Still dark when I left the house, though the city was not asleep; a couple walking down the road talking loudly; drivers chatting in Dell Cars; cleaners in a fast food place.
The bus came on time and was not crowded, but picked up a lot of people on the way. I sat upstairs at the front for the view; after a while, a well-dressed young man sat next to me, and proceeded to fall asleep and loll over me. When the other front seat came free, he moved there.
Not long to wait for the train. It started on time but I had been put in one of the seats with no view and no room in the luggage rack, and travelled backwards to Glasgow. Also, sad to say, the coffee machine was broken and so my first cup of coffee came rather late.
As we pulled out of Euston, the sun rose, and the carriage was filled with orange light.
By the end of the journey, I was quite sleepy; but no chance of a nap, the carriage was filled with noisy children.
We were on time at Glasgow, despite a ten-minute delay at Carlisle. I bought lunch and a paper, and waited for the platform for the Stranraer train to be announced. This happened rather late, and the train was fairly crowded; fortunately I had a reservation. We pulled out on time for the two-and-a-quarter hour journey.
The rain came on to spatter, and Ailsa Craig looked decidedly rainwashed. It is a pretty train journey, but I described it last year and won't do so again. I was near the front, and saw the driver handing the token (a large ring) to the signalman at the end of the downhill stretch. There was a lot of Himalayan balsam in flower by the track. Two girls went to the toilet together (what do they do?), but forgot to lock the door; we had high-pitched screams when a man burst in on them.
Getting onto the ship was a slow process, since the two people checking tickets and handing out boarding cards seemed not to understand what they were doing. Eventually I was through, and a long walk down dark winding corridors brought me to the car deck, from where narrow stairs took me to the passenger deck. I was on in good time, and we were away heading out through the sea loch. Out of the heads, the open sea was a bit rough, and the ship rolled a bit; but soon enough we were in the calm of Belfast Lough.
My bag came quickly and I was out. I had a taxi to myself; the driver took me by a roundabout route so it was relatively expensive. But no problems with check-in this time, and I know that they think "internet access" means "web access", so I won't even try to get proper internet access. Anyway, the web access does at least work.
At 7 I went out to meet Natalia. She and her husband Stanislav and son Sasha were already there. We set off on foot. The first restaurant they tried was full, so we went to another, "Deanes at Queens", run by the University. We had a good meal and a very good talk. Stanislav wanted an example of an abelian topological group with no non-trivial proper closed subgroups. I suggested the abelian group structures on the Urysohn space constructed by Anatoly Vershik and me. (They at least have the property that no proper closed subgroup contains any power of the generating isometry.) I played with Sasha a bit until he started getting a bit manic, so we left.
They walked me back to Elms Village, where I went straight to bed and slept soundly.
With the prospect of an empty day in Belfast, I was in no hurry to get up, so lay in bed and read yesterday's paper.
Because of refurbishment of the student centre, I have to go for breakfast to a small café in a University building about ten minutes walk from my room. It was drizzling when I set out, but soon turned to serious rain. Fortunately I was passing a bus shelter, so waited out the worst of the rain before continuing on my way.
The road in which the breakfast place is located has no sign, so I overshot and had to backtrack. When I got there, it was pleasant. Breakfast is standard enough but good: juice, cereal, yoghurt, cooked breakfast (bacon, sausage, beans, scrambled egg, hash browns, toasted soda bread), and decent coffee. The only glitch came when the waiter asked me if I was vegetarian, and I mis-heard him and thought he asked if I was a visitor.
Back in my room, the prospect of sightseeing on a rainy Sunday in Belfast was not appealing, so I decided to have a lazy day, with maybe some work on my Indian diary. I spent the day editing pictures and putting them in, and reading the rest of the paper. Ironically, the weather turned nice, though there was a cold wind.
By late afternoon I was finished, and I decided I'd better go for a walk. Just past the University, I passed the Crescent Church: not an attempt to reach out to Muslims, but a church in a street called The Crescent.
I headed into town (I knew I was in the centre when I passed the City Hall) and out the other side. Coming back, I went down the road past the City Hall and found the river Lagan. Crossing over a pedestrian bridge beside the railway line, I found a nice wide footpath by the river, so I followed it.
