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.