I have just been at a small meeting in Ghent for the 75th birthday of Francis Buekenhout.
As well as proving some remarkable deep theorems in incidence geometry, Francis was the originator of the field of Diagram Geometry, where he extended ideas of Jacques Tits so that a small diagram could give a concise axiomatisation for a class of incidence geometries. The geometries covered included buildings, together with many operands for sporadic simple or almost simple groups; looked at the right way they include simple (geometric) matroids and much more general structures. In addition, Francis is a large part of the reason for the predominance of Belgians in incidence geometry and finite geometry today, along with Jef Thas in Ghent.
As a cautionary story for how these things work, in 1972 the two (French and Flemish) free universities in Brussels, together with the University of Ghent, applied for funds to set up a contact group in geometry and finite groups. This ran very successfully for 20 years. Then the Belgian science foundation was split into French and Flemish parts, and such cooperation across the language divide was no longer possible. Fortunately, by then, firm links had been established, and a joint seminar continues today.
A bit about some of the talks:
- Arjeh Cohen, in a beautiful talk, gave us a link between incidence geometry and Riemann surfaces, by way of regular maps. This was later picked up by Dimitri Leemans who talked about how his work, on geometries for families of groups, fits in with the work of Marston Conder and others in Auckland on enumerating regular and chiral polytopes by genus (they are now up to genus 301).
- Koen Thas talked about the field with one element. I have heard him talk about this before; this time I learned two new things. First, this is now seen as a potential approach to the Riemann Hypothesis. If various technical problems can be overcome, they will have a set-up where it may be possible to follow the lines of Weil’s proof of the Riemann Hypothesis for function fields. The second, which may be of interest if this comment intrigues you, is that Koen is editing a book on the subject, with chapters from some of the main players, which will be published by the European Mathematical Society next year.
- Pierre-Emmanuel Caprace took us on a trip to the zoo of infinite simple groups, concentrating on the section where the locally compact, compactly generated, non-discrete groups live.
- Ernie Shult told us his philosophy of “tweaking” classical theorems, giving as examples the theorems of Sprague (where he replaced finiteness by a descending chain condition) and Cohen and Cooperstein (where he was able to dispense with a constant-rank assumption).
- Jef Thas talked about his work with Hendrik Van Maldeghem on embeddings of affine planes into projective spaces, where the lines are either affine lines of the projective space or q-arcs. They can determine all such embeddings: there are nine types (four in dimension 3, four in dimension 4, and one in dimension 5 based on the Veronesean variety), but the affine plane is always classical. Then Hans Cuypers showed us a remarkable application to a Buekenhout-type theorem, characterising the partial linear spaces in which planes are dual affine or unitals (with an extra condition), which is relevant to a Lie algebra version of his earlier work on 3-transposition groups with Jonathan Hall.
- James Hirchfeld talked about one of his favourite topics, curves over finite fields. He posed us an interesting open problem. Is it true that, for any prime power q apart from q = 4, the projective plane over the field of q elements contains a 3-arc (a set of points meeting any line in at most three points) strictly larger than the number of points on any non-singular cubic curve? The trivial upper bound for the cardinality of a 3-arc is 2q+3, whereas the Hasse–Weil upper bound for the number of points on a cubic curve is only q+2√q+1; but known examples for 3-arcs come nowhere near the upper bound.
I talked about one of my hobby-horses. There are two approaches to buildings, both due to Tits; the incidence geometry approach (which led to Buekenhout’s work) and the chamber system approach. (A chamber system is a set carrying a collection of partitions satisfying some rather weak properties. We build a chamber system from an incidence geometry by taking the chambers to be the maximal flags, and for each type i, calling two maximal flags i-equivalent if they agree at all types except i.) There are also two approaches to design theory. Mathematicians regard a design as a set of points with a collection of subsets called blocks, satisfying various conditions. If they are thinking very abstractly, they might think of it as an incidence structure with two types, points and blocks, and an incidence relation between them. However, a statistician thinks of a block design as a chamber system where the chambers are the experimental units or plots, and there are two equivalence relations: “same block”, this partition being forced by the nature of the experimental material, and “same treatment”, which the experimenter is free to choose. Rather than requiring an all-or-nothing condition, statisticians optimise one of several “optimality criteria” over the class of all possible designs. There should be more interaction between the theories; in particular, ideas from higher-rank chamber systems and geometries might be useful in cases where there is non-trivial block or treatment structure or some interaction between them.
The main thing about the meeting was that the human side was always very much in evidence, from a 40-year old photograph of Francis Buekenhout and James Hirschfeld, meeting for the first time at a conference in Rome, to Francis’ recent return to active mathematics after a quiet period. There were many very old friends there, some of whom (like Ernie Shult) I had not seen for a very long time. It was also announced that two long-term projects, the publication of the collected works of Jacques Tits (7 years) and the book by Francis Buekenhout and Arjeh Cohen (20 years!) are both very close to a successful conclusion.
Anyway, long may Francis continue proving good theorems!