Potenza and Budapest, 1999

Potenza, 12 - 19 June 1999

London - Roma, Alitalia, 12 June, 12:15

Left home just before 9:00. Uneventful trip to Heathrow. Just one change from routine: I used Barons Court instead of Hammersmith to change from District to Piccadilly. This was because a man was importuning the passenger next to me to sign a petition against greyhound racing, and was then about to give him some anti-vivisection literature; I happen to think there are more serious problems than greyhound racing, so I skipped before he got to me.

At check-in, I found the flight 20 minutes delayed, and it ended up quite a bit more than that. While waiting, I managed to put my coat down and leave it behind. The service not so good: no explanation given for the delay; long wait for lunch, which was a lot of dry bread and no butter, and nothing to drink except a small plastic cup of water until afterwards.

Dina was at the airport. She had left her car at Tuscolana, which is both more accessible and cheaper than Termini. We drove out to her apartment, where I left my stuff and we drank some water (I was quite thirsty by then), and then went to the very good, though cheap and unassuming, fish restaurant near the university. After a huge and excellent dinner (no choices, it just kept coming), we went for a walk around Roma by night, something that Dina doesn't get much chance to do now: Villa Borghesi, Piazza Spagna, Quirinale, Trevi, Navona, etc. Covered quite a lot of ground! And so home.

In the morning I ventured out and got croissants for breakfast from the bar across the road. After breakfast, we went to the station to buy tickets. Just as well I'd done it in advance; the queues were dire, and of course several people desperate to catch trains wanted to go in front. Then we soothed our nerves by walking on the Gianiculo. The wonderful view of Rome from the top was mostly blocked by trees, but Dina knew another place to see it, near the Spanish embassy. We puzzled over the meaningless inscription MDCCCVC on a statue. Then to the station to catch my train.

Train journey Roma - Potenza, Eurostar, 13 June, 15:45

A small amount of trouble finding the train. I was sure it was the right one: the only 15:45 departure from Termini, and a Eurostar, and going to Taranto, which even with my limited Italian geography I know is in the right direction. But there was no list of the intermediate stops, and somebody told me it was not the right train; so I had to go and find someone else to reassure me that it was.

We were out of the suburbs of Roma very quickly. After leaving behind the Frascati hills, we were accompanied by hills rising sheer from plain; behind them an occasional, hallucinatory glimpse of a much higher mountain.

Crops: grapes, olives, peaches, nuts, citrus, loquats, some vine I didn't recognise. The bean harvest is over. Also there were artichokes in full flower, maize, pumpkins(?); and rolled-up straw on harvested cornfields.

A small castle atop a rock pinnacle, seemingly without access. A new high concrete viaduct with the old road meandering below. Escher-like towns covering hills.

Vegetation: tall rushes, some multi-leaved tree I don't know (but very common), figs. As pines and cypresses were left behind, prickly pear appeared. Purple vetch, yellow ragwort coloured the earth.

First view of the sea, from far and high: very blue, sparkling, with small sails. Then the train drove straight at an abrupt mountain wall, and plunged into a long tunnel; suddenly there were no more mountains, and we were in a dead flat plain with slow rivers, and a yard full of Zebu-type cattle.

Napoli from the train: a dreary city somewhat redeemed by oleander, morning glory and bougainvillea, and the looming cone of Vesuvius in the haze. For much of it we were underground.

Second view of the sea: closeup, murky and industrial along the Bay of Naples.

Vegetation now included oak, eucalyptus, and lots of broom. We ran through wooded valleys with fast-flowing streams and chestnuts trees, under towering cliffs; then drier, broken hills. The highway was much more intrusive than the railway, running on long viaducts and shaving off whole hillsides.

Then the best bit of all. A long stretch through half-tunnel, half-gorge. Glimpses of the small river far below and the hills far above, then back into the dark, through a projecting cliff.

Out of the gorge into flatter, drier country with cereals growing, and a large eagle hovering at Bella-Muro.

Potenza, 13 - 19 June

13 June

Arrigo was waiting right where I stepped out of the train. He was picking up two more people as well, and drove us up the hill to the hotel. Dinner at 8:30, just time to put down my bags and wash my face.

Good cooking (pasta, delicious vegetables with pork, a fresh apricot), but rather slow service, and we were barely finished when the waiter turned us out. Most of the summer school had been at the sea and all looked like lobsters. Afterwards we indulged in the local social life and visited the ice cream shop in the lower town.

Potenza is the only town I know where there is a lift, six escalators and a flight of steps to take you between the two parts of the town. The hotel is near the top, just along the street from the Governor's residence. There are probably spectacular views, but I haven't found them yet.

I went back early to prepare my first lectures.

14 June

The air at night is cool but the town is noisy. I woke at 6am when something buzzed, and then went back to sleep until 7am. Showered, made some more notes, went down to breakfast. Nothing is laid but the waiter brings croissants and coffee from the bar. Then I went back, wrote more notes, and headed down to the University, down seemingly endless flights of steps.

