They called us so early for departure that I hadn't even been to the duty-free, though I had got a few things from the chemist (no decent vitamin C, the only ones less than 500mg were mixed with zinc; but I got paracetamol, plasters and shaver batteries too) and a pizza slice and orange juice before I went through the gates of hell. The thought struck me in the middle of the terminal that I don't enjoy travelling any more. Certainly it's true that I don't anticipate it (but that's not the same thing).
Gate lounge full of modernistic-looking pictures, dreary brown and black check carpet, orange and yellow check seat upholstery, I feel a headache coming on.
I'm on the sunny side, by the window but it's hard to see anything because the sun's glare on the frost outside obscures the view. United have a do-it-yourself attitude to safety, expecting untrained passengers to help the crew if necessary. I'm in the seat right by one of the emergency exits; my card tells me how to open the hatch, and also tells me not to let passengers out if there's flame, smoke or jagged edges out there. In other words, they expect me to make life-or-death judgments with no training and no recompense, and don't let me get a drink of water to ease my headache.
Lunch was sole. I dozed before it. My headache is at bay, or resting, at the moment. 2 films to come -- more sleep time?
I can't sleep -- the sleepy phase has passed, and now the tingles make it quite impossible. Another 2 1/2 hours or so, maybe more.
We're now over land, and the direct sun is off this window so I can see a bit better; but actually there's not much to see, since there's a fair amount of cloud, and not much detail on the earth's surface. I saw one very meandering river.
Behind me are two American children with an English mother, if the accents don't deceive me. Actually, her accent is mod-Atlantic though the words are English; presumably she lives in the USA.
Now a more interesting mountain landscape below. Large razorback ridges with grey flanks, some streaked. Patches of snow. Frozen lakes. (Can they really be?) Now a big white splodge of a snowfield. But mostly these peaks rise, not from a snowfield as in Greenland, but from pools of blue shadow. The next lake is definitely unfrozen, green with orange border, contrasting with the earlier white ones. I disbelieve the map when it tells me that we cross the Rockies on about the Canada-US border, heading nearly due west; I think we're much further north than that. (Frozen lakes on the US border in July?) Still range after range of mountains. But coming to an end now, with a road, and less towering peaks.
The shades of blue are wonderful, very pure, going from deep just below us by degrees into the pale horizon sky colour without a break. But there is a noticeable shift some 156 degrees below the horizon. Effect of distance?
Now, for the first time, the patchwork sign of earth worked on by man. The PA is now announcing that we're in central BC. The map suggests that indeed somewhere in mid BC the solid mass of mountains separates into the coast range and the Rockies proper with a flatter area in between.
The kid behind just said, "Mom, how come they call them lavatories instead of toilets?" in a completely American voice.
Big blue lakes, smaller green ones.
I don't think any queues move slower than US immigration. But when I finally got to the desk, behind several people who'd filled in the wrong colour I-94s and one unaccompanied minor who'd lost her passport, I got a friendly official who'd graduated from the University of Oregon some time ago, when they had only 3700 students. He wrote B-2 and then scribbled B-1 over the top -- hope that's all right -- and tried to persuade me to stay longer! Then my bag came just as I got to the conveyor; the customs official was sufficiently fascinated by my business being mathematics that she waved me straight through; and then I made it here on the amazing underground transit system, needing two changes (not difficult if you know what you're doing).
It seems that we go to Eugene on a tiny little plane -- hence boarding shouldn't take too long! Many of the planes coming and going here are Alaskan, with a huge face (Inuit?) on the tail. National borders are incomprehensible -- I'm sure there must be more flights between here and Alaska than between either place and anywhere in BC. The other side is that twice already they've called for volunteers to give up their seats and travel on a later flight, for compensation of a $150 travel voucher. Not a lot of use to me -- I won't jump unless I'm pushed.
To return. Flight down: I was in the very back seat of the 19-seater plane, right by a window with clear views inland. We got a splendid view of Mt. Rainier; we were just a little higher than its summit and quite close. Then several other mountains, of which the only one I could identify was Mt. St Helens, still bare and broken as if by a violent explosion. As we were approaching the Columbia, a boy just in front of me got a sudden nosebleed, and his parents had no tissues. Mine were in my bag in the closet at the very bottom, under everyone else's; but I fished them out. Afterwards the kid seemed OK. After Portland, much quieter country, the farmland of the Willamette valley, patchwork fields of bare earth, green, ripe wheat, something black (burnt stubble already? a crop? who knows).
Bill was at the airport. Getting out was very quick, the baggage came almost instantaneously. He drove me home, gave me food and drink, and had to go; but I was still bright and alert when he and Phyllis got back about 9, and indeed a couple of hours later. I went to bed and slept through until six.
Saturday morning, I started reading two books. Bella Chagall's "Burning Lights" with illustrations by Marc Chagall, and a Faye Kellerman, about an LA Jewish cop -- I didn't even recognize that she had written that silly book about Shakespeare until I saw the blurb in he end, but this book was much better, unbelievably powerful even though it's the end of a series -- and almost they blended into one, talking about celebrating the same festivals in the same way though on opposite sides of the world and in vastly different times and circumstances.
Phyllis and I went into town, to the Saturday market. I'd heard Eugene described as the town of "ageing hippies, and young people who wished they'd been born twenty years earlier", and here it all was: the peace and love, the tie-dye, jewellery, kaleidoscopes, toys (identical duck puppets to the one we bought on that canal boat so many years ago), smoking on the grass, playing guitars at every corner. I got Hester some earrings. But then we went into an area of indoor crafty shops, and all of a sudden the quality was incomparably higher; the same things, jewellery, kaleidoscopes, ... but what a difference! A kaleidoscope in the market was pretty, but one in the shop was like looking at jewelled brocade, to say nothing of the well-made wood and brass mounting. I got a Navajo bracelet for Sheila. Both purchases depended a lot on Phyllis's judgment, which (sometimes after a little reflection) I agreed with. Phyllis got half a crate of excellent strawberries at the market, and various other vegetables which didn't look so good but were undoubtedly organically grown. We ate berries for lunch.
After lunch, some mathematics; then Bill and Phyllis had to go out again. I ran (but how unfit I am! -- the blazing sun, temperatures in the 90s, and fierce hills reduced me to a walk more than once). For supper we barbecued a salmon; a bit underdone but the microwave came to the rescue. Utterly, utterly delicious. More mathematics in the evening.
