wouldn't catch an unwary foot in a crack or root, and if we did fall, we were well cushioned. But when we left the high slope to take an almost nonexistent trail down a steep stream bed, it was much more hazardous. Dugald goes down slopes like these at great speed and with fewer falls than when climbing; but I couldn't have made my legs move fast, even had I been reckless enough. Once I fell and found my foot wedged between two branches, so that it took some extracting.
But eventually we were down.
Apart from the long views, and the sight of snow blowing up on distant summits, there had been some memorable near sights: morning light on a moss-covered rock striking into the heart of the dim forest; afternoon sun between the trunks of the pines; a plant completely encased in ice by a dripping cliff. But the overriding impression was snow: its colours, yellows, whites, blues, and one I hadn't seen before, an icy green-blue in the bottom of deep footholes, suggesting translucent ice caves.
At the bottom, I was wet and icy. The gloves Dugald had lent me had had small holes in the fingers, and were now solid lumps of ice, inside and out; my fingers were frostbitten. My shoelaces were so iced up that I couldn't undo them, but I had to pull my shoes off since they were full of cold stream water. The turnups of my jeans had collected snow which had hardened and rubbed skin off my ankles. And Dugald thought he'd lost the car keys (I thought he'd locked them in the boot) until I found he'd left them in the ignition.
We lingered for a quick emptying of the thermos, then headed for home. One impression: the moon, just over half, magnificently bright, riding over the shoulders of a bare mountain.
Traffic on the way back was incredibly heavy; several times we were at a standstill on Highway 1. But eventually we reached Dugald's house. I don't know if anything has ever been nicer than the hot shower, unless it was the hot tea, lasagne, garlic bread, and newly baked fruit scones and jam, while my wet clothes tumbled in the dryer.
Dugald took me into town to the station. After some delay (the computer was down) I was able to buy my ticket. We boarded, eventually; all of the "book in advance" and "be at the station 1 hour before the train leaves" were just so much hocus pocus. Boarding was something like a rugby scrum. Even after I was on, an Italian family of four, abetted by the conductor, were putting moral pressure on people to give up their window seats so that they could travel together. The train is very crowded, and there is no such thing as a reservation, despite toll-free phone calls to Winnipeg. But now we're rolling east, very slowly, through the dark, and I have tired limbs and a blistered finger to recall a memorable day.
I dozed until a bit after midnight. When I awoke and looked out the window, I saw an astonishing sight. The black pines and a round grey hill were silhouetted against a ghostly symmetrical white mountain, illuminated by the moon, now low in the west on the other side of the train. I was so struck that I sat watching the massive mountains roll by across a deep river valley for quite a long time. But then a mountain was cut in half by shadow, and then only starlight (of which the bear and the hunter were witnesses) provided illumination.
I went back to sleep until just before Kamloops, and woke to quite a different sight, a different kind of hill in a different kind of light. The hills were much lower, rounded, scored with gorges, and quite treeless and barren; and the very first one I saw was lit presumably by streetlights somewhere out of sight, but uniformly and quite brightly.
Kamloops was an ugly industrial town where the train stopped way past the station. I dozed through the stop. I think the may have put on some extra coaches -- a good omen; ours was so crowded that I had been afraid of not getting a window seat tomorrow, though as it happened more people got off than on in Kamloops and there are now several window seats free in this coach.
Now we are back again in forested hills. The light is coming in. (Ominously, no stars before Kamloops, and I fear a foggy or cloudy day.) The bodies strewn about in disarray are re-humanizing themselves. (That comment doesn't apply to the handsome couple opposite me who spent the night entwined in various configurations, proof of one at least of true love, gymnastic training, or long experience of riding on overnight trains.)
