A tribute to Pierre Deligne

This is the first of the issues of Moscow Mathematical Journal dedicated to Pierre Deligne — a great mathematician, a dear friend, and one of the founding fathers of the Independent University of Moscow — on the occasion of his 65th birthday.

It would be completely superfluous, indeed almost preposterous, to try to explain in this introduction who Pierre is and why a festschrift is dedicated to him. Just out of curiosity, one can do a mathscinet search for papers with “Deligne” in the title; on the very first page out of 10 or so, one meets things as diverse as Deligne–Mumford compactification of the moduli space of curves, Deligne–Mumford stacks as a general notion, Deligne–Hochschild cohomology conjecture, Deligne–Lusztig varieties which appear in geometric representation theory, Deligne cohomology, Weil–Deligne group, and the elegant method of Deligne–Illusie in Hodge theory. Browsing through the whole search engine output will considerably enhance this list. Yet this will only be the tip of the iceberg: quite a few things which we owe to Pierre Deligne are his interpretations of other people's work. And, to mention only two random examples, while Hodge theory has been, in a sense, invented by Hodge, and Shimura isomorphism has been discovered by Shimura, the way we think about these subjects nowadays is that of Pierre Deligne.

And then of course there are things so fundamental that they are not even attributed in modern usage, having become in a sense a part of basic grammar of mathematics — such as the phenomenon of weights in cohomology of algebraic varieties, perhaps one of the most important and mysterious mathematical discoveries made in the last fifty years.

What is striking is that in this great diversity, from grand and fundamental things to small treatises on subjects one may even consider esoteric, everything is equally and extremely beautiful, completely regardless of its size. There is not and there cannot be any unity of subject, but there is certainly a unity of style: it is always a pleasure to read a Deligne paper and to meditate on its contents. Thus this festschrift, even more than usual for volumes of this kind, is not so much our gift to Pierre as his gift to us: an opportunity to visit and revisit so many fascinating corners of the great garden of delights that mathematics can and should be. We hope that this is reflected in the quality and wide range of papers in this volume, and in the fact that so many mathematicians, and so many great mathematicians, agreed to contribute to it.

Another subject that we touch with great pleasure is Pierre Deligne and Moscow. From a purely mathematical perspective, Pierre's influence on the Moscow school is more than obvious, including but not limited to the prevalence of homological methods. Perhaps it would not be too wrong to say that Pierre has had at least as many followers in Moscow as in any other mathematical centre. But maybe it is not universally known that this influence also existed and continues to exist at a more personal level. Pierre first started visiting Moscow in nineteen seventies, deep in the USSR era; at the time, such visits from a foreign mathematician, while not expressly forbidden, were quite non-trivial to arrange, and all the more valuable for that. He has continued to keep in touch with Moscow mathematicians ever since. For us, this is maybe even more important now than before: while the political restrictions disappeared completely, so did much of the Moscow mathematical scenery, with people leaving the country in droves and sometimes not looking back. There is a real danger of a break of continuity, in fact of complete disappearance of Moscow as a world-class mathematical center. Everything which helps prevent it is indeed precious. Yet even now, Pierre Deligne comes to Moscow more often than many of Russian-born mathematicians residing outside of the country, and students and young researchers have a opportunity to listen to his lectures and talk to him personally.

In 2004 Pierre received the Balzan prize, one of the most distinguished scientific prizes in the world. Pierre decided to spend it “for the benefit of the struggling Russian school of mathematics”. Thus the yearly “Pierre Deligne Contest for Young Mathematicians” was started. Since then, Pierre has been coming to Moscow each December to take part in the work of the joint Jury of this and the analogous “Dynasty” contests. During the past four years, 16 Deligne fellowships have been awarded. Now Pierre intends to continue the contest by using his personal funds.

It is hard to overestimate the debt of gratitude we owe Pierre; if this volume would repay it in small part, we would be very glad. We wish him many more happy years!

S. Gusein-Zade, Yu. Ilyashenko, D. Kaledin, A. Kuznetsov,
S. Lando, A. Sossinski, M. Tsfasman, V. Vassiliev, A. Vershik