There is far too much, and far too diverse, mathematics going on here for me to describe all or even most of it. Nine plenary lectures on the first day! The talks are being live streamed at https://videocast.fccn.pt/live/idprovider_uab/kinds_mathematics_remind, though the organisers admit that it doesn’t always work (there is only a fixed camera, and speakers do not always stand in front of it).
Henrique Leitão opened the conference with a talk about Pedro Nunes, covering the retrograde motion of the shadow of a sundial, his work on rhumb lines which was very likely used by Mercator as the basis of his map projection, and his algebra book. On the first topic, Nunes wrote,
This is most surprising but cannot be denied because it is demonstrated with mathematical certainty
(in Henrique’s opinion one of the first uses of science to prove the existence of a totally new phenomenon, and one which was not observed for well over 300 years). On algebra, he described a problem easily solved by algebra but where no geometric construction is known. Given the base a, the height h, and the ratio b/c of the other two sides of a triangle, find b and c. Nunes said of this problem and others, “Who knows by Algebra, knows scientifically”.
Peter Neumann told us about Galois’ work on transitive permutation groups of prime degree (in his language, irreducible equations of prime degree), and wondered if we will ever have the classification of these groups without using CFSG.
Then Cameron Freer talked about exchangeable measures concentrated on a single isomorphism class of countable relational structures. The random graph is the prototype, but is too easy; it wasn’t until much later that Petrov and Vershik handled Henson’s universal triangle-free graph, and Cameron with his colleagues Nate Ackerman and Rehana Patel extended this to all structures with trivial algebraic closure. At one point he showed us a picture by M. C. Escher, “Square Limit”, which can be regarded as a graphon for the random graph. The picture was from 1964, which was the same year that Richard Rado gave his explicit description of the graph; should it be re-named the Escher–Rado graph?
Rosemary Bailey told us about weak neighbour balance for designs in circular blocks, which ties in Hadamard matrices, doubly regular tournaments, something that Laci Babai and I called “S-digraphs”, and many other interesting things.
After lunch, Pablo Spiga told us about conjectures of Sims and Weiss and his work on the latter. Dimitri Leemans spoke about the epic battle that he, Maria Elisa Fernandes, Mark Mixer and I had with the maximal rank of a regular polytope with automorphism group the alternating group An, published in the Proceedings of the London Mathematical Society last month.Gareth Jones reported on his constructions for “doubly Beauville Hurwitz groups”, together with a very nice description of the algebraic geometry and group theory behind it.
Greg Cherlin had found an old unpublished paper that Sam Tarzi and I wrote on the structure of the automorphism group of G, the automorphism group of (yes, that repetition is intended) the random m-edge-colouring of the countable complete graph: all the automorphisms are induced by permutations, so the group is an extension of G by the symmetric group Sm, and the extension splits if and only if m is odd. Greg has in mind a far-reaching generalisation of this, particularly for homogeneous metric spaces, with unexpected and mysterious connections with the work of Bannai and Ito on P-polynomial structures on association schemes.
To finish the day, Jarik Nešetřil described recent work on Ramsey classes, and its connection with EPPA.
I didn’t mention the film that the organisers have made about me (based mostly on old photographs). I am really not used to being in the spotlight to this extent!