I went to the bathroom, shaved, and changed back into my jeans, marvelling as I did so at that man Tank who, apart from doing mathematics, shearing bottom stand in a troupe, climbing mountains, owning firearms, hiking in Tasmania, etc., is an excellent cook (but waits at table in his own house because he only eats one meal a day), and was able to take my jeans out of the tumble dryer with a press in them! Then I had breakfast of orange juice, danish pastry, and coffee. (The buffet attendant admired my jersey.)
The scene outside is silver and grey monochrome now, at least that of it that can be seen (a very long train is passing us). The mountains are getting higher again. Now there are trains on both sides of us -- help! Our train runs very slowly, being easily outdistanced even by trucks on the highway.
We passed a little town of low houses, a few illuminated signs, and even fewer smoky chimneys. Now, as the mountains draw ever closer, we pass a long lake which takes up the entire width of the valley, forcing us onto the hillside. According to the map, the lake (presumably a drowned river) goes on a long way, but we leave it and sweep south to Salmon Arm.
I talked to Dugald yesterday about whether the ages of ubiquitous structures have small height, or vice versa. It occurred to me last night that my argument shows that subages of the age of a countably categorical structure are all realised as ages of substructures -- indeed, a wide class of objects satisfies this; is there anything which doesn't? -- and this may give some control, though it can't be worth much, because many countably categorical structures have ages which don't have heights at all! Dugald also raised the question: is there an age realised by a number other than 1, countable, or continuum of countable structures? He believed that such an assertion holds for the models of a first-order theory. If so, this would imply that if an age has only finitely many realisations, they are all countably categorical. Another question: Even if one can't prove that the algebra of a group of structure "finite wreath highly-homogeneous" is a polynomial ring, can one at least show that e is prime?
We struck up a conversation in the waiting room. (Not many people get off in Salmon Arm in the winter, and anyone would be curious.) The upshot was that she offered to show me round the town and get me back to the train the next morning.
That, in a nutshell, was the adventure; there isn't much more to tell. First she made scrambled eggs, with chives from a pot standing in her living room. Then she had a singing lesson. But the singing teacher was late, and when she arrived, she didn't have a key; the people at the music shop downstairs were even later. So they arranged to postpone the lesson until after lunch. We went to some bookshops. (I bought some presents and a book for reading on the journey, if I ever have time.) Then we had toasted cheese sandwiches for lunch. While she had her lesson I walked by the lake on a big flat snowfield crisscrossed by snowmobile trails. It was so flat that it felt like being out on the lake surface, though I didn't try that for real.
Then she drove me out of town to the ski chalet. She was a keen cross-country skier, and had just that morning heard that she'd got a job, instruction and administration on a make-work program teaching kids to ski -- her younger son Sheldon works for the same program -- and had also won some prizes and was hoping to be an official at the next Winter Olympics. On two earlier projects she had helped with the building and finishing of the chalet.
It was quite a long drive round the back of the mountains overlooking the town, the last bit on a forest road requiring four-wheel drive despite having been recently sanded. The chalet was very well constructed, all the doors and windows tight-fitting, a huge furnace in the middle. We lit a small fire in the furnace and sat and waited while it burned down. Then back to town.
I took her to dinner. She suggested a smorgasbord which actually wasn't very good, and we were both too tired to eat much, having sat up and not got very much sleep from Vancouver. So we went back to her apartment and went to bed. I slept on a quilt on the floor with two blankets over me; not the softest or warmest bed imaginable, but I was tired enough to get a good night's sleep on it. I woke once when the phone rang. At first I had that strange slow-motion sensation, where every ting in the ring sounds separately and independently, like someone tapping on a metal tray with a spoon. By the second ring it has collapsed into the sound of a telephone.
In the morning she made bacon and eggs, and then we set off for the station; we left the parking lot at 8:16 (after she'd spent several minutes scraping ice off the windscreen inside and out: the temperature was -12) for a train due in at 8:20 and out at 8:25. In fact we had plenty of time; the train was a few minutes late. Thank you Joy Hart. She was only one year divorced; her husband makes patio doors on 30 acres outside the town (on the way to the ski run) and Sheldon works with him; all the family are carpenters, and she, too, is a good worker, as the chalet testifies.