The large picture: Bill is worried about the overall direction of finite geometry, partly as a result of the purposelessness of many of the talks at Deinze. He wants us to put our heads together on this meta-problem. I absolutely agree but there is no quick solution. In the forward direction, I see a couple of general lines: embeddings and hyperplanes; finiteness theorems. But I think some kind of turning back is necessary to projective geometry and the classical groups. Perhaps our book fits in here. Worth thinking about.
Incidentally, Bill had anticipated (a special case of) my construction for rank 3 geometries using partitions into spreads, in his Como paper. Not at all surprising.
Yesterday we went hiking. We drove out to the south-east, and turned off along an 11-mile gravel road, giving us finally a 1 3/4-mile gentle clime to the summit of Mt. Fuji. Phyllis is recovering from a month-long attack of bronchitis, so nothing more strenuous was possible. But it was a lovely gentle walk with varying terrain: old clear-cut, forest with and without undergrowth, rocky ledges, fine views in all directions. North was a lake and, in the distance, Mts. Jefferson and Hood, the latter scarcely more substantial than a cloud. We picnicked on turkey sandwiches, more berries, and water. Going down, more time to appreciate the near point: seedlings in cracks in huge rocks like bonsai; Spanish moss; spiders; orchids. Back at the car we saw a deer. Lovely time altogether. We drove on to Salt Creek falls. A beautiful waterfall that one could watch for hours, the water coming down making patterns as the different speeds of different parts made themselves visible: a tongue would reach forward from the mass, then attract extra air resistance and be overtaken again. At the end, disaster struck: my film was, once again, not winding on, and I fear that none of the pictures I've taken here will turn out at all. Some lovely things lost. So I must be extra certain to remember them. Leaves of rhododendron and maple glowing with the sun behind them; small alpine gardens on the pumice-strewn slopes; the vertigo of looking through binoculars at a lake in the trees two thousand feet below.
Home again, we ate left-over salmon and lasagne, and didn't stay up late. I've been going through the draft of the paper Cheryl sent me. The wonders of modern technology: emailed one day, received the next, five minutes' work to strip off the headers, compile and print. Basically fine. I can fill in the gaps she's left me. I noticed that there is another subgroup of AGL(8,2) with an orbit of length 210; like the one we had, it also doesn't give a 5-design.
I didn't say about running. In the woods, two small snakes slithered off the path ahead of me.
The girl at the check-in desk at Seattle, calling for "approximately two" volunteers to be bumped.
After breakfast we drove to the university. There's no parking; Bill found a space on the street but apologised in advance for the possibility of a 3/4-mile walk. I saw the secretary and signed a form. I spent some time in the library while Bill did various jobs, but mostly we tackled small-base groups. First we found a trivial argument that, given H, primitive groups with stabiliser H and unbounded conjugacy class size (at least for elements with fixed points) have the property that almost all pairs of points are bases. (The argument isn't quite right but I think it's fixable.) Now a more complicated O'Nan--Scott argument makes no reference to conjugacy class sizes and allows the stabiliser to grow in size (as long as it's o(f(n)), f(n) the inverse function of n -> nn). We thought that the first, but not the second, argument generalises to groups with the point stabilisers of bounded order; but I'm not sure about this at the moment. (Later, yesterday evening, we found examples of primitive groups of order about n3/2 with base size 3, and about n3 with base size 4.) A related problem: If H is primitive, show that H intersect Hg=1 for almost all g in Sn. To prove this it's necessary that the conjugacy class of any element which lies in some primitive subgroup is quite large, bigger asymptotically that the square of the order of any primitive group.
On maths, here is a curious problem. If n is even, then the numbers of elements of Sn with all cycle lengths even and with all cycle lengths odd are the same. Explain. [Now answered by Richard Lewis and Simon Norton.]
Supper was a picnic in a lovely pine-trees-and-rhododendrons park on a hill on the eastern side of Eugene. Phyllis had packed mountains of food; we did well on the crab and sweetcorn, but didn't even touch the salmon, or the potato chips, or even much of the bread. Unfortunately we were buzzed by a lot of yellow-jackets (wasps); they didn't actually attack us, they were too fond of crab meat, but they were a bit distracting. We walked in the park, and then had marvellous ice cream (one of the best in the country, they say, and I would credit that -- I had mandarin chocolate) and then went home to continue working. Tired; to bed before eleven, to worry about the faulty argument.
Wednesday we went to the coast. Bill and Phyllis offered me three options; what we did essentially combined the second and third. We drove through the coast mountains, relatively dry poor country, with the little town of Noti notorious for the racist slogans in its pub. We reached the coast at Florence, wide street catering for tourists, and headed north, out of the sand flats into the mountains and precipitous headlands. We stopped first at Strawberry Hill, a new place that Bill and Phyllis were trying out on me. It was wonderful. The first thing one saw was, on rocky islands inaccessible only because of a channel some 30ft wide, dozens of seals, mostly lying on the rocks, some fighting, playing, falling into the water, or swimming past slowly and gazing at us with their slow, unblinking stares, then diving underwater. The rocks we stood on to watch them were covered in pools, full of starfish, crabs, sea anemones, etc. Some starfish were dining on mussels in the most leisurely way imaginable. Beautiful colours and patterns on these creatures.
Enormous scrum now on the plane as the latecomers try to find seats, negotiate swaps, stow bags, etc.
We walked along the beach, coming to more rocks with deep rock pools, full of sea anemones and sea urchins, and many larger crabs. I was reminded of Monet's waterlilies in the artless art with which they distributed themselves on the floor and walls of the clear sunlit pool. At the back of the beach were caves and cliffs of coloured sand and varied flowers. At the back of one cave, a large twisted driftwood log lay like a fabulous monster in its lair.
Back along the beach, we watched steam drifting inland on the breeze. I hypothesized that the sand, heated by the sun at low tide, was vaporizing the incoming waves.
A little bit further up the coast, we stopped at the Devil's Churn for lunch, picnicking at a table in the car park on sandwiches and fruit. After lunch we walked down to the Churn, a long narrow inlet. The tide was still very low and the sea almost calm, so there were no spectacular water movements in the Churn. But at its back was a cave whose bottom was covered in pebbles; even these small waves were enough to set it reverberating, and the sound of water draining through the pebbles was magnified into eerie music.
