Michel Deza

Luminy

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.

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About Peter Cameron

I count all the things that need to be counted.
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4 Responses to Michel Deza

  1. Dear professor Cameron, are you sure? I communicated with professor Deza two days ago. It is so sadness and I can not stop my tears. He was so kind and answered my email so generously. I publish our last communication here:

    My email:
    Dear professor Deza,

    I really enjoyed by your great and valuable book “Encyclopedia of Distances, 4th edition”.
    Without this book, the publication world of mathematics is not perfect.

    There is a distance which you point out to it (Page 299-Spectral symmetric), but not completely. Actually, we completely solved the spectral distance problem in graph which our paper name is “Distance between spectra of graphs”. I hope you can add this type of distance to your very valuable book in its five edition. The link of our paper is:

    http://www.sciencedirect.com/science/article/pii/S0024379514006892

    With my kindly regards,
    Shahrooz.Janbaz

    and professor Deza’s kindly answer:

    Dear Shahrooz,

    Thank you for message. If I understand well, you, also in 2015, as Gu–Hua–Liu
    considered l_p-metric between spectral vectors
    Their spectral symmetric is between two finite weighted graphs and, in
    general,
    L_p-Wasserstein between two probability measures.
    Both paper were independent and in the same time.
    For l_p -semimetrics only, you fully solved Brualdu problem, I think. ?
    So, I’ll add a line in 5th edition if any. But it can be in 2019 only…

    Best, Michel
    PS. I was in Kashan on a conference about 2010.Best impression about
    Iranian scientists and students.

    I also invited him to Iranian conference for next year and his answer:

    Dear Shahrooz,

    Thank you very much.
    But now I am a retired professor; so, have no travel funds and
    come only when invited with expences paid.
    In Kashan, Prof. Alireza Ashrafi even paid business class air ticket, which
    is, of course, not necessary.
    But I understand that only big bosses have possibility to do such invitations.

    Best, Michel

    I am so sorry for this tribble accident. We lost a great and generous mathematician.

    Shahrooz.Janbaz

  2. I heard the news from Cheryl Praeger and it was confirmed by Patrick Sole. Apparently the police are now investigating the accident, and we may learn more about what happened. I agree with your tribute.

  3. Navin Singhi says:

    Hi! Peter
    I feel very sad to read this news. Just two days back I saw a paper by him. Used to correspond so regularly with him and meet him also often. Can not think easily that he is no more and has died in an accident. I will have to get used to his absence.

    I also have very fond memories of that visit by you and Michel to our home in Columbus. Both of you have been very inspiring not just for me but also for my twin children Nikhil and Nidhi, who have also done Ph.D. in combinatorics- crypto systems.

    Deza was a very closed friend for us and a great mathematician and a great human being.
    He had a deep understanding of Mathematics as well as life. It was so enjoying to learn and understand with him Mathematics as well as other aspects of life.
    regards
    Navin

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