Columbus diary, March 2010

In March 2010, a conference was held in Columbus, Ohio, to celebrate the sixtieth birthday of László Babai, professor of computer science at the University of Chicago. The conference was organised by Laci's student Akos Seress at Ohio State University in Columbus.

I have known Laci for more than thirty years. We have done quite a bit of joint work, and written several papers, one of which (also with Laci's student Péter Pal Pálfy) has been influential in both graph isomorphism and profinite groups. I very much wanted to attend the conference, but because of having a very busy term, had left the arrangements rather late.

Thursday 18 March

Not a good day.

At lunch Leonard said to me that I would have a miserable time getting into the USA, with fingerprints, retina scans, etc. Of course he is right. In the afternoon, I had a bit of time after my lecture, so I printed out the address of the hotel, the basic conference instructions, the e-ticket, etc. There were two emails from Key Travel, more or less identical, but feeling a bit paranoid I read through the fine print in the second as well as the first. Just as well.

Down at the bottom, a new regulation for people travelling to the USA on a visa waiver, brought in this January. You have to pre-register at some website. The process takes 72 hours. At the moment I read it, I had 48 hours before takeoff, a bit more before arrival. But of course, I needed my passport to fill in the form.

So I went home for my passport. Not in the usual place. Not in the second most likely place. I thought the third most likely place was in my briefcase, which I had taken to Lyon in November. (It is from the Tunisian conference; Maurice Pouzet had been there, and had also brought his, which was how I remembered I'd had it.) So back to work, since my briefcase was there. No passport, not even in the secure zip-up compartment.

So home again to do a more thorough search: pockets of coats I might have been wearing, under various piles of paper that got shifted around at Christmas. No luck on the first pass, but I went through again looking more thoroughly, and finally found it. Some good was achieved: I found various other things such as the international adapter that had gone missing, some luggage labels, etc.

So back to work to fill in the form. This asks the usual questions that are on the I-94: street address where I will stay (which I'd already printed out, of course), was I a member of the Nazi Party between 1933 and 1945 (unlikely given my date of birth), etc. I filled it in, and submitted it, and was invited to check the status of my application. This brought the only bright spot of the afternoon: my application was already accepted! No idea why they say 72 hours – just covering their backs I expect.

So, at last, I was off duty. (Apart from tidying up the mess I'd left!)

19 March

A busy day at work: lecture, office hour with lots of customers (one of whom left some lecture notes in my office), class, TLC meeting (at which Leonard got very angry about the proposed policy on teaching materials), Michael and Irene, then Bill talking in the CSG about his new zero-free interval for chromatic polynomials of 3-connected planar graphs. We had a drink at which I got a bit of feedback about the result of the external review (Oscar said that Thomas said that the Principal said at academic board that it had been very successful).

Then home to pack. I have to make an early start in the morning because the Piccadilly line is not running: I have to go to Ealing Broadway and catch a bus to Boston Manor.

20 March

I was up early, to breakfast, do a cursory cleaning of the house, and set off. I was allowing a couple of extra hours to get to the airport. I called in at work to pick up a copy of Laci's paper and read email one last time. No major panics there.

Then I ambled down to Mile End to get the Central line, to find an unexpected delay: the station entrance was closed until 12:30. So back along the road to Stepney Green, where I got the Hammersmith and City line to Liverpool Street, where I changed for the Central. Fortunately, an Ealing Broadway train came in very soon after another train, so it wasn't too crowded.

At Ealing Broadway, a station where I don't believe I have ever gone out to the street, we were directed a fair hike down the road to the traffic lights and along the crossing road. There was indeed a bus, nearly empty, but it closed its doors and left almost as soon as I got on, before I had time to sit down.

We came to traffic lights onto a road absolutely choked with traffic. There was a special lane for right turn, with just our bus in it; but the driver had to let the first change of the lights go by, since a huge lorry had just turned from the other direction and was blocking the traffic. It was a slow crawl to the Chiswick roundabout, where we got onto the A4 to Brentford, where I had walked last weekend. We had to turn right, which involved turning off, doing a U-turn, and then turning left. But finally we got to Boston Manor, where a Heathrow train came along very soon.

