Or, to give the full title, *International Conference on Trends and Perspectives in Linear Statistical Inference, with Celebration of Tadeusz Caliński’s 90th Birthday.*

The conference was in Będlewo Castle, once a private residence for a noble family which had more than its share of scandal, now a mathematics conference centre something along the lines of Oberwolfach, with nearby woods for walking, and in a village with a stork’s nest (now inhabited by house martins) on top of the church tower.

I have been to several statistics conferences before, but this is the first time I have ever spoken at one. And more: I organised a session on *Combinatorics with applications to experimental designs*. I will say a little bit about this later; but here is my team.

It was a very enjoyable conference (or would have been, had I not had so many urgent jobs hanging over my head, including two MSc dissertations to read which arrived while I was at the conference). On Tuesday there was a special session for Tadeusz Caliński’s 90th birthday. His career has been truly remarkable. He is responsible for the close links that have existed for more than 50 years between Polish statisticians and those in many other countries including the UK, France, Italy and the Netherlands. He managed to travel to France even after the declaration of martial law in Poland, because the collaboration had been agreed at high level between the French and Polish governments. (But a couple of years later he was refused permission to travel to Canada, ostensibly for some stupid reason which I do not recall, but actually because he was not a party member but, worse, a supporter of Solidarity.) Many friends he met in the 1960s on his early travels to the west are now dead, but many of those who are still alive had come to the conference to honour him, including Gavin Ross and Rob Verdooren. (But, on the other hand, there were many talented young people there.)

On Tuesday evening there was a concert of music by a string quartet, slightly marred by the fact that people were unsure about when each piece was finished. But the musicians did a good job, and we were regaled with stories about the composers.

On Wednesday there was a choice of excursions, to Kórnik Castle, or to walk in Wielkopolska National Park – I chose the latter. The park was very dry because of the unusual weather, and we were told that the lakes were somewhat polluted (indeed, we saw a machine whose purpose was to help oxygenate the lake water). After the walk they had put out an impressive spread of food for us, but we didn’t fall on it like locusts, since we knew that there would be a barbecue when we got back to Będlewo. The barbecue, as well as huge amounts of food, featured entertainment by a group of musicians in some kind of traditional dress playing violins and bagpipes.

Thursday was the Conference dinner, and presentation of prizes to the young

researchers. A pianist entertained us during the dinner.

I will mention just two talks, which for me were the highlights of the meeting.

Friedrich Pukelsheim, from Augsburg, was a statistican, who wrote a book on optimal design. He switched fields, and is now a sociologist, or political scientist, who looks at electoral systems (with a mathematician’s eye).

Many legislatures have, in some form, a principle known as *degressive proportionality*: the higher the population of a region, the larger the ratio of electors to representatives. But the European parliament is unusual in having this as a formally stated law. Unfortunately, the last election failed to meet this requirement precisely. Friedrich was on a small committee of mathematicians charged with ensuring that the principle is complied with in future. The European parliament is also unusual in trusting mathematicians to do this job, though they don’t trust them enough to accept a formula for the allocation of seats to countries, so the process will have to be repeated for the following election. Friedrich gave us an analysis of the case of Catalonia, where the legal requirements are somewhat contradictory, and in any case are based on population figures from the 1970s. He has written a book, which he recommended as a present for our loved ones.

I have worked a bit on optimality criteria for block designsl the most commonly used of these can be expressed in terms of the Laplacian eigenvalues of the concurrence graph of the design. Thus, A- and D-optimality maximise the harmonic and geometric means of the nontrivial eigenvalues, while E-optimality maximises the smallest eigenvalue.

In a nice but fast-paced talk, young researcher Samuel Rosa used graphs in a different way. We are comparing *n* different treatments, with no blocking factors, and we have a collection of pairs of treatments whose comparisons we are most interested in. A classical case involves a comparison of just two treatments; we should allocate experimental units as equally as possible to the two treatments.

In the general case, the interesting comparisons form the edge set of a graph; our task is to allocate weights to the treatments, giving the proportion of experimental units which should be allocatedd to them, in an optimal way. (Here, unlike the former case, the graph is given in advance.) It turns out that, again, the optimal allocations are determined by the Laplacian matrix of the graph. If I got the details right, assuming that the Laplacian has rank *n*−1, then the D-optimal assignment is uniform; the A-optimal weights are proportional to the square roots of the vertex degrees; and the E-optimal weights are proportional to the vertex degrees.

Quite a few statisticians use Jordan algebras in their work. So I put my neck on the block and talked about the notion of *Jordan schemes* (a generalisation of association schemes). However, from some of the other speakers, I realised that there was a mismatch between the statisticians’ notion of Jordan algebra and mine. After a robust discussion with Roman Zmyślony, things became a bit clearer to me, and I offered to write up some notes on this. I will report when I have done the job.

I had a bit of trouble getting the Polish ogonek on the first e in Będlewo in my slides. The solution I finally used was a package optimised for Computer Modern and Times Roman, but even the package authors agrees it is not entirely satisfactory for other fonts.