A little way along, I passed a man, and remarked on how unexpectedly lovely the day had turned out. (The sun was now sparkling on blue water.) He answered me in a German accent, and told me that a steam train was about to come down the line (or, at least, sometime in the next hour or two).
I decided not to wait, so continued on my way. The path came up onto a very wide and busy road, but with some difficulty I crossed it and found that it continued on the other side of the river. As I passed the station, I heard the train whistling and puffing, and saw the smoke rising.
I continued along a very nice towpath until I saw a cycle track sign to "Botanic Station". There is no station now, but I thought I could cut through the botanic gardens. But when I approached the invitingly open gate of the gardens, two guards came out from behind bushes and started shouting at me that the gardens were closed. So I turned off and found myself in the back of the University. There were many big trucks and vans advertising marquees, catering, etc. parked near the mathematics department, so I guessed that some event was happening in the park.
I went out through the University court and into the street. People were jumping over the low fence and urinating in the rose garden. Across the street, a huge queue had formed outside the Students Union building.
I went down Stranmillis Road looking for a place to eat. The first two places were packed out; the third had spare seats but when I went in I was firmly told they were closed. Eventually I found a Chinese takeaway that had a few tables, and ate there. It was not a very nice meal. After that I walked back to Elms Village. On the way I passed Adam Bohn and stopped for a quick word.
On the way in to the village, I was challenged by a security guard. After I had passed muster, he said "We always check people coming in here", even though he must have known that I knew it was a lie (like the terminal manager in Cochin).
The water in the shower was just above lukewarm; the room still cold. It is clear and sunny outside.
At breakfast, there were several familiar faces; I sat with Robert Wisbauer, with whom I shared an umbrella in one of the worst downpours last year, and David Jordan was at the next table.
Since registration began at 9:30 but the opening ceremony only at 10:30, we were in plenty of time. I sat down and, to my delight, Nik Ruskuc came in and sat down next to me. In the pack was an e-ticket for a concert in the Ulster Hall, to be recorded for the BBC, on my last night; thus the problem of what to do then is solved.
The first talk was a learned oddity by Arnfinn Laudal; though he is from Oslo, his accent seemed South African to me. He was explaining some applications of non-commutative deformation theory; since I am not really at home even with the commutative case, the details were lost on me. He claims that it "explains" something that puzzled him about physics at school, why position and momentum suffice to determine all future behaviour of a particle (somehow his formalism churns out things corresponding to higher derivatives). He began the talk with an appeal for us to help the Abdus Salaam School of Mathematical Sciences in Lahore, which he claims is the only high-quality mathematics institute in Pakistan, and which is threatened with a 50 to 70 percent cut in funding.
At the coffee break, Michael and Tara Brough had arrived. I see from my programme that Tara is giving a talk (and that I am chairing the session).
Then a really lovely talk from David Jordan, moving from mutation of quivers to iteration to Poisson algebras to skew polynomial rings. His recurrence was a fascinating one:
a_{n+4} = (a_{n+1}a_{n+3}+1)/a_{n}(try it with the first four numbers all 1).
Finally for the morning, Vladimir Bavula talked about the algebra of polynomials with differential and integral operators. He has managed to describe its automorphism group (this has not been achieved for just polynomials and differential operators). One feature is that there are n height-1 prime ideals; any prime ideal is the sum of a subset of these; and any ideal is the product of its minimal\ prime ideals. So the number of prime ideals is the nth Dedekind number (the number of antichains of subsets of {1,...,n}).
At lunch (as usual at this conference, rather late, 14:10) I didn't feel like eating (I think the result of last night), so I went looking for birthday cards. The convenience store near the University had none; the selection in the Mace at the BP service station was unbelievably naff, but with little choice, I took the least objectionable. Then I sat in the Botanic Gardens until it was time to go.
There were only two talks in the afternoon. Abdenacer Makhlouf was constructing twisted versions of everything you could think of. The main feature of his talk was the large number of questions, which often led to discussions between four or five members of the audience; so he only got through half of the large amount of material he'd prepared.
Then Robert Wisbauer told us how, although it is essential for a ring to have an identity and modules to be unital (and analogous statements in categories) for many purposes, notably adjoints, he was going to show us how to do without them, to some extent.