So in the end I was at the airport four hours before the flight left. Maybe just as well. The man at the check-in desk said that the flight was very crowded and I had no seat booked; he put me in an exit row seat. Then he didn't like the photo on my passport, and phoned the security person, who phoned the other security person, who finally gave it his OK. The man behind the desk asked me if I wanted to upgrade, for four hundred pounds; I declined. By a remarkable irony, when I got to the departure gate, I found that they had upgraded me for nothing!

I went through security and into the departure lounge. I wanted to have a proper meal, since I have eaten nothing but sandwiches for some time; but one restaurant was absurdly expensive and the other one crowded out, so it was a sandwich anyway. I bought a couple of books.

At the gate, there was the most incredible security. They opened all my bags, made me take off my shoes and undo my belt, and did a thorough body search. (Once I was on the plane, they brought me a tray with two forks and three knives, all metal.) But despite this, we were away virtually on time.

The big roomy seat was more comfortable, though I didn't explore all its positions; they provided Bose noise-cancelling headphones, which do reduce the engine noise considerably, and make it possible to listen at low volume, or simply sit with engine noise much reduced, which is much more relaxing. There was quite a decent meal, and I listened to some music and watched a couple of movies. The choice was very limited; I saw "Where the wild things are", which was under action, not children's (and indeed the monsters had very adult-like hang-ups); and "Bright Star", a real weepie which left me quite cold, and which managed to get the seasons on Hampstead Heath quite confused (wild rose was out before bluebells, and they were worrying about approaching winter while the beeches had just come out in leaf.)

We arrived in Chicago on time, as dark was falling. I had a friendly and helpful immigration officer, and a customs officer who, after being amazed at my lack of luggage, waved me straight through. The train to Terminal 3 was easy to find, and after another struggle with security (they seemed to think it was my fault that I didn't know I had to take my shoes off and put the white toy in a separate tray) and a very very long walk, I was at the departure gate with half an hour to spare.

Then the waiting began. The plane was out there waiting for us, but the rest of the concourse was full of announcements of seriously delayed flights to and from other places. (There was a lot of snow on the ground, and clearly there had been some very bad weather.) One of our flight crew was on a delayed inbound flight, so we found ourselves waiting an indefinite time until he arrived (which turned out to be over an hour). When they did let us aboard, they decided the plane needed to be de-iced, and we had another long wait for the de-icing truck.

While we were waiting, Ron Graham (sitting across the aisle from me) and I noticed each other's existence and had a long conversation, about "To Infinity and Beyond" (the Horizon television programme on the BBC in which we had both featured), reviews of mathematics, nice theorems and problems, etc. (Poor Ron, who had come from California, is due to give the first talk tomorrow).

Eventually we were on our way, and arrived in Columbus not much before 2am. Ron's bag had gone on an earlier flight which he was bumped off, but fortunately it was there waiting for him when we got out of the baggage carousel area. He is staying on campus, so we took different taxis. It was a 25 dollar ride but quite quick, and there was no problem at check-in. So finally I got to bed after 2am.

21 March

I was woken at 5am by some people talking and laughing very loudly. They sounded as if they were just outside my door, but it may just be that the hotel is not very well soundproofed. Finally they were quiet, and I went straight back to sleep, dragging myself awake with some difficulty at 7am. I took a leisurely bath and got to breakfast at about 7:30.

I was sufficiently dozy that I had great difficulty finding where anything was, but after a cup of coffee I became a bit more aware of my surroundings. I was greeted by Csaba Szabó (whom I didn't recognise at first, that is how out of it I was), and breakfasted with him and two of his students. Then Jarik Nešetřil showed up, so after the Hungarians went I had another cup of coffee with him.