Pleasant campus, pines and rosemary. Lots of students at the summer school. I started gently, covering less in my first two lectures than I had meant: classification of quadratic forms, and coding theory as far as the Hamming codes. The students are working hard, solving or debugging my exercises. One of the nicest questions concerned the left and right radicals of a bilinear form. I'd given as an exercise to prove that if one is zero then so is the other if the space is finite dimensional, and had remarked that this is false in infinite-dimensional spaces. I was asked whether, nonetheless, it holds for continuous bilinear forms. I managed to come up with a lovely negative example.

Let V be the space of infinite sequences of real numbers which are constant from some point on (let the limit of sequence v be s(v)), and define B(v,w)=sum(vi-s(v))w. Obviously continuous, left radical consists of the constant sequences, right radical zero.

Then a lecture by Gabor Korchmaros about recovering classical unitals from large subgroups of their collineation groups. He asked me whether a list of maximal subgroups of PGU(3,q) exists (only PSU(3,q) is in the literature). Of course I could only answer: it is possible to work it out.

Gabor was lecturing because Tamas Szönyi (who was supposed to be in harness with me this week) was called back to Budapest because his mother is seriously ill. Then after the lecture fate struck again. Another of his Hungarian replacements, Gyorgy Kiss, got a phone call to say that his mother had died suddenly. He is very bravely going to give his two talks tomorrow, but then will go. He looked absolutely shattered at lunch today.

So another part of the replacement will be a problem session, and I am now going to assemble a few problems before going for a run. I read my email; nothing urgent, but a nice problem from Yair Caro: if the order of an abelian group is congruent to 1 mod 6, can you partition the nonzero elements into triples with sum zero? Yes, if the group is elementary abelian: just take cosets of the group of cube roots of unity in the field.

15 June

More on Yair's problem. There is an easy direct product construction. More generally, if the group has a homomorphism whose image and kernel have the property, then the group has the property. Thus, any p-group works if p is congruent to 1 mod 6; and, if p is congruent to -1 mod 6, then any p-group having a chain of subgroups with elementary abelian factors of order p2 works. A nice combinatorial argument shows that these are just the p-groups where no cyclic factor is bigger than the product of all the others.

The hotel is called "Tourist Hotel". It has a large crucifix above the reception desk.

The dinner situation is that it takes three-quarters of an hour between our order being taken and the first food appearing, during which time there is nothing but water and wine; and then the sweet course (last night, profiteroles and small pastries) is put on the table before we have finished eating the main course. But the waiter took tonight's orders last night, so perhaps they are proposing a more efficient system.

My run was most unsatisfactory. I got caught up on busy roads with no footpath at all, and then going back up the hill, when there was a footpath it was intermittently used for parking, so I was forever dodging and weaving. Despite this, I felt very good after the run.

We went for ice creams again. I didn't do my Cinderella act, but walked up the stairs with the others. As we reached the top of the long 147-step flight, all the clocks in the upper town began chiming midnight.

Several people have mentioned to me that Potenza is in a seismically active area. I trust that the buildings are constructed to cope with this!

I have now five problems for the problem session, most of them triggered by things people have said in lectures or questions they have asked me.

My run did some damage to my right Achilles tendon - a recurring problem at the moment. So I am limping a bit, but I don't think it's serious.

Mid-afternoon, I set out to walk the path by the river that Arrigo told me about. First, it took me quite a while to find it. The only crossing of the railway line I could find was a high wide and busy road bridge with no footpaths. Then I doubled back to the river, and saw the path below me fenced off with a high fence, and went some distance along the road before deciding there was no way in. Finally, I went the other way, and almost at once found the entrance. The path itself was separated from the river by a fairly high bank, and there was a smell of sewage, but at least the path ran without interruption.

It ran through an area of wildflowers, mostly yellow and pink-purple, including mallow and sweet pea, and also lots of fennel.

But not very far. It came to an old stone bridge which had been rebuilt and then superseded by a road bridge, and ended there. I walked over the bridge, and set off for home another way. At a level crossing on an abandoned-seeming railway line, along came a train pulling a German freight wagon. It was a three-man operation: one to drive, and two to stop the traffic on the roads they crossed. I walked on, and encountered them again at another crossing (the train was going quite a bit slower than I was). Finally I crossed the line and headed uphill, eventually finding the hotel after some messing about on top of the hill.

16 June

Before dinner, I had quite a long session with Serena Moscadelli, going through progress on her problem. She is trying to prove that if a group is a product of an abelian and a cyclic factor, then the rank of any chief factor is bounded by the rank of the abelian part. Surely true! She has proved it under a number of different assumptions, most notably that the intersection of the factors is non-trivial. I thought last night about how to finish, so far without success, though I have made a little progress. I also skipped the ice-cream outing to finish preparing this morning's lectures.

Lectures went well. I decided on the fly to put in my "one-line proof" of the upper bound for the number of bilinear forms in a linear non-degenerate set, and it worked; the students liked it, but liked the treatment of extraspecial 2-groups even better. I was quite surprised by how many people hadn't seen this before. The third lecture was given by Guglielmo Lunardon, who arrived this morning.

As we came out for lunch, there was a very slight rain falling, just enough to bring a coolness to the air. I had two cartons of water with my lunch, something I should have done before; I get thirsty in the afternoon, especially if I run! Then after lunch I went back to Yair's problem, and did the cyclic group of order 25 with bare hands. So it works for any abelian group with order an even power of 5. I am now convinced that his conjecture that it is always possible is true.