I got a window seat, lucky to do so; the train is even more crowded than yesterday. The weather in Salmon Arm was clear and sunny with just a little cloud in the northwest. Yesterday had been overcast all day, with the occasional very slow snowflake. The name, incidentally, comes from an arm of the Shuswap Lake, a natural, but very twisting and branching, lake, and the site of several salmon runs, one of which (at 1.5 million in its dominant year, next due in 1986) is the world's largest and a great tourist attraction in October and early November. Many tourists come in summer, for boating, fishing and swimming (we passed some nice beaches on the train between Salmon Arm and Sicamous); but the major industry, logging, is depressed at the moment, and they are hoping that cross-country skiing will build the winter trade. A huge bare area covering most of the hills northwest of the town is where the Co-op (the lumber company) allowed a small fire to get out of control on a windy day; it took quite a lot of houses with it, narrowly missing Joy's brother's. (His wife thereupon refused to live in a forested area ever again.) Incidentally, on a similar theme, I heard the terrible news from India on the radio. What does responsibility mean in cases like these?
Soon after our start, fog over the lake presented an amazing sight: it, and ripples on the lake surface, were absolutely still, as if snap-frozen while boiling. A scene from the Inferno. Following the lake shore, we had patches of view through the fog and low cloud, revealing long lake vistas and fairly high mountains, above the treeline in odd places. Now, after Sicamous, we've left the lake, and it's much clearer, with blue skies slightly hazed, and bright sunshine. I hope it holds through the Rockies. I've waited a long time for this.
I'm not sorry I left my camera behind. A woman opposite just photographed her friend, with flash, against a background of densely-packed Christmas trees -- or perhaps it's really true that she flash-photographed the Christmas trees; she just did it again as we passed a Disney-like painted tower set in the forest edge. The view from my side is somewhat less kempt.
As well as fingers, my shins have suffered from frostbite, or from the rubbing of the ice in my jeans turnups. I should have worn legwarmers that day. There is a patch on my left shin where the skin has come right away. There is also a speckled pattern on my knees.
Night thoughts. A la Pouzet, a countably categorical structure possesses only finitely many points whose removal decreases the age. For if a point x has this property, then there is a finite set F such that every substructure isomorphic to F contains x, amd hence contains the whole orbit of x; so x lies in a finite orbit, and there are only finitely many such. So, if the structure is absolutely ubiquitous, then all but finitely many vertex-deleted substructures are isomorphic to the entire structure.
We've just come out into a valley filled with water and sunlight. The sunny side is almost devoid of snow. We're skirting the margins of a lake, whose picture the woman took (without flash this time). On my side a few trees cling to a precipitous cliff towering over us.
I wonder if she actually burned a hole in the film with those other shots. But it would be an interesting exercise to do them properly. You'd need high speed (to stop the movement) and small aperture (for the extreme depth of focus required) -- I guess ultra-fast film called for.
Brief return: is it true that the age of a ubiquitous structure has only finitely many infinite sub-ages, obtained by deleting special points? No, that's false, as the double star shows. One needs just to get enough control first to show no infinite descending chains (so that the height exists) and then to get small height.
On studying the map I discovered that we crossed the Columbia River in Revelstoke, flowing north(!) out of a very long lake whose northern tip we just touched. After crossing the Columbia Mountains over a 1300m pass between 3200m mountains, we pass the same river again on its way south. I wonder what the geological history of these mountains is. I understood that part of the Canadian Rockies contains very old fossil sediments; but presumably the mountains themselves must be fairly young, or the sediments would have long since disappeared. (The gaps in the fossil record seem much less perplexing from this perspective; it seems a miracle that any continuity is possible.) Ayway, we cross a 1600m pass just a little further on.