The hotel desk provided me with toothpaste, and sewing kit (to mend my torn bag). Since it is Sunday, with not too many buses, they had put on two special buses for us; I made it downstairs in time to catch the second of these, and found myself next to John Dixon.

At the conference I checked in and got username and password for the university's guest network. Then it was time for the lectures.

There were two parallel sessions in the afternoon but only a single session in the morning. And what a splendid collection of talks I heard! Ron Graham, as fresh as ever despite his hard day's travelling and late night, on some beautiful new combinatorics of permutations (involving both descents, defined as usual, and drops, which are numbers which are mapped to smaller numbers). Then we had Anna Gál on 3-query locally decodable codes (a new branch of coding theory for me, clearly inspired by Laci Babai's ideas), and Subhash Khot, on "Grothendieck inequalities" for arbitrary sets of vectors, reminiscent of (but quite different from) some of Aidan's work.

After coffee, where we needed a bit of intelligence to find the way (it was in a different building all of whose outside doors were locked, but there was an internal entrance from the conference building), we had John Dixon on finding representations close to approximate representations (an old question of Ulam, which in other guise involves lifting representations from characteristic p to characteristic 0), and Bojan Mohar (quantifying Babai's structure theorem for large vertex-transitive graphs).

For lunch we walked towards the high street and found an open sandwich bar ("sandwich" meaning something covered in molten cheese). Nearby there was a bank with the intriguing title of "Fifth Third Bank", which suggests that five times the economy has grown enough to support three banks, and in each of the first four it crashed back again.

After lunch, I decided to stay in one room the whole afternoon; the talks there looked slightly less technical and more expository. It was a great collection of talks once again: Zoran Sunic using the Towers of Hanoi to construct the first example of a branch group having the infinite dihedral group as a quotient; Delaram Kahrobaei on public key exchange systems based on non-abelian groups; Gábor Hetyei who gave a wonderful talk explaining my observation on the relation between John Dixon's formula for the number of pairs of permutations generating a transitive subgroup with the number of connected permutations (via the notion of "Bernoulli games"); Tom Hayes on why "stirring" can make a Markov chain converge faster; Gabriella Pluhár (one of Csaba's students) on the maximum number of "islands" in rectangular or triangular grids (these are families of rectangles or triangles such that for any two, either one contains the other, or they are disjoint and don't touch); Daniel Štefankovič on a variant of the rank generating function (where, instead of the matroid rank of the graph formed by a given edge set, you put the mod 2 rank of its adjacency matrix), which has specialisations counting matchings, perfect matchings, and (in the case of a bipartite graph) independent sets.

At tea Cheryl Praeger, who is the editor of the Australian Mathematical Society Lecture Notes, told me that she has had referees' reports on the counting book, mostly favourable, which she will send me soon.

After the talks were finished, Laci wanted to talk about a paper we should write. He has observed that the methods we both used ages ago to prove that almost all Latin squares have no non-trivial automorphisms also show that for almost all normalised Latin squares the rows generate the symmetric or alternating group (and so, by Häggkvist and Janssen, actually the symmetric group). I had arranged to go to dinner with Csaba and his students to talk to them about infinite permutation groups; Laci and John decided to come too.

We went to a Greek restaurant, where we had a decent meal and a lot of good mathematical discussion (as well as a lot of talk about Tom Lehrer etc.). The only down side is that they appeared to add 20% service charge, which seemed a trifle excessive. I will have to find a hole-in-the-wall sometime soon!

Back in the room I tried out the internet connection. No luck. But a phone call to the desk established that they change the password every week on Sunday; with the new password it all worked flawlessly. So I copied yesterday's diary to London and read my email, before writing up today's and going to bed.

22 March

I slept well but woke up about 5, thinking about nice presentations of homogeneous structures. Raindrops were streaming down the window. Eventually I decided to get up and type up my thoughts, and email them to Laci and Csaba. I found that it wasn't even necessary to log in; the system remembered me from last night.