I tried to count the steps this morning, but took a wrong turning somewhere. This afternoon, I made it 389 steps.

Two generalisations about Italians which contain some truth: they walk slowly, and they eat quickly.

I ran down to the river and along the path. I tried the other direction: the concrete blocks stop but the (very overgrown) path continues. I got stung and scratched but managed to avoid putting my foot into a pothole. There was some real running in places, a change from recent form! The final complement: back at the hotel, Philippe and Alice asked me a question about my lecture on the way upstairs, clearly not considering me too out-of-it to answer.

Before dinner I had another talk with Serena; I told her what I'd done and, at her prompting, managed to settle the case where the cyclic factor has prime order. Then the long-awaited storm came. There was a lightning strike very close, a terrific crash, setting off various alarms and the phone in the hotel lobby and crashing their computer.

17 June

Seven down, two to go. I did better on quantum codes than at Brunel, mostly because I had twice as long and I had done all the background. Now I won't really have time to do Z4 codes properly, so I will have to end with some handwaving, I think. Of course quantum codes raised a lot of discussion! Then I presented five of the nine problems at the problem session.

Another refreshing shower at lunchtime. The lunches are really excellent for a student canteen: today, lasagne, and roast chicken pieces in rosemary with beans and green salad.

After lunch, we had a little working group: Gabor, Arrigo, Angelo, Domenico and I talked about the problem of 1-factorisations in which every 1-factor is an automorphism. Gabor had almost proved (and I finished off) that if n is congruent to 2 mod 4, then n=2. My conjecture is that one of the 1-factors is central in the group they generate. This would allow the possibility of induction. If the induction works, at least it would show that n is a power of 2, and at best it could give a complete classification. Anyway, if the group is imprimitive, then any block of imprimitivity has an induced 1-factorisation with the same property, so as a first attempt we should consider primitive groups. Some serious difficulties remain!

Then Arrigo and I walked up the hill together: he had business in the upper town.

Before dinner I went for a little stroll around the old part of town. Some buildings in a sorry state - no doubt damaged by the earthquake and never repaired - but many nicely restored. I saw a church with a lovely old carved wooden door, with figures of monks, naked women, pipers and devils along with the geometric floral patterns. At another church, a priest was blessing passers-by. At one fountain a boy was drinking; at another, a boy was washing his hair. All, of course, ruined by cars, driven aggressively or parked obstructively in the narrowest of streets.

After dinner we went for gelati again. Just down the street there was a show: just a couple of karaoke artistes in spangles and a magician quartering a woman, low-grade stuff, but we jigged and congaed around and enjoyed ourselves until they shut down for the night. This time, as we reached the main square in the upper town, the clocks were chiming half past midnight. We were adopted by a dog, who saw us safely back to the hotel.

18 June

The corridor where the students sit to work has a big sign saying "Vietato fumare" on one side, and a row of ashtrays on the other. Which message do the students notice?

I got quite a round of applause at the end of the course. I do think I understand the stuff a bit better myself now! Several Italian students thanked me for my clear explanations, from which I deduce that Italian professors don't always bother with explanations (true in some cases!)

After lunch, and another inconclusive session on 1-factors (in which at least we decided that, if Baer's theorem says what we think, and we have to reduce to the primitive case and hack through O'Nan--Scott, at least we don't have to worry about the affine case), Gabor took Angelo and me for a drive.

As we were leaving the University we saw Serena and offered her a lift. She wasn't going anywhere in particular, so she came too.

Impressions of the trip:

At dinner, there were just nine of us left: the four Belgians, four Italians, and me. With an indiscriminate mixture of French, Italian and English, we got on very well, and the two waiters joined in the fun. The meal finished with watermelon.

After dinner we went for ice-creams. The same performers were on in the street, but the show was quite different this time. There were two live musicians, a keyboard player and an accordionist, and the lady in spangles was acting more like a dance caller, even going so far as to play mazurkas and quadrilles. The atmosphere was so informal that small children would climb on stage and ask her for her photo, and an old man went up and wandered aimlessly around the stage playing his mouth organ: nobody could hear him but nobody seemed to mind. We enjoyed ourselves greatly, joining in everything.

They stopped just after midnight, and we went back to catch a few hours' sleep. (The Italians and I are on the 07:48, the Belgians are taking a later train. There seems to be just one taxi, which will make two trips for the five of us.)

Return, 19 June

And departure. I set my alarm for 6:30 and, of course, woke at 5:30 from dreams in which I was very angry with various people (for good and not-so-good reasons), but slept again until 6:20, which was time to turn off the alarm and get up. Breakfast was available already, but by 7:10 there was no sign of the taxi, so an old guy from the hotel drove the first group to the station. The two girls paid him but wouldn't let me contribute.

It turns out that all of us (including another girl and a friend of one of the Potenza team) are in the same carriage. So, in a gentle, sleepy way, the party continues. It is a clear day except for haze on the hills, and the views are very fine.

A good sight of Picerno: a main ridge and two outlying hills, one hill crowned with a round tower and the other with an old village, new building spilling down the slope and along the ridge.