Short pause there for the safety announcement. We've left the bay two minutes early. I'm in a center seat and may just see something of the view, but it's the wrong side for the real mountains.
After the Churn, we walked along the beach to the Spouting Horn, which wasn't spouting because the waves were too small. On the way we picked and ate blackberries and saw crows perching in a small tree. We returned to the car park on the trail, parts of which (out of sight of the sea, with tall pines and ferns) could have been mountains rather than coast.
Now airborne, banking left, so I have only a minimal view of ploughed fields. It's very hazy.
Back to the coast. We drove back down to Florence and out the other side, to Honeyman Park in the dunes. Bill and Phyllis lay in the sun and slept while I took off my shoes and ran in the dunes. It was a beautiful run. First impressions: the sand surface was almost strong enough to take my weight, but at the moment of push-off I would break through the crust. This made it hard work, not even with the rhythm of really soft sand or deep water, and encouraging a light elf-like gait. I passed a thick clump of forest on the top of a sandhill and then it was open dunes. After a while I crested a high dune to see a wide stretch of flat sand, across which I headed diagonally (directly into the sun) towards the scrub bordering the sea. Coming through the gap I was at first dismayed to find the beach crowded; but it was only a family of five, seeming like a crowd after the isolation of the dunes. The real disappointment was that the beach looked like a race track, every inch covered with tyre tracks. I took a small piece of seaweed for proof and headed back.
Returning, I took a different path.
In flight, snack just arrived: turkey and cheese sandwich, cheese and crackers, two strawberries, and chocolate-coated macadamia nuts. Coffee'll be here soon.
To return -- actually it was hard to draw my memory away from that beautiful scene. I came to a stream, just a little trickle by the path. As I went on, the stream grew stronger and wider, until I couldn't avoid getting my feet wet, and splashed happily along. But there was one startling thing about the stream: it was flowing backwards! Presumably some source of surface water in the dunes, and as it flows into the scrub towards the sea it soaks away. The banks were lined with yellow flowers and grasses.
Back to now: here's coffee. The cheese was Monterey Jack Vermont; I suppose that it is as generic as cheddar now. The strawberries are tasteless; a shock after those I've been having. (For breakfast, we had warmed-over blueberry pancakes and maple syrup smothered in fresh peaches -- enough to render these snacks superfluous.)
Reaching the dunes a bit south of where I left them, I found that there were many forest-topped sandhills. I correctly identified the one I'd come from, but decided to aim for the next one down. This was a mistake: I found myself in long grass and brambles, barricaded by a tongue of impenetrable rhododendron forest. Somehow I hacked and slithered my way through the forest; was I relieved when I saw sand through the leaves! Up a hill, and there in a sheltered hollow were Bill and Phyllis.
Adjoining the park (which had a boating lake as well) was a large campground. We drove in looking for showers. By the time we were desperate enough to ask someone, we were almost on top of a sign pointing to the showers. They were free, clean, and hot -- wonderful. I had brought a towel and there was soap in the car, but I blew it by leaving the soap behind!
Back to Florence for dinner. We poked around the shops a bit, including a wonderfully colourful kite shop with a box kite about ten feet long hanging from the ceiling. Then we passed up the famous fish and chip restaurant in favour of something more classy, and very very good. In a high-ceilinged old building with lots of plants and baskets, we sat at a table near the window. It was spacious, cool, and light, and the food was really outstanding. I had blackened redfish, a dish which I gather has exploded in popularity to the extent that the Louisiana redfish was nearly wiped out, so now it is made with other fish, red snapper in this case -- a strong enough fish to take the spices but not so strong-tasting to distort the result.
Back to now: a view of hills and what looks like coastline except that there is cloud where the sea should be. Some of the cloud spills through the gaps. So I'm not quite sure. There are no obvious coastal features hereabouts. The hills are quite high and bare on top.
Driving back in the near dark, we pulled off at a tiny parking place by a mountain stream, one often used by Kantors on trips to the coast with kids. The soft evening light over the water was lovely, but most striking were dozens of bright red crayfish glowing in the water. We enjoyed the peace for a while before heading for home. We all agreed that such a perfect day could hardly have been bettered in any respect.
I've just discovered I missed a day somewhere. Could it really be that we went to the coast on Thursday? Anyway the trip was certainly after my talk -- I could scarcely believe the attendance, about twenty people in late July! There are summer courses for teachers and presumably people appreciate a change, but even so ... ! I ranged over quasigroups, Latin squares, primitive groups, random elements of Sn, fixed point free permutations, Peter Donnelly's heuristic, and the formula for the limiting proportion of fpf elements in Sn on k-sets. Afterwards I got some useful feedback, from Charlie Wright on loops, and from Sergei on Russian work on the limit of the uniform distribution on Sn.
My run up Spencer Butte -- now, apparently, an inevitable part of a visit to Eugene. I headed out along the road on the east side of the mountain. Much further than I expected, I struck the trail, on the far side of a small valley which it soon crossed by a bridge. After some time through the forest of pine and ferns, an old broken sign offered me the choice of Willamette in 1.2 miles or the summit in 1.3 miles. Of course, I took the latter, and made not too bad time up the hill. (I'm not as unfit as all that!) The trees thinned and I remember two grey trunks beside the path looking like standing stones. (This is without my glasses.) Many variants of the path lead up to the final stretch to the summit. I didn't worry about making for the highest point, but traversed the ridge to the north end. Wonderful views of blue openness, town and country, stubble fires to the north (hence those black fields). A clear trail led off the north edge. perhaps I should have known better: it turned into a very long loose dirt slide, about 1 in 1 slope. I was very worried about my knee, which seemed to take a lot of strain as I braked against trees and bushes, sometimes from quite considerable speeds. Finally the slide reached the bottom and became a trail which joined another trail. I had to make a decision and I think I got it wrong, because finally, almost out of the woods, it veered the wring way, and I took a much smaller trail which led onto a drive and down to the wrong side of locked gates and an electric fence. I was able to slide under the fence in a small gully into the roadside ditch; I was indeed on Willamette, but about half a mile south of where the trail meets the road. In any event, getting home from there was a doddle; journey time 80 minutes, and no ill-effect with my knee. That was the morning before my talk.