When I had finished, I had breakfast. I was joined by Chris Godsil who had just finished. Our conversation drifted to such dreary topics as research assessments, lists of approved journals, and so on. We were rescued by the appearance of Cheryl, but she had brought bad news: her part of Perth had just been hit by a cracker storm which had caused enormous damage. Of course she was worried about her house, of which she had no definite news yet.

After breakfast the rain had stopped but it was still cloudy and cold. I set off a little after 8 to walk to the University, not knowing how far it was. I was taking a photograph when up came Cheryl and John Dixon. It turned out to be about a 3/4 hour walk altogether; at one point when we were crossing against the lights, John remarked on the damage that would be done to permutation group theory if we had all been wiped out.

Another day of extraordinary talks. Satya Lokam talking about matrix rigidity (I knew the basics of this since my student Taoyang Wu had got interested in it at one point and we had discussed it); Lajos Rónyai on polynomials and Gröbner bases, unfortunately much too fast but he ended up with a generalisation of the Frankl–Wilson intersection theorem from primes to prime powers; then an absolutely barnstorming talk by Gene Luks. He told a story, new to me, of the time when Babai had shown him a letter I had written giving bounds for orders of primitive groups using CFSG. Laci had left to take a bus to New York and a plane to Hungary; Gene had phoned the New York Port Authority to get them to put out a call for Laci to call him, to verify that the theorem really did say what he remembered it saying.

I went back to the same place for lunch and was joined by John, who like me is averse to new experiences. I saw my former student Robert Bailey (now on a postdoc in Regina) outside the lecture room; he wanted me to think about the metric dimension of dual polar spaces. I said a few rather muddled things, but afterwards thought about it a bit more sensibly, and found that I could remove the log factor from Babai's upper bound but no more. Not really worth doing.

After lunch, Bill Kantor told the same story Alex had told, from a slightly different point of view; then Bob Beals talked about polynomial-time matrix group computation. One thing that happened in the session was that my suggestion to Laci, that speakers put their slides on a web page, has been taken up, and we have been asked to mail our slides to the conference address.

Two talks not quite up to standard, and then Laci's talk. It was a talk with the Erdősian title "Some of my favourite problems". He wrote on the blackboard so that the problems would remain visible. The first problem was whether the number of self-normalising subgroups of a group of order n is at most n^c. He promised to come back and tell us why he wanted to know this but never did. The remainder was an assortment of problems about improvements to the Frankl–Wilson bound, graph spectra, vertex-transitive graphs, polynomials with coefficients 0 and 1, etc.

At various points during the day I had run into OSU mathematicians Ron Solomon and Neil Robertson, both of whom have threatened to come to my talk.

I walked to the bus stop with Robert and waited there until the bus came, then walked back to the hotel. While I was at the bus stop, Csaba and his students came by and suggested we go to dinner and talk about homogeneous structures.

We went to a pub/microbrewery very near the hotel and had a pub meal and a few pitchers of beer – Csaba is still as good as ever in that respect. We had a very good conversation about many aspects of homogeneous structures and many related things. Finally we went back to the hotel. I flopped down and was asleep almost immediately.

23 March

I woke with a burning feeling in my throat. Is it a cold coming on, or just the hotel atmosphere? Who knows? Anyway, after breakfast I felt more or less OK. I breakfasted with Cheryl and John again, and after breakfast we met up and walked in together.

It was considerably colder than yesterday; I wished I had wrapped up more warmly.

We had György Turán talking about algorithms for Horn formulae (popular in artificial intelligence, it seems); Cheryl, on the great improvements she and Akos have made to the analysis of the black-box recognition algorithm for general linear groups (depending on some nice enumerative results for semi-simple elements); and Harald Helfgott on growth for small subsets of groups (i.e. showing that if A is a not-too-large subset of a group of Lie type, then A³ is substantially larger than A; he did assume a bit too much of his audience, I thought). In his talk I found out what has happened to Jan Saxl's Australian student Nick Gill: he is now Harald's postdoc.

