It’s been a long time since posts, mainly due to the fact that logistical issues caused all my year’s teaching to be compressed into first semester (that’s late-Feb to early-June for any readers not used to Southern Hemisphere habits). It was pretty hard, especially as one of my units is a 550-student first-year Engineering Maths that I had not taken before.

But after many weeks of weekends, evenings or nights spent desperately trying to finish lecture notes, tutorials and solutions for the next day’s lectures, workshops and tutes, the semester ** eventually** ended.

So rather than stay home to attend to the vast number of overdue non-teaching tasks (admin, refereeing, bureaucracy) that I’d had to resolutely ignore during the semsester, instead I flew straight to Germany for a week-long meeting on Graph Polynomials at Schloss Dagstuhl (in Saarland, southern Germany).

Schloss Dagstuhl is an old stately house that has been converted and extended to serve as a residential research centre for week-long workshops, seminars and conferences — primarily in CS / IT, but obviously an occasional Maths one sneaks in as well. It occupies a fairly substantial site surrounded by fields and woods, although it is nominally in the village of Dagstuhl. Wadern is the nearest small town about 1km away.

I’ve never been to either Banff or Oberwolfach but those who have told me that Schloss Dagstuhl has in some ways modelled itself on the latter, though I don’t know which of its features are original and which are adopted. We each had an ensuite room with decent sized desk and single bed (though with the crazy German (European?) custom of having no top sheet, but just a quilt. If you need to sleep covered, then it’s either the whole quilt (= rapidly wake up in a lather or sweat) or pulling the quilt out of its cover and then just using the cover! No keys for the doors though – you can lock it from the inside at night, but rooms are all open when vacant.

Downstairs there is a kitchen / cafeteria at which we were served our meals. For lunch and dinner, complementary versions of our nametags are distributed randomly on the tables to encourage (enforce) people to sit with different people every time. Also a beer fridge full of beer and white wine, and a wine rack with the red wine. All on the honesty system – just take what you want and tell them at the end of the week how many bottles you had. I was taken by the beer – 500ml bottles of good quality German beer for a flat rate of 1.50 euros. So AUD $2.25 for a beer better than one that would cost AUD $12 in a Perth pub! I marked my consumption down on the back of my name tag so I wouldn’t lose track.

The breakfast was great, with a variety of mueslis (Bircher etc), yoghurts, smoothies and an array of spectacular fresh German bread and bread rolls. Beautiful multigrain rolls lavishly covered with a variety of seeds (pumpkin seeds etc). There were also eggs, sausages, “heart attack bacon” (thus christened by Joe Kung who had a few slices each morning) but I avoided those. Even James Oxley had to yield to the temptation to have a bread roll!

The other meals though were pretty bland and a bit stodgy. I remember years ago

being in Berlin when all the restaurants were advertising their “asparagus specials” with famous German asparagus because the season hasd just started. I ordered some of course, only to be rather disappointed when I got long spears of fat, white and very soggy asparagus, that had clearly been on the receiving end of a significant boiling. To me, asparagus is a thin green spear, briefly blanched and served nice and crunchy with some light sauce (maybe lemon juice, maybe a little butter) and perhaps a few slivered almonds. Because white asparagus requires more work (heaping up the soil around each spear as it grows, or growing under plastic to prevent the chlorophyll forming), it is more expensive and often considered a gourmet speciality. But then having lovingly tended an asparagus, heaping up the soil every few days etc. it is immediately given a solid boiling. The white asparagus at Schloss Dagstuhl was served like that.

Other than that, everything else that an academic could want for research, collaboration and down-time is laid on in spades. Seminar rooms of course, internet, 60000-volume library, lounge area with coffee machine, ice-cream chest-freezer, a second beer fridge and a second wine rack, a music room with instruments, a games room (table tennis, darts, foosball), a sauna (if you could pry Martin Loebl out of it), yet another games room with board games and of course a third beer fridge and a third wine rack.

The workshop itself was very interesting with a mix of big names – (Bollobas, Courcelle of MSO logic, and Kauffman of the bracket), old friends, postdocs and grad students. It was also different from a standard conference, in that there was a mix of research talks, expository talks, tutorial sessions and collaboration time. As the topic of the workshop was “Graph Polynomials: towards a comparative theory”, it was pretty much compulsory to have at least *one* talk on characteristic polynomials. Not the characteristic polynomials of matroids, but the usual matrix characteristic polynomial for the adjacency matrix. As nobody there was a hard-core researcher in spectral graph theory, the organisers had asked me (a few weeks ago) to give an introductory talk on it, and given me a nice Monday morning timeslot to present it. Preparing the talk reminded me of how many fundamental open problems remain in the area – perhaps one of the nicest is the question of whether almost all graphs are determined by their spectrum (in other words, share their characteristic polynomial with no other graph). For the first time, I also properly understood Allan Schwenk’s proof that almost all *trees* are cospectral and was extremely impressed. I’ll post some thoughts on characteristic polynomials in the next month or two.

Also unusually for a conference, we spent quite a bit of time thinking about, and discussing, exactly why the <INSERT YOUR FAVOURITE> polynomial is interesting, relevant or useful. The bar is not set too high – it doesn’t have to be “How many pound Sterling is your study of the roots of the five-variable whizzbang polynomial estimated to contribute to the GNP of the UK over the next five years?” but if the answer is “purely for my own interest, I made it up myself and it has no apparent connection with or ramifications to any other area of combinatorics, and me and my students are the only people who have ever studied it or cited my papers” then perhaps some thinking needs to take place.

With my characteristic polynomial hat on, I was on relatively firm ground because the spectrum is easy to compute, yet provides bounds on many graph parameters (e.g. size of maximum independent set) that are NP-hard to compute. Chromatic polynomials and their complex roots have connections to statistical physics via the Potts model, though it is perhaps difficult to justify the study of ** real** chromatic roots on this ground. But at least real chromatic roots do have a substantial history and, as I will report in a future post, Thomas Perret gave an intriguing talk on one aspect of these. There is a lot more to be said about the relative merits of studying various graph polynomials, but I won’t say it here and now.

So all in all a very pleasant trip; although it is far from Perth, the flights and connections are very good, which helps a lot. I could take off at 625am from Perth and put my head down on the pillow at the Frankfurt Airport Hotel at 1030pm, and basically just slot immediately into the new time zone. On the way back, I could take off 10pm from Frankfurt and arrive back in Perth at 10pm, again ready to sleep at the right time of day.

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