That evening, Bill and I ate in a restaurant with Gary Seitz and Martin Liebeck (who's here for the summer). Sheila was ill and couldn't come; Phyllis used the opportunity to cry off as well and stay home to get on with things. It was quite a good restaurant; nice food and wine, very plain salad --
-- Interlude, landing in San Francisco. From the air, both land and water were drab colours. A very long low bridge over water was opposite the sun from us, so cast no visible shadow on the dead calm but non-reflecting water; wispy clouds below us strengthened the illusion that the bridge was floating unsupported in the sky, with the cars crossing it and the pylons floating by its side. Technology crossed over into magic as we came down into the sci-fi city of the future. Of course, the ground soon dispelled the illusion! I'm now in the departure gate lounge waiting for the SF-LA shuttle, which seems very frequent; there is another one to go before ours! Everything is very crowded for Sunday lunchtime: a popular travelling time here. The 737 from Eugene had credit card operated phones, one for every three seats, but I didn't see anybody using one. --
-- back to the restaurant in Eugene. The dynamics were curious. It was like two couples. Gary and Martin had their shared jokes and stories, Bill and I had ours. It seems that Gary and Bill have grown apart during their time in Oregon. In a way, Gary is a producer, Bill a consumer, of a certain kind of mathematics; and the farmer and the townsman can be friends but never completely share sympathy.
Also before the coast trip was Gene Luks' lecture. I had been to bookshops and comic shops in the morning, after presents for the boys but inevitably ending up with stuff for me, such as a book about the life and teachings of Naropa, and for just over a dollar a second-hand copy of the Upanishads (which seems to be a text-book for a course of some kind here). I walked around campus, looking at a lovely coloured pavement inlay being installed by two people (workmen? craftsmen? artists? in Eugene the distinction seems less clear than elsewhere). Gene was in the middle of a series of talks showing that he can "handle" solvable matrix groups over finite fields in time which is polynomial in the dimension, the logarithm of the field order, and the largest prime divisor of the group order (the last has to be put in because of the discrete logarithm problem). Charlie Wright has a bad back; he managed to sit through two hours of the talk but then had to stand. He'd brought me a copy of an old paper of his on the multiplication group of a loop. I think I must extend my quasigroup result to loops. Incidentally, Sergei suggested looking at the Russian work which (for example) shows that (in the limit) 99% of all permutations have the property that 99% of all points lie in the largest 9 cycles. Now all one has to show is that the probability that some sum of cycle lengths of the largest 9 cycles lies between 49% and 50% of the total is small -- surely not too hard.
Now on the plane for LA. They were also boarding a plane for San Diego simultaneously through the same gate -- I hope I'm going in the right direction! I have a window seat but it's over the back of the wing, not ideal for viewing, and also on the seaward side. I don't expect too much in the way of spectacular scenery. I wonder if anything will be recognizable. There is a prominent cape between here and LA, so that Santa Barbara looks south out to sea; but I think it will be too far away to see.
Anyway, where are we? Friday, I went in with Bill, we talked for a while about the possibility of asymptotic finite geometry -- the most promising thing would seem to be quotients of buildings, e.g. affine buildings -- and then I took my film to be developed; we had lunch in the department, where they joked amid the stringencies about abolishing the philosophy department -- the President and his wife are both philosophers -- then I picked up my film and we went over to Computer Science to talk to Gene Luks again about matrix groups. Some of the most interesting questions involve GL(n,Z). For example, let G be a finite subgroup of GL(n,Z). Does G have a "small index" subgroup? (Index <cn, perhaps?) And can one find such a subgroup (in time polynomial in n)? I understand that Babai and Beales have a polynomial time algorithm for testing finiteness, which actually finds an invariant quadratic form. This is interesting because of course there's no possibility of averaging over the group, and raises the possibility that other averages over the group can be found (e.g., for the square of the character, this would test irreducibility over C). The difficulty is that one doesn't have, even implicitly, a presentation (as one does in the permutation group case); so, even given a putative homomorphism to the cyclic group of order 2, one couldn't verify that it really was. For this we sat in Gene's huge, air-conditioned office, with two sofas, a Sun on a computer table, and another expensive machine too (not even on the desk, which was almost clear except for a bowl of nuts from which birds can fly in and help themselves -- though none did while we were there), a wall of books, abaci and magazines on the coffee table; the other half, without a doubt. Using Bill's terminal to telnet to London to attend to my mail was feasible, but difficult because the keys of the terminal were so unresponsive; often I hit the spacebar and found later that the keypress hadn't registered. But I did learn the Unix command stty erase ^H which should solve my problems with the backspace key. (I think I'll put it on a function key when I get back to my office.) By the way, one of the messages was rather important, instructions from Irvine about what to do at the airport.
On Friday the Aschbachers were in Eugene. Pam was giving a talk at a conference on "Education 2000", on assessment -- in essence, persuading teachers that there's more to marking essays than just reading them through and saying, "It feels like a B to me." We went round to Seitz's for dessert to see them Before, we ate at Bill's and Phyllis's favourite restaurant in Eugene, the Zenon, which set its theme well by having a "Periodic Table of the Vegetables" prominently displayed. Clearly, to them, vegetables are not just an afterthought. I had a large and delicious yellowfin tuna steak, with yellow zucchini and red potatoes done in some way in herbs and salt -- exquisite. We had an Oregon chablis with it, rather more like an Alsace wine, very fruity.
Now pulling back from the bay, virtually right on time. One curious thing is the prominence the security announcements give to the instruction not to tamper with the smoke detectors in the lavatories. Is this the result of behaviour by nicotine-starved addicts on non-smoking flights? I think I'll watch for a bit.
Dry Californian hills, with a dirty white cloudbank behind like an impossible snow-capped range. Just about on the runway now; we'll be off soon. Then back to the story; and it looks as if this book will be plenty big enough. Certainly I couldn't diarize in this way if it were not for the time spent in limbo while travelling. An Alaska plane bakes on the tarmac. Meanwhile, a TWA comes round, jumping the queue in front of us. Otherwise there's little movement. Now a United plane is also jumping the queue. In the distance, airport machinery crawls round on its inscrutable business. Now we seem to be going. A single-engined propeller plane is also sitting on the runway but we cut in front of it. In the distance, four planes, two facing two, seen in silhouette, like some] confrontation; they change places and one takes off. Now we've stopped again; maybe we weren't as close as I thought. Curiously, those planes are using a runway nearly perpendicular to ours. Now we're off.