We've come to a stop just outside Revelstoke. I hope we are not too delayed -- it would be a pity to pass through the Rockies after dark.
The scenery gets better by the kilometre. Earlier there had been white summits like broken teeth sticking out of the tree-covered rounded mountains. Then a small swiftly-flowing river. Then the track, climbing, rose above river height and crawled along a ledge above a sheer drop, with the road similarly situated on the other side of the ravine. Old tunnels and bridges showed the route of the railway in bygone days, the telegraph line still following it. Now we are in deep snow with high mountains towering over us.
At lunch I was put at table with two girls who had met on the train, one from Kamloops, one from a small town in Saskatchewan. One girl was surprised when a "beef patty sandwich" turned out to be a hamburger. (I had a salmon-burger.) I shared a half-bottle of Blue Nun with the Saskatchewan girl; it wasn't chilled, and came out of a plastic cup. The waiters are highly specialised: one for tea, coffee and milk, one for pickles, etc.
I overheard a conversation which I found most offensive. Several people, more than one of them railway employees, were talking about a certain passenger. They all thought he shouldn't be on the train. Most thought he should be thrown off; but the conductor insisted that he had a valid ticket (issued last night in Vancouver) and couldn't be thrown off until he infringed a rule. His crime? Disturbing other passengers' travel by being dirty. I'd seen him (I'm sure it was him) while I was queueing for the toilet; he'd tried the door, found it locked, failed to respond when I addressed him, drunk two cups of water, and then gone away. He seemed a harmless eccentric to me. One of the speakers in the conversation made great play of the fact that he might have been a millionaire.
We're now travelling through the only substantial tunnel of the trip so far. Earlier ones have just been a hundred yards through a ridge, but I just overheard the conductor say that this one is five miles long. I guess it is the one marked on the map as the Connaught Tunnel in the Glacier National Park.
Here is a puzzle. According to the brochure, most salmon have four-year life cycles; but some, the "jacks", return a year early and others a year late. If this occurred without some controlling mechanism, there would be no dominant run; over evoolutionary time things would have evened out, with a slow drift of genetic lines between the separate four-year cycles. Obviously then, there is some control, but quite how does it work? I can't see how anything like this could work unless there is an evolutionary advantage to being in the majority, or the salmon "know" whether they are early or late. In the first case, the jacks and latecomers would have been weeded out by evolution unless there is some Maynard-Smithian game-theoretic explanation. But this also seems to presuppose that the salmon "know" where they are in the cycle, so that they can "choose" their strategy accordingly. Of course the brochure also suggests that man-made disasters and artificial plantings have changed the cycles somewhat. It would be interesting to know more, especially how the proportions in the four years of the cycle have changed in recorded time. The brochure says that many Fraser River salmon runs have been wiped out by deliberate or accidental river blockages caused by lumbering, construction, etc., and that the Adams River run in Shuswap Lake nearly went that way but managed to recover.
Out of the tunnel, we had to stop to pass a long grain train. This line is "single track with passing places", and the conductors have walkie-talkies.
The weather is still holding; the sun is dazzling on the mountains, especially in the gashes left by logging, but the sides are too steep for it to reach us down here.
Both lake and river were frozen over. The snow-covered ice formed a neutral matt background for the trees, the most arty of which were isolated deciduous trees with some leaves still clinging. In places, especially in the lake, logs were embedded in the ice, visible only as relief patterns in the otherwise unmarked flatness.
We followed the river for a while. It grew narrower and some quite high mountains (the highest so far, I guess) on the other side drew closer. Then we crossed to the other side. We've now stopped in Golden, a town of decrepit mobile homes, one motel, and a sawmill emitting smoke which catches the western sun in a demonic glow.
The scenery on this stretch was the best yet. Out of Golden, we went under cliffs laced with frozen waterfalls. We had mountains, the first yet for which the prospect of walking up them seems totally unreasonable. Chocolate-box Rockies. We went through several mile-long tunnels. Each took us from a valley into another valley of quite different character.