Luca Giuzzi told me that the first summer school he attended (as an undergraduate) was at Bella-Muro. Since the Eurostar stopped there, he had assumed it was a large metropolis, and he had been somewhat disconcerted to find nothing but the station and the conference centre in the middle of nowhere. He was reassured by the arrival of a car with Hungarian number plates.

Different impressions this way. The gorge, a mixture of intimate and panoramic glimpses between tunnels, the mountaintops covered with soft cloud; fields of grey-green olives on opposing hillsides; cliffs topped with red-roofed houses; a girl having a very loud conversation on a mobile phone and then (to everyone's relief) losing the signal in a tunnel; the Bay of Naples, and the muddy-brown colour explained (an optical illusion caused by the muddy-brown netting and railings between train and sea).

By a bit of seat-swapping, Serena came and talked to me about the possibility of doing a Ph.D. in London. I chatted with her and Luca the rest of the way to Roma (hence not so much observation).

We were about ten minutes late into Termini. I phoned Dina (under some difficulty: the phone was clapped out and her voice very faint, and the announcements of departing trains were almost non-stop). It turns out that she is having a new floor laid today because of the leaking pipes, and has to stay at home.

So, after buying my ticket from a machine (which asked me for too much information, in English), I went to the station restaurant for lunch. What a difference from Britain: from a very wide choice, I took penne with salmon, beef and olives with zucchini, a creme caramel, and a can of beer. Extremely tasty and filling, for about ten pounds. Then I went to find the train: a long walk to the end of the last of 22 platforms on the station, past building works. Only one train an hour! But at least it is non-stop - though that is because it hardly gets started, limping and crawling along and taking 40 minutes for the journey.

Two strange events at the airport. At check-in, the girl looked at my passport, typed "Sydney to London Gatwick" into the computer, got no response, and finally gave me back my passport. Then at the security check, the man in front of me was repeatedly setting off the metal detector. So I waited to put my bag on the belt until he was sorted out. The man behind me got very impatient and tried to push me through. The result: when the time came, I went straight through, he triggered the metal detector. Revenge was sweet.

I had looked at belts in London, hoping for a replacement for my disintegrating collection, and found that the cheapest to be had in the duty-free shops was 25 pounds. So I looked again in Roma, and found it was twice that. Roll on the end of duty-free: that is an utter rip-off. Eventually I found one for twelve quid and bought it. The cashier put it in a bag which he closed with tape saying "Do not open until after takeoff". Stupid - I slipped it out of the bag and put it on.

A long wait for boarding. I finished "Mariette in Ecstasy" and yesterday's Times crossword that someone had abandoned. The flight was delayed an hour and a quarter. The airline blamed traffic congestion, but the pilot told us that the crew had been delayed on an incoming flight and apologised on the airline's behalf.

Visibility was good when the clouds weren't covering. I saw clouds reflected in the sea, the Alps with almost no snow, farmland near Paris, and both shores of the Channel. The food, though, was lousy. Despite my passport having been inspected three times by Alitalia staff, I had to ask for a landing card.

Budapest, 3 - 10 July 1999

3 July

Woke at 6:10, quite tired after a heavy week. I was out of the front door by 6:35. Everything very quiet; little traffic, few people on the Tube. It all changed when I changed to the Piccadilly line, crammed with travellers and luggage. But still to the airport in good time, and nothing worse than a long queue at the security check.

Once we were on the plane, they announced a delay for air traffic control, but I had two crosswords to keep me occupied. The Guardian crossword's theme was "written by du Maurier, shot by Hitchcock", so we had "The Birds", "Jamaica Inn" and "Rebecca". But I do think it's a bit unfair to expect us to know where Rebecca came from! (Seems it was Manderley.) I didn't quite manage to finish it, and I dropped and lost the plastic bit of my pen, without which it is not much use.

Not bad lunch; better than Alitalia managed last month, with French Chardonnay. Clouds over England and France, but fine views once we reached Germany, mixed fields and woods with the occasional road or small town, mountains in the distant haze to the south, and flocks of small clouds safely grazing. We flew high (37000 feet) and fast (with a tailwind) trying to make up time. Further on, more clouds, fields and towns, less forests, and first sight of the Danube.

We landed and I got through the airport without any hassle. I like the way they have exchange, hotel reservations, and minibus ticket counter all in the baggage hall. The minibus takes you to any address in Budapest for the princely sum of 1200 forints (slightly more than three pounds), or less if you buy a return. The down side is that you may have to wait for a minibus - about twenty minutes in my case. Then off along the airport road a mixture of old boneshaker, new superhighway, and detours for the bits in between. All of a sudden, the 49 tram to Deák tér; the patterned tile roofs, characteristic bulging church spires, and narrow streets of downtown Pest. I was second last off: we crossed the Petófi bridge, turned down a couple of side streets, and there I was.

The Professor's Guest House occupies the top floor of a large student residence belonging to the Technical University. It's quite luxurious: my room is split level, with the bed on the gallery and everything else downstairs.