Up over the water, dry hills across the bay, then over the city, a landscape of buildings and coves, until the clouds white everything out.
So, to the Seitz's house, one of the most artistically uncluttered places I've seen. (Phyllis has nicer things on the walls but her lounge doesn't look like a high-class art gallery.) Kirsten and Meredith were there, briefly; they took their ice cream cake upstairs and left the company to its own devices. After we'd eaten we went for a walk up into Hendricks Park where we'd eaten crab before, to watch the last daylight fading and the lights of Eugene reflecting in the river, and to see the amazing hillside houses, one like a sailing ship with lots of outside stairs, several on very high stilts. Back to the house for a drink and a chance for the LA-bred to compare high schools, to the disgust of the others, and to hear the true story of the hike Gary and Martin took.
And so I reach yesterday, and our trip to the mountains.
Pause -- nice views of coast, beach and hills, under thin cloud.
We took minimal lunch and headed up the McKenzie river, through the richest land I've seen so far, where everyone had a little plot of corn, and orchards grew apples, berries, and hazelnuts. The hills on either side grew higher, the river rougher. Soon after a short stop at the ranger station at McKenzie Bridge we turned off on the old McKenzie highway, a road so winding that I think every variation on the curve sign must occur on it somewhere. At first the forest was full of leafy undergrowth dappled by sunlight striking through the pines; but as we got higher, it became more alpine. Our destination was a parking space by the side of the road opposite a small alpine meadow. Though the main wildflower season was over, we saw a good display as we walked in, including lovely deep blue gentians, with their elegant deep star shape. We hiked in at the edge of forest and stream until we came to Hand Lake, lying under a wall of lava -- an old lava flow which had just stopped dead. Our route took us round the lake, making long detours to cross arms of the lake (in fact not really necessary; the level was down so far that the arms were dry). Past the pine-scented shelter, through the woods, and we came out at the lava. A dramatic sight, dark grey, knife-edged, virtually devoid of vegetation except for a few hardy trees. We went up at the interface between lava and forest; it resembled the coast in being an abrupt interface between two totally different systems. After a quarter of a mile, an old pioneer wagon trail had been constructed, at great labour, across the lava field. It was almost smooth underfoot and gave us easy passage. We stopped to eat plums and drink water in the shade of a tree on the other side. Then we walked down past the lake. Normally the trail should have followed a narrow beach just below the forest edge; but there was lots of room so we walked on the lake bottom. Storms of butterflies, drinking from the moist earth, erupted before us and settled behind: an amazing sensation. A few ducks swam on the water. Finally we reached the trail back to the road.
At the car, a short re-fuelling stop for turkey and cheese sandwich. (And by coincidence, our in-flight drinks and peanuts arrive.) Then on our way; and if I thought I'd seen lava fields, I had a shock coming, as we reached the area between Mt. Washington and the Three Sisters: a huge expanse of lunar landscape, the older flows supporting a few tenacious pine trees whose dead white trunks contrasted with the black lava, the newer flows completely bare, the mountains floating above the desolation. We stopped to look, and then drove on to the Observatory at the crest of the pass, a mile high, a tower built of lava, from which one could see from Mt. Hood in the north (scarcely more substantial than a cloud resting on the shoulder of Mt. Jefferson) to I don't know where in the south. Back to the car, we found three chipmunks playing hide-and seek among a pile of rocks. We tempted them with bread and enticed them out but not into our hands.
Then down through the much drier forest on the other side. Out of the mountains the land was dead flat and carried a huge forest of ponderosa pine. We drove into the town of Sisters, stopping to look at the llama farm. This is a town which structurally is still much as in Wild West days, but commercially has become a purveyor of not very good artistic souvenirs to the tourist trade. we went to a produce store and got some good tomatoes, and had ice creams from a café in a small modern shopping mall full of knitting and ceramics, but went to the big out-of-town supermarket for food for supper. Phyllis and I got a feast: barbecued chicken, cheesy bread, cherries, watermelon, cookies.
Back along a different road through miles of ponderosa forest to the mountains. This is a skiing area, where you need a permit even to stop at the roadside from November to March. Higher, we saw a lot of dead trees; but the forest was live and green again when we reached the Mackenzie.
Coming into LA, and it looks as if I won't finish the story on this leg. Mountains more rugged, no sigh of sea. But we then crossed the coast and made our approach over cloud-shrouded water, with little wave-rippled pockets through gaps in the clouds. Now we're going down over the LA street grid.
Los Angeles: twelve-lane freeways crossing and re-crossing; low buildings covering vast areas (land cheap, earthquakes common); the blanket of smog, through which the city centre buildings seemed less substantial than the ice-blue mountains in the distance. Now on the ground, taxi-ing to the terminal.
Past huge quantities of machinery. Two Qantas planes, one called Longreach. A car, light flashing, speeds down a runway; I wait for it to sprout wings. Then a stop, waiting for our gate to be free.
Back to Oregon yesterday. We pulled off the road to walk the waterfall trail on the upper McKenzie. The trek in didn't give me much idea what to expect; we scarcely saw the river until we came to the observation platform at the top of the first spectacular fall. We could sit on the rocks and put our hands in the water just before it took the plunge; or look up at a series of small cascades. A series of platforms further round gave a series of views of the water, leaving the rocks, suspended weightless for a never-ending moment, then thundering into the pool.
The walk took us down beside the river as it tumbled over rapids and down small falls. We came to a pool enclosed in a rock wall, with only a gentle current, inviting us to swim. I took versions of a couple of photos that were lost at the waterfall last weekend: sunlight through maple leaves, etc. The river leapt over another big fall, and then into a reservoir, which we crossed by a long bridge. There was a path back on the other side, they assured me, and there was at least the start of one, though it soon petered out. I went to the end and explored a big ancient lava flow covered with colourful maple bushes. But no sign of a trail, so I was forced back. In fact it went from further back. It had a different character from the outward trail: less developed, no built-up viewpoints, so we had to see the falls from the top of a sheer cliff, holding on to trees. Everything was very dry so high above the spray, even the moss was dying. Finally we came to the little bridge where we crossed back over the river, and watched it chattering along in its rocky bed in the fading light; a beautiful, peaceful scene. Getting back to the car involved leaping sure-footed over a big log blocking the trail.