Field, our last stop in B.C., was little more than a railway station, with towering mountains above. (Just before, there had been some jagged crags which , despite their angular edges, preserved the continuity of horizontal strata for quite a long distance. They must have been upthrust with minimal bending and then weathered away.) I begrudged the 25-minute stop there, as darkness was drawing on. Many people got out and played on the ice, but I felt sufficiently threatened that I kept my place. I did some sums on polynomial generators for the wreath product of C2 and a highly homogeneous group, obtaining results which are totally inexplicable, even if not quite correct, and far from monotone. The other feature of Field was a steaming stream, whether natural or artificial I couldn't tell. However, it did give lovely reflections of the last sunlight on the snowy peaks. (This had been the main source of light in the valley before Field, as the sun was far too low to reach us.)
After Field, dim fading daylight until Lake Louise where darkness closed in. The darker the sky and the higher the moon, the clearer the moonlight on the snow. But I was pretty much sated with mountains (at last), and the payoff wasn't worth the effort of peering out into the gloom.
We seem to have left the mountains behind. For the first time an entire range is visible, at some little distance, behind us, and nothing in front. Certainly signs of civilisation have been much more in evidence since Banff. The map suggests that we've been in the same river valley since Lake Louise, and continue in it until Calgary. The guard says we'll be fifteen minutes early in Calgary. Those two have swapped addresses (she made a play of refusing at that point when he asked her outright) and gone to the bar for fond farewells. Another distinctive group of three ladies opposite are playing cards. The one with the flash is the boss. The other old one's act is the scatterbrain who plays Rafferty's rules (they are Australian, or perhaps New Zealanders). The middle-aged one respects her elders and says little. Rich pickings here for a novelist.
I tried to phone Sheila in the morning but couldn't get through. So I bathed and breakfasted and set out for the University, where I found the department easily enough thanks to Robert's directions. They sat me in the Visiting Professor's office and plied me with coffee. At lunchtime I went running with Ivan Rival. Quite fast for me in the cold air (about -10) but it felt good, even though I wasn't bundled up in layers of water- and wind-proof allover suits like the others -- I think they were quite impressed by my toughness or foolhardiness. It was a bit more work than usual because of the snow; I had to watch my footing rather carefully on the icy pavements. We ran out across an ice skating rink under construction for the Winter Olympics, and then by the river.
He reminded me of the problem: does there exist a partially ordered set with elements x and y for which the proportion of totally ordered extensions in which x < y lies between 1/3 and 2/3? What's the significance of this? If no such exist, then does putting x < y if that holds in the majority of extensions give a linear order? (Certainly if x < y and y < z each more than 2/3 of the time then x < z sometimes!) But I must have the problem wrong -- clearly you can get 1/2. Is it "does there exist one in which the proportion is never between 1/3 and 2/3?"
Richard Guy mentioned the problem of for what n there is a regular n-simplex in Zn. He said that examples exist for n = a2+1 and also 7 and 1057. But I thought they exist whenever a Hadamard matrix of order n+1 exists. (Normalise and delete the first column.)
After that Robert and I went to the Faculty Club (where he seemed to know everyone) and on to Richard and Louise Guy's for dinner. A most enjoyable evening. Chicken in apricots and almonds, and pavolva! An assortment of wines, never the same from one glass to the next. Back to bed at midnight.
In the course of the evening Robert mentioned that the number of countable models of a complete theory can be any positive integer except 2. The non-occurrence of 2 is a theorem of Vaught, strengthening Ryll-Nardzewski; he outlined a proof which I won't attempt to reproduce. Other numbers are produced as follows. The theory of a dense linear order (without endpoints) having a countable set of constants forming an increasing subsequence has three countable models, depending on whether the sequence is convergent, bounded divergent, or unbounded; to get more, simply colour the elements with any finite number of colours, with each colour class dense, so that convergent sequences are further distinguished by the colour of the limit point -- we presumably require all terms of the sequence to be of the first colour.)