After the talk, Nik and I decided to go to dinner. We went to the Giraffe (a one-off, not part of a chain), which did us a pretty nice meal, and had a good talk about mathematics, its history and philosophy, and many other things. Nik told me that, as a native speaker of a Slavic language, he had had trouble with articles in English, and had formed the impression that they were unnecessary, until he saw a sign above a hospital receptionist's desk: "Please be patient".
Then home to work for a while before bedtime. I did a small search for news about the funding cut at ASSMS and found nothing, though it is clear that the rest of the story is true: it is a very good institute, and both the University and the country seem proud of it.
This morning, when I woke, there were five magpies outside my window.
There was a huge quantity of information in the morning talks; let me single out the best two.
Tom Lenagan talked about Grassmannians. I don't believe he is capable of giving a lecture without explaining clearly what he is talking about. In this case, after a brief and lucid discussion of the ordinary Grassmannian, its decomposition into Schubert cells, and their parametrisation by Young diagrams, he went on to an extraordinary relationship between the totally non-negative Grassmannian (all minors non-negative) and the quantum Grassmannian: prime ideals of the latter invariant under scalar multiplications of coordinates are bijective with non-empty cells of the former.
Then Peter Jørgenson talked about cluster algebras and cluster categories. It was clear that his work (which he would have liked to have talked about) was on the infinitely generated case, but he was too honest not to give us a very clear explanation of the general case first. I never really understood this stuff before, and I have to say that, while I find the cluster algebras absolutely fascinating, the motivation for cluster categories completely escapes me: it seems to be an attempt to replace straightforward combinatorics with complicated algebra.
For lunch I got a sandwich from the Hope Café and sat in the sunshine (with my coat on – it is still very cold, even in the sun) to eat it.
Then I got to thinking about clusters. The mutation transformations are involutions, one for each vertex; so, when the Dynkin diagrams made an appearance, I confidently expected the groups generated by these involutions to be the corresponding Coxeter groups. But they are not! In the case A_{2} (two vertices joined by an edge), the Coxeter group is dihedral of order 6, but the mutation group is dihedral of order 10. What's going on?
This problem took up much of my attention during the afternoon. I don't think I missed too much. The first talk was by Travis Squires, who was trying to replace something conceptually simple but with complicated equations by something conceptually much harder, but he ran out of time. I learned that the terms "semi-strict" and "hemi-strict", applied to 2-vector spaces, are not the same. (A 2-vector space is like a category but sets are replaced by vector spaces and the various maps such as source, target and composition are linear.)
After tea, Natalia talked. She really has developed well; she has something to say, says it well, deals with questions well, and is master of the material.
The final talk was cancelled since Vladimir Dotsenko hadn't been able to get a visa in time. So we had the conference photograph, after which Nik and I decided to go to the pub. Soon Adam came in, and the three of us had a nice talk. The others didn't really believe me when I said that 7:15 for 7:30 meant that we should sit down at 7:15, so the three of us got the last three seats and were forced to continue our discussion. It was a very nice dinner, but we didn't stay late since both Nik and I have talks to prepare.
The day dawned cloudy, but the water in the shower was decently hot. I wrote the birthday cards, and then thought about the mutation group for A_{3} until breakfast. It acts on the 14 quivers imprimitively with 7 blocks of 2, and is 2^{6}.S_{7} (if I am not mistaken). The blocks correspond to reversing all the arrows.
Without sunlight to beguile the eye, it was clear that the floral display in the Botanic Garden was past its best (especially the marigolds). A leaf detached itself from a tree as I passed, and drifted ever so slowly down to the lawn. A very autumnal start to September!
It was nice to have some real group theory for a change. Shou-Jen Hu told us about checking the Noether property (rationality of the invariants, connected with the inverse Galois property) for groups of order 64; altogether, it was a very nice summary of the subject. Alexander Lichtman talked about non-abelian valuations, which he uses for non-standard purposes, such as describing the multiplicative group of the skew field of fractions of a group algebra. Unfortunately he didn't write very much, and what he did write he almost immediately erased.
At coffee time I caught Peter Jørgensen and asked him my question. He said he had never thought about it and didn't know if anyone had. I think he was almost as surprised as I was.