We headed back up the road a little way for Clear Lake, our supper stop. There was no running water in the public area, but we went into the resort and found a tap, where we washed ourselves and the produce from Sisters. The other side of the lake was still in bright sunlight, and there were a few rowing boats out, with people fishing or just enjoying the calm evening (no motorboats allowed on this lake!) but our side was in shadow. There was a noisy generator running in the resort, so we went back to the car and carried our food down to the lakeside, where we sat on rocks at the water's edge to make our feast, listening to the ducks and the fish jumping, and quite untroubled by insects of any kind.
Replete with dinner, we got back in the car, and headed down the highway for home, about an hour and a half's journey. The time sped by and we were home. We had a beer and Bill asked me about a wooly assertion of Laci Babai's, that n random elements in a permutation group of degree n almost certainly generate. We had another beer and decided that, in an elementary abelian group of order pn, n+f(n) elements almost surely generate if and only if f(n) -> infty as n -> infty. We also considered the linear analogue of my question about random elements of Sn: is it true that, if g is random in GL(n,q), then almost surely any irreducible subgroup containing g contains SL(n,q)? On the basis of n=2, we decided that the answer is no, but is yes if you strengthen that to absolutely irreducible and primitive (for any n); there is an algebraic reason for this, viz. in the algebraic group, almost all elements (in the Zariski topology) are semisimple (all eigenvalues distinct).
So to today, most of which is already told. I got up in time to do my washing before breakfast. We had the leftover pancakes (made yesterday because we ran out of eggs) toasted, with syrup and lots of chopped fresh peaches. We said goodbye. It was an exceptionally good week, as a holiday; mathematically we didn't get as far as we might have liked, but it was important that we discussed the direction of finite geometry (this discussion will bear fruit, I hope, sometime in the future). Then I said goodbye to Phyllis, and Bill drove me to the airport.
The rest, up to arrival in LA, is recorded history already. Off the plane, it was a long walk to baggage collection, and then for the first time ever I was asked for my baggage check, which I had to fish out of my bag. After some delay, I found the right Super Shuttle bound for Orange County, also carrying a Japanese man and a family of three just back from a trip to Japan, where they'd managed to lose something vital (passport?) A long journey and I was last off, so it was an hour after landing when I got to the hotel.
Then my troubles started. I had to phone Duncan Luce. The phone didn't work, so I called hotel information. they put me through to the desk, who told me the phone hadn't been switched on yet (then a quarter of an hour after I checked in) but it would be if I waited two minutes. After five minutes, nothing had happened, so I went down and used a pay phone. They gave me the address and said, Come on out, Take a taxi. So I asked the desk to call me a taxi. OK, they said, it'll be here in ten minutes. It took 25 (I had almost given up on it). It had gone to the Radisson Suite, not the Radisson Plaza. Then the driver thought I was going to UCLA. What a relief I caught that one! It turned out that he didn't know where the address was, and neither did his dispatcher, who had to call the house for directions, and then give them to the driver bit by bit over the radio; he still managed to get lost several times before he found his way to the house.
Then, finally, it was all OK, with lots of food and drink, people I knew (Manfred Droste), people I should have known (Charles Holland), and people I had come to meet (Duncan Luce, Louis Narens). After the reception, I went back with the OPG (Ordered permutation groups) crew to the hotel in a hotel van (which at least knew the way), and then we had a little meeting to plan our strategy. I have to talk a bit about things I know, but I don't have much idea about what the audience knows. Anyway, we shall see.
This morning we heard about the foundations of measurement theory. What interests me most is the historical antecedents. The Greeks had proportion, tantamount to a kind of measurement. Two objects a, b define a Dedekind cut as the set of rationals m/n for which na<mb. (We assume order and composition, i.e. the extensive case.) After Pythagoras they knew that not all proportions were realized exactly. Thus measurement theory leads directly to the construction of the reals by Dedekind cuts (the reverse of the contemporary view). Everybody referred to Hölder, but one speaker only to the axiomatization of geometry. The difficulty there was that the axioms are straightforward in dimension at least 3; Desargues is required in the plane, and a whole new approach on the line, rendering the geometric approach less good than Hölder's.
In the case of Hölder, the interesting relation is the implication from Archimedean to commutative. Narens pointed out that, assuming commutativity, an ultraproduct of the finite structures gives immediately a representation in the non-standard reals, and the Archimedean property then restricts this to the standard reals. It was also pointed out that commutativity is more satisfactory experimentally than Archimedean-ness, not just in practice, but in theory (as a single universal statement rather than a disjunction of the negations of infinitely many universal statements ny>x, n in N+.)
Luce and Narens are both clear expositors as well as nice people. That as much as anything should make this a successful meeting.
I guess Phyllis is on her way to Florida now.
Much uglier too. Irvine is a new town, the Milton Keynes of Southern California. Unlike Milton Keynes, you are not meant to walk. I walked from the hotel to the campus this morning with Mike Cohen (who does nice stuff but doesn't explain it all that well). It was further than I thought, and we ended up ten minutes late (though by myself I think I could have easily made it on time). The walk went past some shiny modern buildings, along roads which more often than not had no sidewalks. One place was a wall covered with modern art of sorts, with some nice sculpture where the entrance should be, but behind the wall was a bulldozed wasteland. In the nature reserve I saw rabbits and some squirrel-like creatures among the dry grass. On the watered grass in front of the new buildings were lots of toadstools. There was a hawk over the nature reserve and some smaller birds. But basically all I saw was ugliness; the river is a storm drain, bulldozers build up its artificial banks. There was, however, an arboretum. In Southern California style, I expect that carefully managed nature is far superior to wild nature (i.e., not a desert).
The message of Duncan Luce's talk this morning is that you can expect uniqueness results for concatenation structures over any order, but for relational structures only over Dedekind complete orders. I was a bit surprised that nobody even knew that a group satisfying finite uniqueness is closed. (Better, a group in which some n-point stabilizer is trivial is the automorphism group of a 2(n+1)-ary relation; this works in general, using the existence of a rigid binary relation on any infinite set, which is much easier to show than I'd imagined.)
Walked back with Mike. The new buildings are something else again, especially the foliage around them (banana groves, Japanese gardens, etc.) -- the contrast with the nature reserve could hardly be greater.
Charles Holland said today that Hölder's Theorem isn't explicitly in his paper, though something equivalent is; moreover, Hahn attributes it to an earlier Italian paper on the foundations of geometry, which also doesn't have it explicitly, but does have about as close an approximation as Hölder has.
