Andrei Zelevinsky passed away a week ago on April 10, 2013, shortly after turning sixty. Andrei was a great mathematician and a great person. I first met him in a combinatorics conference in Stockholm 1989. This was the first major conference in combinatorics (and perhaps in all of mathematics) with massive participation of mathematicians from the Soviet Union, and it was a meeting point for east and west and for different areas of combinatorics. The conference was organized by Anders Björner who told me that Andrei played an essential role helping to get the Russians to come. One anecdote I remember from the conference was that Isreal Gelfand asked Anders to compare the quality of his English with that of Andrei. “Isreal”, told him Anders politely, “your English is very good, but I must say that Andrei’s English is a touch better.” Gelfand was left speechless for a minute and then asked again: “But then, how is my English compared with Vera’s?” In 1993, Andrei participated in a combinatorics conference that I organized in Jerusalem (see pictures below), and we met on various occasions since then. Andrei wrote a popular blog (mainly) in Russian Avzel’s journal. Beeing referred there once as an “esteemed colleague” (высокочтимым коллегой) and another time as  “Gilushka” demonstrates the width of our relationship.

Let me mention three things from Andrei’s mathematical work.

Andrei is famous for the Bernstein-Zelevinsky theory. Bernstein and Zelevinsky classified the irreducible complex representations of a general linear group over a local field in terms of cuspidal representations. The case of GL(2) was carried out in the famous book by Jacquet-Langlands, and the theory for GL(n) and all reductive groups was a major advance in representation theory.

The second thing I would like to mention is Andrei’s work with Gelfand and Kapranov on genaralized hypergeometric functions. To get some impression on the GKZ theory you may look at the BAMS’ book review of their book written by Fabrizio Catanese. This work is closely related to the study of toric varieties, and it introduced the secondary polytopes. The secondary polytopes is a polytope whose vertices correspond to (certain) triangulations of a polytope P. When P is a polygon then the secondary polytope is the associahedron (also known as the Stasheff polytope).

The third topic is  the amazing cluster algebras.  Andrei Zelevinsky and Sergey Fomin invented cluster algebras which turned out to be an extremely rich mathematical object with deep and important connections to many areas, a few are listed in Andrei’s short introduction (mentioned below): quiver representations, preprojective algebras, Calabi-Yau algebras and categories,  Teichmüller theory, discrete integrable systems, Poisson geometry, and we can add also,  Somos sequences, alternating sign matrices, and, yet again, to associahedra and related classes of polytopes. A good place to start learning about cluster algebras is Andrei’s article from the Notices of the AMS: “What is a cluster algebra.” The cluster algebra portal can also be useful to keep track. And here is a very nice paper with a wide perspective called “integrable combinatorics”  by Phillippe Di Francesco. I should attempt a separate post for cluster algebras.

Andrei was a wonderful person and mathematician and I will miss him.

jerusalem93 Andrei Jerusalem 33

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5 thoughts on “Andrei

  1. The Russian web math community is struggling with the loss of Andrei with more and more people feeling that they have to share what they remember about Andrei. Completely leaving apart his mathematical legacy which will definitely be discussed), all people invariably mention Andrei’s incredibly soft personality which was coupled with a sense of humor which never was offensive and most principled stance when appropriate.

    Accidentally, in my previous comment to your post I mentioned the Free Jewish University in Moscow. Andrei was one of those who taught at this university: he was not a rebel, but a person driven by the feeling of justice. You can (if you read math Russian) try to get an idea of how he was dead serious about looking for the best way to teach the “outcast” Jewish students who were barred from the Moscow University from his memories.

    Another story is self-descripting. In 2009, short after the Cast Lead operation in Gaza, many academics (including, unfortunately, our fellow mathematicians) tried to revive the academic boycott of Israel. Here is the copy of the email that Andrei sent to his colleague (I delete a few identifying details) with whom he was not merely a close co-worker, but enjoyed personal friendship.

    Dear Xxxxxx,

    I hope you are doing fine. I was delighted to learn that you’ll be
    delivering the Emmy Noether lecture at Xxxxxxxxx-2010.

    Unfortunately, the reason I am writing to you is sad. I have just
    learned that “the university of Trondheim may become the
    first university in the West to adopt an academic boycott of Israel,
    if a majority of its board votes in favor of the move at a meeting on
    the subject next month” (the quote is from I very strongly
    believe that regardless of people’s political views, an academic
    boycott is a horrible idea. I am afraid if such a decision will be
    made, it will do very serious damage to the international scientific
    (in particular, mathematical) community. On a personal level, if your
    University adopts this measure, as a show of solidarity with my
    Israeli friends and colleagues, I will be unable to visit Trondheim
    anymore (and I am pretty sure this will not be just my individual
    reaction). I urge you to do everything in your power to defeat this
    very unfortunate initiative.

    Please feel free to use this message any way you find appropriate.

    With best wishes,


    יהיה זיכרו ברוך…

  2. Gil, you (or some of the readers of your blog) could find a few characteristic pictures of Andrei and the math people who were around him in the photo page of his daughter-in-law Karen. Click the links, (Katya’s eulogy – Andrei’s daugther)

    Sorry, – this might look as an ads campaign, but it is not. He was a really exceptional person, very much different from his teachers, Bernstein and Gelfand, who he absolutely admired, – as a human being he was not.

    I promise not to spoil your blog with these personal emotions anymore, but Andrei’s untimely demise was really a shock to all who knew and loved him. Probably the last time he was visiting us was Dima Kazhdan’s special program in HUJI about 2007: we rushed to Jerusalem to meet him and walked all the day in the Old City.

    Be his memory blessed. He was one of the iconic figures for many of us for many long years, regardless of mathematical specialization…

  3. Thank you so very much for your post! And also thank you for posting on the Memorial Page we have set up for people who knew and respected Andrei.

    It is still extremely difficult to come to terms with the terrible blow we have suffered, but posts like this one definitely provide some desperately needed support.

    Andrei’s friends and family would really appreciate it if people come visit Andrei’s blog (which we are trying to turn into a Memorial Center of sorts), point us to links we have missed and – especially – speak about their memories of Andrei’s on our Memorial Page.

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