After checking in and changing, I went for a walk up the Danube and back. First I walked to the Academy of Sciences, partly to see if anyone was around, partly to see how long it takes (just over half an hour). Then I carried on up the Pest bank until my way was blocked by a tributary just above the Árpád bridge, came back and crossed the bridge, and passed many places I knew: the train to Szentendre, the Victoria Hotel, the trendy shops of Pest, the Mathematical Institute, the University. I checked the restaurant opposite the Institute, where Laci and I had invented the Pisa conference, but it was very crowded.

By then my tongue was hanging out. Just after the Institute I found a little shop and bought a litre of water, which I polished off in a little park opposite. Then food was the next problem. Just before Burger King became the last resort I found a little restaurant, completely empty. I had goulash soup, then chicken stuffed with cheese and apricots, with chips and a beer. Good but not perfect: the goulash was tasty but the meat wasn't hot through, and the chicken was a bit dry. Very cheap too: 1820 forints the lot.

When I came out, lovely soft twilight colours were covering the Buda hills. I walked back to the hotel, meeting Jarik in the corridor on the way back to my room.

Observations: I've not been here in summer before; a surprising number of people go naked. Also, when I first came, the third McDonald's had just opened; now there is everything from an Irish bistro to an Australian restaurant.

4 July

It was hot when I went to bed. Later a cooling breeze came, but stopped before dawn.

I woke early and couldn't sleep. So I got up and watched the sun rise huge and red over the rooftops. Then, of course, I fell asleep and had great trouble waking up for breakfast - I kept dreaming that I was getting up. But two cups of strong coffee at breakfast (when I finally got there) did the trick.

After breakfast, once I had gathered up the energy, I set off up the hill. I walked round the back of the citadel and joined my familiar route by the underpass. Then it was a steady climb, past the hairpin bend under the cliffs of Sas-Hegy with the little shop on the corner; past the tram circle at Farkasréti tér, with its row of flower stalls; walking up the hill past the cemetery to the 53 bus terminus. (I remember this walk for the fact that I was here when the first hard snow came one night, driven on a biting wind. What a contrast today, the sun blazing and the berries ripening on the rowans.) The old bus shelter with its R.E.M. graffiti was gone, replaced by a new one in modern public transport style. But the hazy panorama of Budapest was not much changed.

I tried to find my way to 49 Kazmir utca, where I lived for three weeks in the garden room. The roads seemed to have altered, with all the new building going on. In particular, the steps seem to have fallen into disuse; presumably the nouveau riche don't walk to the bus stop! I did find it eventually.

Then up along the forest path beside the house, a long climb among the trees until it emerged into an alm rich with flowers and butterflies, familiar and strange. The most noteworthy wildflower had green pointed leaves, shading to a rich blue-purple approaching the growing points; but these were not the flowers, since yellow petals grew from the leaf-stem junctions further down.

Almost immediately I was among people again, and soon came to the restaurants and shops at the top of the cogwheel railway, where I had pork in paprika with noodles, sauerkraut, bread, and beer: very good!

After lunch I sat in the shade by the water fountain for a while, sweaty but relaxed. I thought about Serena's problem, and realised that my progress had been illusory: any subgroup containing one of the factors of G trivially is itself a product.

There was nowhere to buy tickets for the cogwheel railway at the top, so I walked down, following the line of the railway, along a street shaded by yellow-flowering acacia-like trees and white-flowering bean trees. Then through a park where, along with the boys playing football, there was a model being shot in a provocative pose pressed up against the netting, one of the acolytes holding a big drumhead-like reflector to illuminate her from below; and old men playing cards at a couple of tables.

To Moszkva tér, with its banks of tram stops, and up to the castle hill. I remembered as I walked that I had made notes on this part of Budapest on my first visit, but they were in my bag which got stolen from my office, and I never tried to reproduce them. Anyway, this is a very picturesque area!

I walked past the Hilton Hotel engulfing the old priory, and past the Matthias church, to Fisherman's Bastion. But the steps down were closed, so I had to go on through the really touristy parts until I found another way down, which took me right to Clark Ádám tér at the end of the Chain Bridge.

I crossed over and found registration in full swing at the Academy, with various old friends including P cubed and Vera Sós. They were selling 7-day travel passes so I bought one, and tested it out by taking the tram back to Szabadság bridge. In the hotel, I had a very welcome and refreshing cold shower and lots of water.

Back on the tram to the party: very hot and stuffy, very crowded, lots of people there I know. The food and drink didn't last long, but a rumour came round saying that on the next floor up, in the art gallery, it was cooler and there was more to drink. So I tucked in again. In total, not really enough for dinner, but enough that I didn't feel the need to sit down to a meal. So back to the hotel. The twilight was fading, but the lights on the Chain Bridge hadn't yet come on. In a street near the hotel, two bats who couldn't fly very well.

A custom I've seen twice in two days: wedding parties drive in convoy, sounding their horns.

5 July

The conference organisers have forfeited some of the goodwill by various shambles. The opening session started late; then coffee was re-scheduled twice, and when it came, the serving facilities were quite inadequate for the numbers of people; there were no signs for the lunch room, and I and others were directed all over the building before we found the right place; lunch itself was somewhat on the light side, and then they charged for coffee; and every lecture room has two different names in English, the one in the program bearing little relation to the one on the signs. (Algebra is in the "Round Hall" or "Lecture Room", for example.)