A question of Narens: Can Ohkuma groups be measurable? Can Fraissé-type relations on Q be continuous? (Surely no.)
Afterwards, I couldn't face another big expensive meal, so I went looking for something lighter. After a long walk seeing nothing but huge modernistic banks and offices and their perfectly tended grass and trees, I had given up and was going home, when I saw a pink neon sign in a window in the distance. I walked there and found a tiny row of shops, the first I'd seen except on the campus. At first I thought only two were open, a copy shop and a fast film processor; but right at the end of the road was a small grocer. I got biscuits, dried fruit and nuts, and a soft drink, and ate them in front of a big building before going home.
A couple of lessons in principle from today's lectures on dimensional analysis. First, traditional dimensional analysis in physics allows two quite different processes to be confused: change of dimension (i.e., units of measurement), and change of scale (the magnitude of the physical quantities). For example, dimensional analysis of a physical situation enables you to describe the behaviour of an aeroplane in terms of that of a scale model, and so has real predictive power which it has no right to have. This says something about the nature of physical quantities. It cannot be assumed in other situations. Second, psychology actually extends physics in its treatment of quantities: e.g. you can ask experimental subjects to compare lights and sounds, and they will do so consistently. Third (this wasn't stated but is my inference from a discussion) dimensional analysis can derive the general form of the solution to a problem without solving it, but you shouldn't expect a real simplification; however, there is a practical value in that the dimensional constants are stripped out at the beginning and put back at the end, instead of having to be carried through, so the possibility of error is less. We were told that there is a case where dimensional analysis reduces a PDE to an ODE; I can't see this unless there is a direct relation between the quantities which could presumably be put in directly.
I phoned the Aschbachers last night. Pam will collect me on Saturday afternoon and put me up overnight. So one less night in the hotel, and an easier journey to LAX on Sunday.
d2x/dt2 = g-(k/m)(dx/dt)2.Even before this is derived, one can see that if length, time and mass are multiplied by a, b, c respectively, then g and k are multiplied by a/b2 and c/a respectively. Thus, the solution x=f(t,m,g,k) satisfies
f(bt,cm,(a/b2)g,(c/a)k)=af(t,m,g,k).The solution of this functional equation is
f(t,m,g,k)=(m/k)phi(sqrt{gk/m}t).[Proof: Set phi(t)=f(t,1,1,1). Putting M=c, G=a/b2, K=c/a, and T=t sqrt{M/GK}, we get c=M, a=M/K, b=sqrt{M/GK}, and (M/K)f(T,M,G,K)=phi(t)=phi(T sqrt{GK/M}) as required: three lines!] Then phi satisfies the differential equation
phi''=1-(\phi')^2.I will amuse myself with the functional equation for associativity later.
Today I saw the backs of some of those immaculate buildings. These consist of huge car parks! Also, A sign to Irvine Town Hall and "Police Facility" (not station). Another sign warns that the air intake for the University administration will be polluted if you park or idle your car in a certain spot. I imagine someone with a grudge against the administration poisoning them all by parking his car outside; Mike Cohen said that's probably not how the sign is meant to be read!
I'm in my hotel room now after being caught short needing to go to the toilet. I left my film for processing at the place I found last night; it should be ready before dinner tonight.
What I haven't photographed, and may not now get, is the vivid coloration of the sky after sundown, contrasting with the airport lights.
My leg doesn't seem permanently damaged; just stiffness in the Achilles tendon area.
Interestingly, one of the principal references for dimensional analysis is Birkhoff's "Hydrodynamics".
On the subject of names: I have a vague impression that the Handbook of Mathematical Psychology into which I dipped years ago was edited by Luce and two others (Bush and Galanter?) Nobody mentioned it here yet. But Janos Aczel did mention the name H. Peyton Young in connection with a theorem about equitable taxation (is there such a thing?)
From the applications session on colour vision yesterday, a point of current controversy. Everyone agrees that the space of colours perceived in a fixed context is 3-dimensional. (This means: the equivalence classes of the equivalence relation "give rise to the same percept" on the set of intensity distributions are cosets of a subspace of codimension 3.) Surprisingly, the equivalence relation remains the same as the context is varied (within limits which are not too wide). Now does this mean that we have a 3-dimensional manifold of colours for which each context gives a coordinate patch? Niederée claims "no", reasoning as follows. Let the context C be a uniform spot of light surrounded by total darkness. In this context, he says, it's impossible to perceive grey; white, dimmed, is just a dimmer white. But, for any intensity distribution beta, let Cbeta be the context where we look at a uniform patch of colour, surrounded by a uniform annulus of beta, surrounded by darkness. Now grey is perceived; indeed, for t<1, Cbeta(t beta) is perceived as a greyed-out version of beta. Thus, Cbeta(t beta) is not in a 3-dimensional neighbourhood Cbeta'(beta') of Cbeta(beta) (which is a uniform, larger disc of beta, so presumably like C(beta). Thus at least one more dimension is required.
On meaningfulness: Fred Roberts discussed some questions in combinatorial optimization. Meaningfulness is defined as invariance of the solution under re-scaling of the data. For example, shortest path is meaningful for ratio scaling but not for interval scaling. Travelling salesman is meaningful for interval scaling, essentially because the number of edges is fixed. But minimal connector is meaningful for ordinal scaling, in view of its solution by the greedy algorithm. He posed a problem about frequency selection which seemed a bit silly because it admitted integral solutions only, so ratio scaling didn't seem at all appropriate. Still, an interesting say of thinking about problems.
Another piece of trivia from Fred Roberts: There is a bibliography of 104 papers in the use of social choice (for reaching consensus) in molecular biology! So much for scientific objectivity.
President Bush is in the news as saying that the Southern California defense workers should vote for him because his defense cuts will be only 1/4 as much as Clinton's, and the latter's will throw x people out of work. (Presumably, one infers, his own will throw x/4 people out of work, so the case seems a little weak.)
After the meeting, Marley and Narens said they'd visit me in London, and Marley and Niederée want copies of my paper. Several people thought my stuff was a highlight of the meeting. Good.
Louis Narens asks about the complexity of Fraissé structures and automorphisms. In particular, in an n-unique structure on Q, every point should be nameable uniquely in terms of n parameters. But how complicated is the description?