Nik was going to use his own laptop, for reasons he didn't reveal, and offered me the use of it for my talk. Apart from the fact that the laser pointer gave out halfway through, it went well. I wasn't expecting to talk about more than a small part of the material, and so it turned out.
Nik's talk was lovely. He had a clip from "Waiting for Godot" to describe the different types of growth rates for direct powers of algebraic structures that he was talking about. It was a very fine summary, illustrated with lots of detail. After the talk, we got sandwiches and sat in the Botanic Garden to eat them, talking about things that are on his mind as he begins his stint as head of school (research assessment, appointments, recalcitrant administrations, international reviews, etc.), until he had to go and work. I sat a bit longer before going back for the next lecture.
In the afternoon, we had a more or less completely incomprehensible lecture on vertex operator algebras from Alexander Zuevsky, followed by a session I was chairing. Tara Brough talked about her PhD work under Derek Holt on groups whose word problem is a finite intersection of context-free languages. The conjecture is that such a group is virtually a direct product of k free groups, where k is the number of languages. She is close to a proof in the soluble case (where the conjecture says that the group is virtually abelian). Then Stanislav Shkarin talked about dense orbits of linear operators on Banach spaces (these cannot occur in finite dimension, but he has some strange infinite examples.)
After the talks, Nik and I decided to go and eat; Nik was being very encouraging to Tara, so we asked her and Michael to come along as well. We went back to the Giraffe, where we had been two days ago, and had a pleasant meal. Tara does talk a lot!
On the way home, I stopped for cash at the BP service station. It was all in Northern Irish notes. Nik told me the story of a colleague who had tried to pay a taxi driver in London with a Scottish note; the cabbie had complained, and eventually called the police, who fined him fifty pounds for wasting police time! He capped it with another story of an officious policeman in Perth, who did things such as, on seeing a man drop a twenty pound note in the street, had picked it up and returned it to him, and proceeded to fine him for littering.
I had breakfast with Peter Plaumann, a German now retired and living in Mexico. As natural for a pair of expats, we spent a long time swapping stories about visa and passport problems, of which we both had a huge supply.
He was the first speaker. His talk included two classical theorems, the first of which I didn't know at all, the second not in this form:
Then Sergei Sylvestrov gave a talk at which he took the situation of topological dynamics (a map from a topological space to itself) and translated it into a sort of "skew polynomial ring" over the ring of continuous functions of the space, so that properties of the dynamics translate into properties of the ring; this allows him to treat many questions in a purely algebraic manner.
After the break, a surprisingly nice talk by Lucas Fresse on Springer fibres. One of these is the set of fixed points of a nilpotent matrix u in the flag variety. Such matrices are determined (over the complex numbers) by the sizes of their Jordan blocks, a partition of n; Springer showed that there is an action of S_{n} on the cohomology of this space, so that the top cohomology group is the Specht module corresponding to the partition (so all the representation theory of S_{n} in characteristic 0 is here). Of course it all became very combinatorial; components of the Springer fibre are parametrised by tableaux of the given shape. One striking puzzle emerged. They had found for which partitions it is the case that all components of the Springer fibre are smooth. They had also found for which partitions the centraliser of u has a dense orbit in each component. The partitions in the second theorem are precisely the conjugates of those in the first. They don't know why; it is just an observation!
Finally Miguel Ferrero talked about "partial actions" of groups on rings. I might think about this, but I would prefer to replace the rings by something simpler such as abelian groups.
Then it was over; we thanked Natalia, said our goodbyes, she took a copy of my tickets, and we left. Nik and I again got sandwiches and sat in the sun in the Botanic Garden. I told him I had to go and read a Masters project, at which he came out with a wonderful story, which apparently happened to Victoria Gould at York. She got an email from a Masters student, attaching a project in applied maths. The email said, "Dear Dr Gould, I believe you are the only reader in the department. Could you please read my project by the end of the week?"
He left to work (he is writing up a paper which contains some rather hand-waving geometric arguments which he is having trouble making rigorous), and I went back to read my email and Sigita's project.
After working for a while, I went for a walk. With some difficulty, after following a zigzag path, I found the cycle track, and walked out along the river Lagan. The river bank had lots of Himalayan balsam, with some giant hogweed and other rank late summer growth; there were lots of gulls, ducks and crows and a solitary coot. The path left the river to follow a disused canal with reeds and grass in it, cutting off a big loop of the river.