I learned almost nothing from the preliminary talks about Erdös. Predictably, Bollobás didn't like Hoffman's biography; he doesn't realise that it is no easier conveying the creation of mathematics to the general public than that of music or art. Dvoretzky gave a more balanced judgment: he said the title, "The Man who Loved Only Numbers", angered him until he realised that it was at odds with the content of the book.

Then Noga Alon gave a very clear talk on the probabilistic method. (I'd never seen the probabilistic proof that graphs with arbitrarily large girth and chromatic number exists, but Noga fitted it onto a single slide.)

After lunch I was on first. Not my best talk, since I had no real fireworks to show, but it went OK. The room had a skylight, and the sun played havoc with visibility of the OHP screen until we moved the furniture around. The walls were lined with nineteenth-century academicians (very off-putting). Then Peter Paul Pálfy, and Laci Pyber (who has tentatively agreed to come in with me on the paper).

A nice question, from a Hungarian whose name I didn't catch. Is every residually finite (countable) group embeddable in the group of permutations of the integers which move every element only a bounded distance? (Perhaps one has to allow the integer lattices of arbitrary dimension.)

I got through the entire afternoon without falling asleep (only just!) After the last talk I read my email (two of which I knew about already), then wandered into town.

At a corner I ran into Laci. We had an ice cream, and he told me about a world music festival in the open air tomorrow and Wednesday. Earlier in the day I had been to the record shop looking for something by Makám, without success - and here they were. I shall have to go. We tried to get a ticket but the agency had just shut. I will try again tomorrow. Laci seems to be getting cold feet about the survey, unfortunately.

I went to a posher-than-usual restaurant for dinner. The food was good: lamb and tarragon soup, and medallions of pork with ham and asparagus. But they disgraced themselves in a couple of ways: they forgot to bring my beer; and they overcharged me. I revenged myself by leaving a 1% tip. (Of course, the sums were so small that it was hardly worth worrying.) After dinner, the sun was sinking behind the hills and its rays were catching a knob of cloud high in the sky.

6 July

A mulberry tree near the hotel drops unripe white fruit on the street. Further along, on the riverbank, ash-like trees carry heavy clusters of red and yellow seeds.

Peter Borwein gave a lovely talk interspersed with cartoons. My favourite was the research institute with two signs: "Unanswered questions" and "Unquestioned answers". He asked, "If Erdös had got the job at the IAS in Princeton, would he have touched our lives in the way he did?"

At lunch I went to the agency to buy a ticket for the World Music Festival. On asking for a ticket, I was a little disconcerted to get the reply "How much?" I thought they would know that! Turned out he meant "How many?" It also turned out that the park is very near the hotel (in fact, I think it is where I used to change for the alternate route to work.) So I will spend Friday evening with Csaba, and, with the banquet on Thursday, that fills my social programme.

After lunch I read my email; there was one from Willem Haemers requesting an abstract for the Seidel meeting. Fortunately I had written one, and merely had to re-type it via telnet. Two curiosities about the computers: they are called "Coax" (is this how they have to be treated?), and the keyboards have sun and moon buttons (I suppose for using them by day or night).

From the afternoon's lectures I learned, in a lovely talk by James Taylor, that I should know more about Galton-Watson trees. (The trees I am interested in are almost never G-W trees, but the techniques might extend.) But he described things I had believed but sweated over as "trivial".

Hungarian drivers only stop at pedestrian crossings in two situations: in traffic jams, and when there are so many people on the crossing that to continue would cause carnage.

According to the daily bulletin, there is a session on combinatorial number theory tomorrow afternoon, the first talk being on sum-free sets. That simplifies things; instead of my having to decide between Szentendre and a run, the choice is now made.

After the last talk I sat peacefully in the square with the ice-cream stall, drenched in late-afternoon heat under listless plane trees. I sat on the steps of the fountain. A breeze sprang up and cooling spray drifted over me. I watched people go by: girls in platform shoes, youths in trainers, two men in identical dark trousers, white shirts and red ties eating ice cream, a girl reading "War and Peace" in Russian, a pigeon and a sparrow scrounging.

Then to P cubed for the party. And plans for tomorrow have changed again: I have agreed to spend the afternoon drinking beer with two group theory students, at Csaba's instigation. So what did I do but spend half the party preaching to them on the random graph, random sum-free sets, etc. Here is an interesting question: is there a transitive torsion-free permutation group in which every cycle of every element is finite?

The other change of plan is that I am invited to lunch with Csaba and Eva on Thursday. I finally got to meet Eva at the party; we'd both heard so much about the other, but despite that we got on like a house on fire.

And so home on the 49 tram from Deák tér, no change needed.

The weather forecast for tomorrow is cooler, chance of rain.

7 July

Definitely cooler today. Rumours of a storm at 16:00 this afternoon; I took a coat! I walked over the Petófi bridge for a change, and all the flags were standing out straight.

The tram took me past Nehru Park, where a girl was lying on a bench with her leg in the air, and the old warehouses where there used to be a flourishing market (now completely defunct it seems).

