Victor Ginzburg

This issue (as well as many articles in the succeeding one) is dedicated to Victor Ginzburg on the occasion of his 50th birthday.

Victor is one of the brightest mathematicians of his generation, whose style is quite characteristic of the Moscow school: broadness of interests, approach to mathematics as to an indivisible, almost living being. The number of research problems he puts to students around him is exemplary.

The first major theme of Ginzburg's research may be called Geometric Representation Theory. Some part of it is covered in his brilliant and already classical book with N. Chriss ``Complex geometry and representation theory.'' His works (some with coauthors) in this realm include the geometric study of the representations of affine Hecke algebras; the D-module approach to Lusztig's character sheaves, the development of Drinfeld's program of geometric Satake isomorphism; the geometric study of representations of quantum groups at roots of unity; the discovery of Koszul duality patterns in the representation theory of complex semisimple Lie algebras.

The second major theme may be called Noncommutative Geometry. It includes his works on the representation theory of symplectic reflection algebras and related geometry of Hilbert schemes and Calogero–Moser spaces; McKay correspondence and noncommutative instantons; Koszul duality for operads (classical by now); Calabi–Yau algebras.

His vivid interest and energy is highly appreciated by everyone who contacts him.

We wish Victor to harvest many a ripe fruit for the benefit of mathematics and congratulate him with his birthday.

A. Beilinson, R. Bezrukavnikov, M. Finkelberg,
Yu. Ilyashenko, A. Shen, M. Tsfasman

*   *   *

Last year, Victor Ginzburg turned 50. This being an anniversary issue, I probably ought to start by listing Ginzburg's mathematical achievements. But that there seems to be no point—everyone can do a MathSciNet search; besides, you can't really list the achievements of someone who does something new every few months, can you? maybe in 2017, or better yet, in 2027. Fifty years may be ripe old age for some; for others, it is just a curious and bizarre observation—is Victor Ginzburg really 50? oh. Who would have thought.

Let us treat the occasion, rather, as an opportunity to publish some papers and swap some gossip.

I first met Victor Ginzburg in about 1987... was it really twenty years ago? oh, well. I was an undergraduate at that time, and I had just realized that whatever they were teaching me at my Institute had very little to do with actual mathematics. I was totally at a loss as to what to do. In retrospect, this was idiotic—Moscow in mid-eighties was probably the best place to be for a math student, ever. You just had to go to seminars, and talk to people, and not be intimidated. I was very good at being intimidated. But I had friends.

So, I was given a phone number and told to call Vitya and explain my situation. But ``Vitya''? this seemed awfully disrespectful. So, I spent a month or so trying to learn what his patronymic was. To no avail: no one in Moscow seemed to know.

I think I only learned it in 2000 or so.

I never was Ginzburg's student in any formal sense (mid-eighties Moscow wan't high on formality anyway). I did write a paper under his supervision, a decidedly uninspired piece of work which I never published; then I left for America, met other people. What I came to realize in a while was that everyone of about my age in Moscow was a student of Victor at some point, to some extent. As far as I know, this went on in the early nineties, too, for as long as he still lived in Moscow at least part-time, and only ended when he finally had to take a real job and moved permanently to Chicago. As far as I know, now that Victor is settled in a new place, it is the same story with Moscow replaced by Chicago. Well, so much the worse for Moscow... sadly, it is notoriously bad at keeping the people it can't really afford to lose.

What Victor taught me was not so much mathematics per se—although he certainly did teach me quite a few things—but rather, an attitude to mathematics. This has to do with the specifics of the Russian school. It has always been an article of faith in Moscow that mathematics is fundamentally not random—there is some grand scheme of things, The Book as it is sometimes called, slowly being revealed by the work of generations of mathematicians, the ideal realm where all the definitions are ``correct,'' all the maps are natural, and the proofs are reduced to the level of tautology. This may be quite a productive way of looking at things; after all, standing on the shoulders of giants does raise your eyesight. But the perch is precarious. You're bound to feel slightly embarassed at being effectively reduced to a flee, you're bound to wonder if maybe some of the giants have clay feet... and so on, and so forth. The second or the third time I came to see Victor in his Moscow appartment, I told him in my best tones of adolescent despair that I can't really do anything, because I have a feeling that whereever I look in my knowledge of mathematics, there is a gap. So what, he replied. Welcome to the club. Everyone who is doing math feels the same way.

A couple of years later—he probably forgot it completely, but I do remember—he also mentioned quite casually that the difference between this or that giant and yourself is not qualitative but quantitative, and so should be the difference in your output; not being a giant is certainly not an excuse for doing mediocre work.

What's really great about having Victor as your teacher is that he is very obviously not superhuman: he has exactly the same limitations as you. And then again, he's a very successful mathematician who did quite a sizable amount of astonishingly original research. When you are young, this looks like an unsurmountable paradox; watching someone who has surmounted it without acquiring pomp or losing the sense of humour certainly helps you along the way.

What's really great about having Victor as a friend (and a ``colleague,'' for lack of a better word) is that, to put it briefly, he is interested. Or rather, he doesn't like to be bored. When you have something you want to share with the world (or, as it happens, with someone who'd listen), you can always count on Victor to listen to you; and this won't be out of some sense of obligation, either. If whatever you say is trivial, don't worry about it (rest assured that you won't get far anyway). If there is something in what you say, he will be sure to notice (and remember, and think about it later).

Among the authors of the papers in this anniversary volume all are Ginzburg's friends, and many, if not most, are his students. If there is a general theme to the collection, it is this fundamental unity of mathematics which is so dear to everyone from the Russian school. The concrete topics are quite diverse. This reflects—and only to some extent, at that—the diversity of Victor's own interests: he was never the one to find comfort and safety in being bound to a fixed ``field.'' At a risk of not being quite politically correct, I would term Ginzburg's mode of operation ``creative tresspassing,'' a nice phrase which I stole I think from someone's description of Arthur Koestler; to the extent that this stolen phrase applies, I wish he would go on in exactly the same way for years to come.

D. Kaledin