I had resisted two entrances to Lagan Meadows (a nature reserve), but just after rejoining the river came one I couldn't resist, up a steep wooded hill. I followed this path right around the reserve. First it was pleasant, undulating mixed forest; then, passing two rowans with astonishing red berries shining in the sun, it went through grassland, fenced off by hedges of hawthorn, bramble, gorse and honeysuckle.
Back to the river in plenty of time, I retraced my steps and crossed the disused canal into a small forest of poplars, with Himalayan balsam and nettles in the understory. The balsam pods were huge but not yet ready to pop. Crossing back to the towpath, I decided the time had come to turn back.
The path took me right into Belfast, though I turned off into the gasworks area (now modern offices and shops with no street life) and had a bit of trouble finding my way through. Eventually I saw the BBC building and, with a bit further meandering, found the Ulster Hall. It was just seven, and queues were already forming; but soon they let us in.
The Ulster Orchestra were practising (and recording) their Proms programme. It was typical Proms fare: some English pieces from the Henry Wood era (by Bliss, Bax, Dorothy Howell and Parry) and some favourites (Rachmaninov's piano concerto, the Karelia suite, and some well-known excerpts from Eugene Onegin. The Bliss and Bax were mostly "sounding brass and tinkling cymbals", and the Parry not really to my taste either (it could have been good but needed much more precision than it got) but Howell's Lamia was a pleasant Impressionist piece with Faun-like qualities. But the familiar pieces were very well done.
On the programme, they were advertising some concerts featuring Roddy Williams. I noted this to tell Hester later.
Nine "survivors" were at the concert (Natalia and Stanislav, Tara and Michael, Sergei, Miguel, Abdenacer, Arnfinn and me), and we decided to look for a place to eat afterwards. A lot of watering holes had closed at ten, but Italian restaurants continue until eleven, so we found one very near the University and had a good meal. (I had lasagne with garlic bread, very filling.) I ordered a beer with mine, but Arnfinn, sitting opposite, got two bottles of rosé d'Anjou in the course of the evening and insisted on putting some in my glass.
After dinner, and some goodbyes, everybody walked very slowly to the Botanic, where more of the party peeled off for a last drink; Michael and Tara went to their B&B, and Arnfinn and I continued back to Elms Village. It was well after midnight when I got to bed.
I woke at the usual time, showered, and went out for my last breakfast, an identical copy of all the others. On the way back I bought a paper, so I would have some puzzles to do while waiting round.
I packed, checked out, and waited for the taxi. Sergei and Miguel were taking a taxi to the airport a little before mine, so we had another round of goodbyes.
My taxi came a bit early, and I was at the ferry terminal in good time. After not too much waiting, we were allowed onto the ship, where I found a seat and took it. Nobody asked me to check in my bag. Two truckers came to sit opposite me. One got out his laptop and started surfing the web. I've no idea what he was looking at.
Arriving in Stranraer, I found my way out of the docks and turned along the road towards the guesthouse. It was a little further than I had remembered from the map, but I found it before I had begun to worry. The landlady Elaine showed me the room and gave me the key.
I stopped at the tourist office. They were very friendly, and gave me the new public transport timetable (out today) and a town map. I also picked up some leaflets about walks and gardens. One of these was the Lochryan Coastal Path, so I decided to walk a bit of that; I had a bit less than two hours each way before Ro's train was to arrive. Had the bus timetable been different I could have tried to walk the whole thing and get a bus back; but that was not to be.
The weather was beautiful, and the birds were astonishingly varied: wagtails, finches, crows; ducks and swans; small brown waders and larger black waders; gulls; large flocks of what may have been curlews. There were many different wildflowers (and a few garden escapes) among the gorse and brambles. The sun sparkled on the water of the loch as the ferries came and went.
At Cairnryan, just beyond the P&O terminal, it was time to turn back. I bought a cold drink at the village store and carried the bottle four miles before finding a bin. When I did, five yards before the bin someone had left a cider bottle.
Coming into Stranraer, a gull on the water swam through the sun's track, and left trailing fire.