At coffee, I had a chat to Ian Leary, who works here now. (I later saw his photo on the faculty board.) Afterwards I began showing my age, nodding off in the talks by Miklós Abért (who had little respect for his audience) and Madhu Sudan (who was a bit slick). Sudan's talk was how you recognise that a map between groups is close to being a homomorphism; I tried to imagine a situation where you would be given a map which was close to but not a homomorphism, how it would be defined, but gave up and closed my eyes.

I decided that I would skip lunch; large plates of greasy food no longer suit me. Probably this is why I am sleepy. So I logged on and read my email, including a letter from the Principal which didn't actually say very much, and tracked down a Bertrand Russell quote that John had reminded me of this morning. Through all this, the white toy connected flawlessly to the wireless network and everything worked without a hitch.

I had to meet Akos before the afteroon to sort out expenses. The front door of the main building was open, so I went in. What you are not allowed to do in the building is smoke, cycle or carry a gun. In the foyer, there was a lovely polished stone sculpture, a shell-like thing that might have been a strange attractor. (On the steps outside, they have various graphs and knots inlaid.) Apart from the fact that Akos had far too much bureaucracy to do, it all went fine; I got on well with the secretary, and the business was done quite quickly.

After lunch, Gabor Lippner gave a much clearer explanation of graph limits than Miklós Abért had done. Then Igor Pak gave a rather odd talk, ending up with a little rant to the effect that combinatorial enumerators want an exact result to a counting problem, but computer scientists think an approximate result is more exciting. He had earlier explained that these were his two hats, and showed a picture of a white and a black cowboy hat; but he didn't actually say which of the two was the good guy, though I got the impression he thought it was the computer scientist.

Then I swapped rooms to hear Steven Butler talking about joint work with Ron Graham on a variant of shuffling, and then Csaba talking about solving equations in groups (a bit uncompromising). Finally, after coffee, Avi Wigderson talked about Cayley graphs as expanders. I think he explained the basic theory less clearly than I recently explained it to Alex O'Neill. The talk was full of imprecision, and was all done in Comic Sans (I think), where the variable x was completely indistinguishable from a multiplication sign, and he had to use a "proportional" sign to simulate the semi-direct product symbol. Lots of very good arguments for not doing mathematics in sans serif fonts (a battle we are currently fighting with College)!

There was to be a party at 8pm in the common room, after dinner. Robert and I headed off for an oriental restaurant he had found the day before. We ended up leading a crocodile of six people. The restaurant was a bit of a shock: the food was nothing special, and the television was playing an absurd cartoon quite loudly. (What teenager would agree to the proposition that "cool is in the genes"? It would mean admitting that their parents were cool!)

Then back to the party. Akos had rounded up several students and collaborators of Laci's to talk about him, and Laci responded with a long speech in narrative mode. I was glad that he mentioned the BCP paper as one of the highlights. Then we had cake, and the promise of champagne; but things had gone on rather long, and I had a talk to prepare, so I skipped the champagne and took the first bus home. On checking my talk, I found an annoying misprint; but it was no trouble to edit and recompile the file in London and fetch it back to the white toy.

24 March

I slept later this morning. Checking my email I found that Bhalchandra Thatte had at last sent a title and abstract, so I had to post these in two places on the web. After breakfast I went down and found Bob Beals ready to walk, but he suggested waiting for Cheryl, who came along with John soon after.

On the way I talked to Bob about the construction of the sporadic simple groups. He feels, and I agree, that someone should document this in a more detailed way than either Mark Ronan or Marcus du Sauty have done, not being afraid to talk about the mathematics. He felt from my comments that the people who actually did the computing are being written out of the script. For a simple example, the groups that were called HHM, HJM, and LyS are now called He, J₃, and Ly. He said that Joe Gallian has gathered some material about this. He also talked about NSA, GCHQ, etc., where he has worked, and even told me a non-classified story about something that happened in GCHQ (but was unable to tell me the mathematical result of the computation).

