The final day of the conference was devoted to a satellite workshop on “Symmetry in finite and infinite structures”. The morning was a solo performance by Laci Babai, in which he explained in detail his quasi-polynomial algorithm for graph isomorphism. In the afternoon there were short contributed talks by participants, at which I had to make invidious choices and miss many talks I wanted to hear. I did listen to Milan Stehlík using root systems in statistics (in quite a different way from my one venture into this), and also celebrating the fact that, in New Zealand, the Maori have succeeded in having the Whanganui River declared a legal entity (with the rights, duties and liabilities of such) after 140 years. Then Shayo Olukoya introduced us to the rational group (homeomorphisms of Cantor space induced by transducers acting on strings) and some of its subgroups; Fatma Al-Kharousi gave us some combinatorics related to semigroups of order-preserving maps of a finite set; Justine Falque described nearly complete work to show that if the number of orbits on n-sets of an oligomorphic permutation group is polynomially bounded, then the orbit algebra is finitely generated, and so the orbit-counting sequence is polynomial on residue classes; Shichang Song showed that the automorphism group of Philip Hall’s universal locally finite group has ample generics; and Wilf Wilson talked about algorithms for finding maximal subsemigroups, which will be included in the Semigroups package for GAP. What a feast!
Altogether an amazing conference, certainly one of the most memorable experiences of my life. You can see the short film that was made to open the conference; if you watch, you will see João Araújo’s sense of humour shining through. Thank you, João, for this, and for so much else you contributed to the conference, and to your many kindnesses to me over the last ten years; the theorems, the crazy trips all over Portugal, and much more! (João is my equal top coauthor; we have ten papers published or in press, even though the first one only appeared in 2013.) And thank you to all the other organisers as well: Teresa Oliveira (Aberta), Mário Edmundo (Lisboa), Maria Elisa Fernandes (Aveiro), Gracinda Gomes (Lisboa), Ana Paula Santana (Coimbra), and Ivan Yudin (Coimbra). I hope that everyone who was there took away good memories and good new mathematics to think about and work on.
As I said in my talk, in seventy years time I want these problems solved!