Dinner last night at the tandoori restaurant -- extremely mild and adapted to the American taste. One disappointment: the Kingfisher was imported from England! I should have had Taj Mahal, which did apparently come from India.
I went there with blood pouring out of my head. I went to pick up my photos, and the man (after first offering me someone else's) explained that my film was badly overexposed, and he was trying again to make decent prints. He'd made three copies of some, getting tolerable results, but he hadn't quite finished, and could I wait twenty minutes? I did, and he finished, but then of course I was running late. So I was looking at the pictures while walking (fast) back to the hotel, and I walked into a temporary road sign planted right in the sidewalk.
I finished the novel about P. T. Barnum's American Museum. It was OK but not great; having started, I went on as a duty, but by the end I was skipping paragraphs so as to get finished sooner. The Ursula K. Le Guin should be better though it has to be taken in small doses.
Check-out time is 12.00; Pam promised to fetch me at 2.00. In fact, I might just as well sit in the lobby as here, so I will probably go down fairly soon. There's a message on my telephone which is gabbled at such a rate that I can't understand a single word except my name, and I'm at a bit of a loss what to do with it.
On the subject of book, Oregon brought three authors home to me vividly: the dunes, Frank Herbert; inland of the Cascades, Richard Brautigan; and of course Ursula Le Guin.
I wonder whether it might be true that, for almost all g in Sn, G intersect Gg=1 for every primitive group G except Sn, An. Certainly this would require much better estimates than we have at present.
My hair and fingernails are much too long. I want to get home and cut them.
A couple of quotes from the conference.
The definition of a qualitative economist: one who knows no mathematics and what he knows is wrong). To which (from my own experience) I add the self-definition of a mathematical economist: ``Your theorem is true, but it's unsatisfactory.''Catalonian oath of allegiance to the Aragonian kings, ca. 15th century:
"We, who are as good as you, swear to you, who are not better than us, that we do accept you as our king and sovereign lord, provided that you do observe all our liberties and laws -- but if you don't, then we don't."(Quoted by J. Aczel.)
I left my suitcase in the bell closet and took my green bag with me, and walked out of the hotel. Soon I came, unbelieving, to a seat, surrounded and shaded by trees, in a little flowery garden. This is part of the Douglas Plaza, named, improbably, after McDonnell Douglas, who have a building opposite the airport. The plaza has one unassuming restaurant and a florist well hidden from view. Nobody ever seems to go there; not surprisingly, since there is nowhere for anybody to live round here.
A man with a big bunch of keys just walked past. but he didn't tell me to leave.
How different this place is from Oregon. And why? The "Save every drop" signs up over the bathroom taps are signed "Irvine Ranch Water District". I assume that reflects history, i.e. there was nothing here but a ranch once, and it was sold for development; they put in an airport, banks, other huge buildings, but left out houses, flats, shops, pubs, churches, parks, pieces of history, or anything else. You can't live here, you have to go somewhere else to work, somewhere else to eat, somewhere else to sleep. I bet even the police facility is personned by commuters from some distance away. Then so much energy goes into gardening (but where do the gardeners live?) and nobody sees it. By the end of my stay I was even fed up with the jazz radio station for its stupid repetitive news and inept announcements. Even the announcement welcoming new listeners via cable in Bakersfield was made over and over again.
A hummingbird just came to try his luck at the red flowers. After a while, thinking better of it, he was gone like an arrow. How clumsy the aeroplanes are beside these and the butterflies!
One of the banks I passed yesterday had bronze-tinted reflecting windows, that made the grey of a smoggy sky look like the gold of a remembered summer.
The hummingbird is back, at the bottlebrush flowers this time. One of the trees has sunlight speckles superimposed on the dappled colours of its trunk. They have been carefully planted in the notches of a quarter-cogwheel round the circumference of the paved area, at whose centre is the flower garden with three gum trees and three rocks distributed around it.
That called to mind a discussion with Rick Ball last night. Apropos of earlier comments on the difference between the American and European systems, I pointed out surprise that unleaded petrol is more expensive here than leaded. His reply was that the American way is to rely on free enterprise and market forces; if people want unleaded petrol, then economies of scale will bring the price down. This contrasted with the European system of managing the market by punitive taxation (some punishment, making unleaded cheaper!) What it really contrasts with is the other kind of American interventionism, requiring catalytic converters by law, confident that technology will be able to provide them (and, I think, confident really that American technology will be able to provide them first, so that foreign imports can be excluded).
I was also surprised to hear him say that, even in the late 70's, almost all European intellectuals were communists. I wish he could have heard my colleagues at Merton discussing the Just War at the time of the Falklands war. Bush has sent missiles and an aircraft carrier to Kuwait, and yesterday it was announced that he was sending troops as part of his election campaign (or words to that effect, from an official Pentagon source!) The next few months won't be pleasant.
I'm now in what might be a park, little but bare grass surrounding a huge mirror-glass building. Probably just to give it an impressive facade -- but what is the water cost in keeping all this green?? Nearest the busy road there are a few small trees.
Someone said that, if there are massive layoffs in the defense industries, Orange County may get more intelligent taxi drivers.
I can feel the palm at my back rocking in the wind.
"I prostrate myself before your speech
Which is inseparable from its innate emptiness."
Michael and Pam picked me up from the hotel just after 4 -- I collected my case and went out to wait in the sunshine. Quite a long drive took us to Pasadena. Pam made delicious prawn-and-scallop spaghetti, Michael made margaritas; we watched the Olympics on TV, seeing the women's 100m won by a girl who came within a week of having her feet amputated because of a misdiagnosed disease of the thyroid, and then had to have radiation because the chemical treatment of the disease involved substances banned for Olympic athletes! There's the Olympic ideal for you. No doubt the old fogeys turn in their graves when American stars appear in the commercial breaks advertising everything from Reebok to Macdonalds. But there you are, it's a TV show now; a tear-jerker story is an excellent subplot, and tying the commercials into the main plot is clearly effective for the advertisers. Today I'm promised dim sum before I get on the bus. I bet it's one or two cuts above the Iranian restaurant!
The smog has been very bad while I've been in Pasadena, much worse than Irvine (because of geography). The mountains have been almost visible, looming through the haze; I wish I'd seen them clearly.
Meredith was a bit stroppy yesterday after an all-night party, but more herself today. Kirsten now drives, and has free use of a car -- perhaps Hester had better not find that out!