The view across the river was unusually fine: the Technical University (which I'd only seen locally before), Gellért hill, the Castle, the Matthias church, were shining in a splendid way. What a beautiful city!

A lovely lecture by Joel Spencer, ending with his personal tribute. The other two lectures were similarly excellent in different ways: Lubinsky exuding enthusiasm; Elliott, an extraordinary performance getting inside Erdös' achievements in probabilistic number theory. So far, the plenary lectures have attained a very high standard.

After lunch I went to the combinatorial number theory session. Unfortunately they had reversed the order, so I had to wait to hear Yuri Bilu talk about counting things like sum-free sets. (He described the problem as the Cameron-Erdös conjecture, but said it was due to me but Erdös had got interested in it and talked about it.) Nothing new on the original problem, but he can solve one of the generalised versions.

This made me late meeting the group theory students, so we didn't set out until after 3:15. We went to a bar near the University where they only had Amstel on draft, but we ate there (they hadn't had lunch yet, and I had a plate of goulash and dumplings to keep them company). After that, we went looking for (and found) the Crazy Café, where there is a much wider choice of draft beer. We had a good time, and I suggested a couple of problems which they found interesting. I also explained the basic facts about the random graph, as well as the B-group problem.

Then it was time for the music festival. I arrived a few minutes late, Makám had just started playing. There were eleven people on stage, including the singer (somewhat in the Marta Sebestyen style) and a didjeridu player who was only on for the first number. The others were multi-instrumental, playing a range of things from violin and saxophone, through cembalon and twelve-string guitar, to all manner of strange percussion including a terra-cotta water jar and something that looked like a sink plunger. The first number was four square, but I needn't have worried; they hit their stride with a number which started in 5 and moved seamlessly into 13, and we also had 7, 9, various 3 against 2 rhythms, and so on. Lovely stuff. There were two of their CDs on sale; completely unable to choose, I bought both.

Next was a Zimbabwean group, Stella Rambisai Chiweshe, three men in matching Hawaiian shirts and dark trousers and a woman in all-over shiny white. They played mbiras, a standing drum, and maraccas. Interestingly, the drummer improvised rhythms in much the way that a jazz musician improvises melody; the rock-solid foundation was provided by the maraccas.

Soon after they started, the sky began to darken. Then what had seemed to be camera flashes turned out to be lightning. The wind sprang up, and the backstage crew ran round furiously securing things (even climbing in the scaffolding). Then the rain began bucketing down. The musicians had played on bravely through most of this, but the downpour brought them to a halt.

I decided that there wasn't much prospect of the storm passing quickly; so a bit reluctantly I decided to miss the chance of hearing Mynta (Indian-Swedish group including Zakir Hussain's brother), and walked back to the hotel through the dramatic lightning and rain, miraculously managing to turn down the right street.

8 July

A sparrow, sitting on the bridge under the railings, takes off and plummets down towards the water far below ...

I took the Petófi bridge route again, hoping that the storm might have freshened the city and made it even finer. But thin cloud veiled the sun and made everything look a bit dim.

Ron Graham's lecture was a first: it was the first time I had seen a good lecture given using a computer. Two things were specially nice. In the bulk of the lecture, he used the computer like an OHP, except that, instead of covering up parts of the slide, he could make them appear as required; each paragraph appeared in large type and then shrank into regular type when the next one came up. Also, he did geometric demonstrations very well, the picture growing with the argument.

Richard Guy's introduction was also excellent. As he described it, a bad joke, and two Erdös stories, one trivial, the other not. But it was clearly a heartfelt tribute.

The other two talks were not so much to my taste, for different reasons. Miki Simonovits talked about extremal graph theory, not my favourite topic; and Andras Hajnal's talk was given by Peter Komjáth, and involved a lot of reading from Hajnal's article in the Graham-Nesetril volume, which I have read (even the story of Erdös climbing the church tower in Szeged).

After the talks I went to Csaba and Eva's place, right on the edge of Budapest (in fact, Csaba said that the Budapest authorities denied its existence for some time). It is right next to a conservation area; there is a large road which goes nowhere, since the rich folk on the other side of the Danube have successfully resisted its incursion.

We had a most enjoyable afternoon (and evening) - delicious food and drink, good talk, a not totally successful attempt on my part to charm their 14-month-old daughter (I had better luck with the two boys, who played Three Little Pigs with me in the role of the big bad wolf), and even a little mathematics.

When I left at 23:00, I had almost missed the last bus. Fortunately not quite, and it turned out that, after going to the metro stop on its regular route, it turned into the night bus replacing the metro, and took me all the way to Kalvin tér, (though very slowly), from which it was only twenty minutes walk home. I got back about 00:15. Sad to say, I had drunk so much that I didn't sleep very well.

9 July

Rain this morning at breakfast. It had stopped when I left the hotel, so I decided that I needed the exercise and walked to the Academy.

I noticed that the Chain Bridge is Lanczh¡d on the masonry supports, but Lanch¡d on modern maps. Maybe Hungarian spelling has changed.