It was a beautiful sunny morning, much warmer than previously, and showing the beginnings of blossom on the trees.

The first three talks, by Lance Fortnow, Chris Godsil and Scott Aaronson, were all very good. Lance confused his audience, who all understood the acronym PCP, by using the word "begat", which several people didn't know. (Later in the day, Paul Schupp talked about Garden of Eden patterns, those which do not arise from any transition.) Lance's talk was about Arthur-Merlin and related complexity classes. I didn't know that, if graph isomorphism is in P, then PH collapses.

Chris talked about some quantum things related to graphs: perfect state transfer, walk-equivalence, controllable graphs (a stronger condition than having trivial automorphism group). Of course he went very fast; I would need to see the slides at leisure. Scott Aaronson talked about quantum versions of Arthur-Merlin-type protocols.

During these talks, I decided that I should put at least a little bit of complexity in my talk; so in the coffee break I wrote a couple of frames on testing for, or finding, fixed-point-free elements in permutation groups. The latter is easily shown to be in RP for a transitive group, but the proof that it is in P requires the Classification. The white toy performed flawlessly, and I had compiled the file and fetched it over just before the start of Sasha Razborov's talk.

Sasha talked about a subject which got a lot of airing today, expansion properties of subsets of a group. He showed that, for free groups, all the difficulty is in the abelian case: a finite subset A of a free group which is not contained in a cyclic subgroup satisfies |A³| ≥ |A|²/(log|A|)^c. He also showed that this is essentially best possible, and explained more clearly than anyone else why in a nonabelian group we have to use A³ rather than A².

Then it was my talk. I talked a little about working with Laci, then went through my fixed-point-free example. This meant I was a little short of time for the main stuff, and went a bit fast at the end, but the whole lecture went over well; I think different people appreciated different parts of the lecture (always a good sign).

Robert and I went off to lunch talking about metric dimension. We ended up back in the same place I'd gone to the first two days. I had a chicken salad sandwich, but it still comes with greasy melted cheese. I do like the advertising slogan they use for their lemonade. After they explain that they squeeze real lemons, they say, "Why do we use real lemons? Well, fake lemons don't grow on trees, you know!"

The afternoon also brought some very nice talks. Gábor Ivanyos talked about finding a matrix of maximal rank in a subspace of the space of m×n matrices, and its relation to module isomorphism. After his talk, Laci got to his feet and gave a not-too-short appreciation of Gábor. Apparently, among his achievements, he has given a test for the completeness of a set of quantum gates, using the Classification of Finite Simple Groups (if I understood it right). Martin Kassabov talked about expansion properties in Coxeter groups. a beautiful theorem with lots of geometric content, clearly the right way to go about it. Paul Schupp talked about cellular automata (like Conway's Game of Life) on arbitrary Cayley graphs, whose properties are closely related to things like amenability of the group. Finally Robert Bailey talked about metric dimension, and did a good job.

I was a bit suspicious of one of his theorems, but didn't catch him during coffee, after which Laci Pyber gave his talk. Another bit of history is being re-written here: he has been described at this conference as the person who proved that the number of groups of order pⁿ is p^f(n), where f is a polynomial. Of course a much more precise statement than this was proved by Charles Sims and Graham Higman ages ago. What Laci has done (and this is no mean feat) is to get precise asymptotics, but ... His talk was on expansion properties in groups of Lie type. Apparently, by coincidence, he and his collaborator, and another team including Terry Tao and Ben Green, posted papers on the arXiv within four hours of each other proving exactly the same theorem (Laci's was first).

After the talk, I caught Robert, and we walked into town together. I persuaded him that his theorem was wrong but could be patched (to give a weaker but still good bound). We agreed to go to dinner and talk about it. While we were walking, Cheryl caught us up. She has heard more about her house: many windows, and possibly the whole roof, will need to be replaced. She doesn't get back for another three weeks because of an IMU meeting. We invited her along to dinner. Waiting in the lobby, of course several people asked if we were going to dinner, and we picked up one more, Bob Beals. I thought that if the worst came to the worst, the other two could talk while Robert and I worked.