This morning there's a chinook. The air is very much warmer (about +9) and the sky is filled with ragged clouds. The tips of distant mountains are just visible above a ridge on the other side of the river.
No, of course, the third isn't the same -- consider infinitely many disjoint edges.
Talk was OK. Not quite the sense of excitement I got at Santa Barbara or Eugene. And I went a minute overtime and held up an incoming class. Afterwards a probabilist was sure that zero-one laws of random walk theory could be brought to bear on random sum-free sets. I should really try to show that the probability of getting only finitely many (not zero) even numbers is zero -- that would give a "tail event" with probability other than zero or one.
Robert can do his "dense linear order with increasing sequence" trick with two binary relations -- an equivalence relation such that one class has size n > 1 and all others have size 1, and a dense linear order on the set of equivalence classes so that the elements of the distinguished class come in the correct order. Probably the structures are far from absolutely ubiquitous -- yes, certainly so.
If I had a rocket launcherSo it goes.
If I had a rocket launcher
If I had a rocket launcher
Some son-of-a-bitch would die.
This is soon after getting home from dinner at Robert's, and what a dinner -- that guy can cook it as well as appreciate it! Shrimp bouillabaisse, ratatouille (baked, not stewed), tongue with carrots and riced potatoes, cheese (a very English-style cheeseboard with Stilton, Boursin, etc. -- is this indigenous Canadian?) and then lime soufflé -- I hate to quibble but it was slightly overdone, but the pastry was sublime (literally).
Off tomorrow morning at 8:00, assuming that I wake up in time. It seems quite incomprehensible that the time has all gone; I can't have been away for two weeks! But the richness of this trip-within-a-trip quite leaves the Pasadena experience in the shade. This is what travelling is really all about -- the slow-moving immensity of the mountains, the great happiness of seeing Bill and Phyllis, the sheer exhaustion of ploughing up the Chief through waist-deep snow, to say nothing of the mathematics, the running, ...
On that subject, a fairly long conversation with Ivan Rival. At the same time, he challenged my prejudices ("You compress your sinews by running, don't you think that they need to be stretched afterwards?" -- he might be right, how should I know?) and seemed to read much more into my casual pronouncements than I ever intended ("You worked on your stride? How?") I used to be dogmatic about running (witness those Road Runner articles and that passage in my famous unpublished novel about the blurred hand) but I am much less so now. And I can't help seeing parallels with a snippet of conversation tonight, in which someone was expressing puzzlement that the Inuits and the modern bomb-generation conservative youth could both do something as foreign as "live for tomorrow" (a telling slip, that) with such different results. Which reminds me, again, of a recurrent phrase in A Life Full of Holes: "Today and tomorrow, today and tomorrow". Whenever he says that, he is totally resigned to doing the same (unwanted) task for as long as need be. This is the answer to the Reagan youth, I guess.
What can I do about the world? It's only in very loose moments that this thought can ever strike home, but certainly the idea has its seductive charm.
Today is even warmer. While I waited outside for Robert, the horizontally-layered clouds were painted bright orange-pink by the dawn, dimming the neon signs of Motel Village and the flashing Christmas tree lights in the lobby of the Sun Bow Inn. Great gaps in the clouds let in clear sky. No mounds of soggy snow, just damp patches round the packed ice on the pavements.
As we drove out, and while we waited in the airport looking out the window at the little cluster of Calgary skyline against a backdrop of white mountains, we talked some more about mathematics. Robert has been trying his hand at Dugald's question on the spectrum of countable structures of given age; I told him about my plausible guess about absolutely ubiquitous graphs. Also I told him about my permutation representations of free groups.
The long-awaited confrontation with immigration presented no problems at all; the man was actually friendly and helpful. I'm now on board in the Connaiseur class and, for once, well forward of the wing. I like travelling!