I have just heard that Michel Deza died in an accidental fire in his apartment in Paris. Michel was one of my earliest collaborators, and a good friend. This is not an obituary, just a few words to mark his passing.
When I met him, Michel was interested in matroid theory, a subject rich in connections. I think it was in Oxford that we met: Oxford was at that time a hotbed of matroid theory, with Dominic Welsh and Aubrey Ingleton, though I was not really part of that group.
One of Michel’s big ideas was that under certain conditions on intersections, an extremal family of sets would be the hyperplanes of a matroid, and even a perfect matroid design. This is a matroid in which the cardinality of a flat depends only on its dimension. Perfect matroid designs include many of our favourite geometric structures, such as truncations of finite projective and affine spaces. One of his favourite unsolved problems was the existence of a PMD of rank 4 on 183 points, where the lines and planes have cardinality 3 and 21 respectively. This structure would be “locally” a projective plane of order 9, in the sense that the quotient by a point is a projective plane. To my knowledge, its existence is still unknown.
But another of his big ideas at the time, one which reeled me in, was the idea that there should be structures in the semilattice of subpermutations (partial 1-1 maps on a set) analogous to matroids in the lattice of subsets. He called these structures permutation geometries, by analogy with combinatorial geometries (another term for matroids advocated by Gian-Carlo Rota). In particular, he called a permutation group geometric if the intersections of sets of permutations in the group form a permutation geometry. This includes examples such as affine groups, and projective groups over the 2-element field. This was very much to my taste, and my first paper with him in 1977 was on this topic. This led to the classification of all finite geometric groups by my last Oxford DPhil student, Tracey Maund, for which (unfortunately) the only reference is her thesis. The subject also connected with logic; in this context Boris Zil’ber gave a determination of geometric groups of rank at least 7 using elementary but highly involved geometric methods. (Tracey used the Classification of Finite Simple Groups). This, and a push from John Fountain, led me to my paper with Csaba Szabó on independence algebras.
This collaboration led on to many others, with (among other people) Laci Babai and Navin Singhi. After a conference in Montreal, Michel and I made a trip (on the now-defunct airline People Express – quite an adventure!) to Columbus, where Navin was visiting; we spent hours sitting in the local Wendy’s restaurant (I almost said Wendy House) proving a theorem on infinite geometric groups.
A permutation geometry arising from a geometric group has some additional properties. The “hyperplanes” are the permutations, and below any permutation we have a matroid. Also, the permutation geometry has an algebraic structure: it is an inverse semigroup. I thought at the time, and still do, that this is an interesting class of inverse semigroups worth further investigation, but to my knowledge this has not happened.
If we have a bound for sets of permutations with prescribed intersections, it is natural to ask whether this bound can be attained in a permutation group, or whether a better bound can be found. Michel Deza put me in contact with Masao Kiyota, with whom I had a fruitful collaboration on this.
Michel was also a founding editor of the European Journal of Combinatorics, along with Michel Las Vergnas and Pierre Rosenstiehl. A special issue of the journal has been proposed.
The last thing I wrote with Michel was a chapter on designs and matroids for the Encyclopedia of Combinatorial Design.
Michel’s interest moved on to polyhedra and distances, and he became an important worker in discrete geometry. In particular, the Encyclopedia of Distances, published by Springer, is not just a mathematics book, ranging through biology, physics, chemistry, geography, social science, and medicine, among many other things. He worked with chemists on the structure of fullerenes.
As hinted by this, Michel’s interests were always extremely wide. He grew up in the Soviet Union where he was regarded as a poet as much as a mathematician. (I don’t know if any of his poetry has been translated.) His apartment in Paris was full of parrots, which were not caged but had the run of the apartment. One had to move carefully!
The picture is from a Luminy conference on distances; Michel is next to me in the second row.