Beautiful entertaining lecture by Carl Pomerance - the clearest exposition of Erdös-Kac I have ever heard, and the famous Hank Aaron incident (which was his own introduction to Paul) illustrated with a photo of the honorary graduands on that occasion. Peter Komjáth then gave his own talk which was mainly a catalogue of theorems but included one stunner at the end. A result proved using CH is then applied, via a forcing result which puts any model into a CH model, to showing the existence of a finite K4-free graph such that any 2-colouring has a monochromatic triangle. He has no idea how large this finite graph is from his method. Surely a challenge here for proof theorists). Ron Graham mentioned in his lecture that the smallest such graph known has billions of vertices, but it is not implausible that one with a hundred vertices exists.

After the talk, Yair Caro brought me up-to-date with what he has done on the abelian group partition problem, and Laci Pyber gave me his comments on my conference paper (and explained why, as things are in Hungary now, it is better for him to have four citations than one joint paper (citations are counted, but not in your own paper!)

Then I headed back to the hotel for a nap. On the way I stopped off to look at the market in its newly refurbished hall (the building with the green and yellow tiled roof by the Szabadság bridge). Wonderful displays of dried peppers, garlic, and sausages, their parchment shades completed to the Hungarian colours by heaps of beautiful vegetables. I succumbed and bought some paprika and some goulash cream.

The hotel lift has parallel mirrors, so I always feel there are crowds of people in with me but keeping out of sight. (But when there is someone else too, his clones don't hide.)

Of course, I didn't sleep, so I started working through Laci's corrections, accepting most of them. Then it was almost time for Miki's party, so I showered and dressed, and decided to walk over Gellért hill.

I'm told that there is a field somewhere outside Budapest where the statues of the Communist era have been put (lots of Lenins, etc.). No doubt it will become a big tourist attraction, if it isn't already. But they decided to leave the one on the top of the hill - it is certainly the most prominent Budapest landmark.

Down the other side takes longer than you think, since the paths wind so, and many were covered with washes of mud from the storm. So instead of continuing past the Castle, I walked over Elizabeth bridge to Astoria and took the Metro.

Arriving at Heroes Square a bit early, I sat in the park under a willow, seeing the waiters bustling among huge flowerpots serving people on the lamplit terrace opposite, and twilight playing among the labyrinthine clouds, and hearing the sound of the fountains and the peeping of a lone duck.

A super party. The food was delicious and in vast array. Among the guests was someone I met half a life ago, Ethel Rathbone, secretary and mother-figure to visitors at Ann Arbor when I was there in 1973, now running the Number Theory Foundation and travelling with John Selfridge.

I congratulated Ron on providing me with a first. Other people around joined in, and liked the pictures growing in stages, but Ron himself singled out the shrinking text.

10 July

Checked out after breakfast. There was some delay: I asked the guy in the office to book the minibus for me; the girl wanted to know what number in Roosevelt tér was the Academy! He offered a time which would get me to the airport in time; I took off half an hour; and the minibus girl took off another fifteen minutes. So it should be fine.

I took the tram, left my bag in the cloakroom, and decided to be a set theorist until coffee break. Shelah gave the clearest talk I've heard from him, though not the clearest transparencies! Jech struck me by talking of something holding "in ZFC", rather than "in any model of ZFC" - specially striking because the property of the Boolean algebras he was describing was most easily defined in terms of the corresponding forcing extension. Then Jean Larson gave a very entertaining illustrated talk about Erdös and ordinals.

I read my email - very slow today, and several people were having the computers freeze on them - then said final goodbyes and went outside to wait for the minibus.

A hot, humid day, with enough breeze to stir the flags. The Castle in full face behind the Chain Bridge, the lady with the wreath grey in the haze, the statues in the square green with verdegris. It turned out that Professor Kanamori was also catching the minibus, and had been told the same time; it came right on time, had only one more (nearby) pickup to make, and got to the airport in half an hour (and then my flight was announced fifteen minutes late), so there was no panic at all.

Kanamori asked me to give Wilfrid a gentle nudge about an article on the history of model theory. I think the best way may be to ask him for a copy.

At the airport, through customs and immigration, I settled in for the long haul with a large lunch. Not quite my most expensive meal on this trip, but massive amounts of food: tuna salad, huge pizza, watermelon, and a bottle of Leffe (why not float out of Csaba's hometown on Belgian beer?).

Off at last, a bit late (I'd almost finished the Telegraph crossword before takeoff), in a table-less seat by the emergency exit. It was pouring with rain as we left, and quite bumpy at first, until we came through the first cloud layer into a little slit illuminated from the west. Then up through the second, gentler, clouds and out into brilliant sun - the magic of flying, I never tire of it.

Visibility later was better: western Europe blue with little clouds floating; the Thames estuary razor-sharp, from the Goodwin Sands like some giant smear in the blue water, Canterbury where I will be tomorrow, Southend with its pier and two stations. We flew to the north of central London, but the major features were clear: the M25, the Thames Barrier, the Dome, Tower Bridge. And approaching Heathrow from the west, a splendid view of Windsor Castle.

I did my good deed for the day by helping Frank Harary get to Victoria: minding his luggage while he went to the gents', giving him the fare to avoid joining a horrendously long line. (Of the two machines which can take notes, one said "Exact fare only", the other "Closed". Back in London all right.)