We tried the North Market, which had been recommended as full of good eating places, only to find that it closed at 7, two minutes after we arrived. So we went to a posh Italian restaurant just up the street. It was not ideal for our purpose, since it was very crowded and the level of conversational noise was deafening. So after a while Robert and I decided to finish our work in the hotel, and we made general conversation after that. Cheryl and Robert swapped stories of having to deal with nutters. Cheryl won the competition, having once had to deal with a Vietnamese boat person whose parents had died of thirst and been thrown overboard after pirates boarded the boat they were escaping on. He reached Australia, and at one point, went looking for acquaintances who, it turned out, had just left that address. He went into the house. The owner returned and called the police. The police challenged him. He decided that the incident had brought such shame on himself and his family that he should commit suicide, so he drew a knife to slit his wrists, upon which of course the police shot him through the shoulder. Cheryl had to go bail for him, and this committed her to getting him to the court. A long and quite horrific story altogether. We also got round to talking about Peter Johnson, who now apparently is married and happily settled in Brazil.

The food was excellent, especially the bread pudding, one of the nicest I have ever had, and I treated the others to a bottle of Snoqualmie syrah from Washington State, very nice wine it was too.

Back to the hotel, Robert and I talked about dual polar spaces and got nowhere. Then Laci Pyber came along. Robert had mentioned a paper of Liebeck, Macpherson and Tent about small orbital diameter; I had said that Laci's result was probably relevant to this. So we asked him: he knew the paper, and yes, it was so.

It was getting late so I went to the desk to book a taxi and a wake-up call, and then up to bed.

25 March

As always before an early start, I had a disturbed night. At 5:10 I mis-read my watch, thinking it was 5:30. As a result I was able to get dressed and finish last-minute packing in a relaxed way. When the call came at 5:45, I went down to check out. The girl asked if I wanted her to call the taxi to ask it to come now instead of waiting until 6. Why not? I had only time to fill in three numbers in the USA Today sudoku before it arrived. By just after 6 I was at the airport.

There was a huge queue for Delta, and first time along I missed the American check-in. It was much shorter, but moving quite slowly. A flight had been cancelled and passengers were being re-booked; passengers on flights whose departure was imminent were being called to the front. But after an hour I was through, checked in to London, and on to the next stage. Security was also slow-moving, and took half an hour; I arrived at the gate only ten minutes before it was due to open.

We arrived fifteen minutes early. No need to change terminals this time, but it is a very big terminal, and I had quite a long walk; but no need to go through security again. I was at the gate in very good time. Several other people on my flight were making the same connection. The plane was going to be very crowded.

But we started boarding in good time, and pushed back only five minutes late; the passengers were aboard but we had to wait for the baggage. I am a long way back (I was boarded in the first group), and it is like a time warp: no individual screens, no free headphones, terribly cramped. I have revised my opinion of American Airlines downward.

"Breakfast" came well after 4pm London time, and did at least include a bottle of water and a goody-bag of snacks. I finished the puzzles in the in-flight magazine and read for a bit.

The afternoon and evening drew on. Dinner came: a badly-cooked pizza and a boring salad. I finished one book and started another. The occasional glimpse of our progress on the screens showed that we were running about a quarter of an hour late. But eventually it was over, and we were on the ground in London.

A very long walk to passport control. Fortunately there the queue was moving quite fast and I was not long detained. The immigration officer was very friendly; he said that he tried to persuade his children that mathematics underlies everything and they should take an interest, and he warned me that getting a stamp in my new passport would cost a thousand pounds.

Given the lateness of the hour, I decided to go on the Heathrow Express. Better eighteen pounds to them than 50 to a taxi! By great good fortune, I caught the last train; it came in on platform 10 and I was near the back, and as I ran up the stairs and down the stairs to the underground, a Barking train was just pulling in.