I was born in 1954 in Kharkov (now Kharkiv, Ukraine, formerly a part of the USSR). My mother was a professor of Latin, my father a professor of mathematics. When I was a child, he saw that I had some mathematical abilities, and started teaching me mathematics; importantly, he did it in a way which was pleasurable to me.
As I grew up, I realized that I had to find a field in which I could excel. Mathematics was a natural choice for me, and I invested as much energy as possible into studying it. During 1965–1969 I attended School 27 in Kharkiv (this is a physico-mathematical school founded by N I Akhiezer, a renowned mathematician). In 1969 I took part in the International Mathematics Olympiad and won a gold medal.
During 1969–1977 I studied at the Mathematical Department of Moscow University (first as an undergraduate student and then as a graduate one). In 1971 I became a student of Yu I Manin. During 1971/72 Manin, jointly with I I Piatetskii-Shapiro, organized a seminar on modular and automorphic forms. I was strongly influenced by that seminar and by Piatetskii-Shapiro’s course on automorphic forms on GL(2). Informally, Piatetskii-Shapiro became my second scientific advisor. During 1974/75 I was mentored by D Kazhdan (before he emigrated to the US).
During my student years I mostly worked on the Langlands program for function fields. My main result during this period was a proof of the global Langlands conjecture for GL(2) over a global function field. The proof used a new notion of “shtuka” and was based on the study of the moduli varieties of shtukas. Later the moduli varieties of more general shtukas were used by L Lafforgue and V Lafforgue in their works on the global Langlands conjecture for reductive groups over global function fields.
From 1978 to1980 I worked as an assistant professor at Bashkir State University in Ufa, a city near the Ural mountains. In Ufa I met my future wife. We have a son, Andrey.
In 1980 I returned to Kharkiv. During 1981–1998 I worked there at the Institute for Low Temperature Physics and Engineering. This was a good place to do research in mathematics.
During 1980–1989 most of my research was devoted to algebraic questions of mathematical physics. One of the challenges was to understand the algebraic mechanism underlying the theory of quantum integrable systems. While studying a paper by E K Sklyanin (a student and collaborator of L D Faddeev), I realized that the key algebraic structure is that of Hopf algebra and that the Hopf algebras relevant for the theory of quantum integrable systems are neither commutative nor cocommutative. Such Hopf algebras are now often called quantum groups, and they have Poisson—Lie groups as their classical limits. It turned out that the universal enveloping algebras of semi-simple Lie algebras (and more generally, of Kac—Moody algebras) have canonical quantizations; they were constructed independently by M Jimbo and me. These quantizations turned out to be very important for representation theory, as demonstrated later in the works of G Lusztig, M Kashiwara, and many others.
On the other hand, in the early 1980’s I wrote a paper, which became a starting point of the Geometric Langlands program (a series of works by G Laumon was another starting point). The main idea of this program is as follows: given a local system E of rank n on a smooth projective curve X over any field k, one should try to construct a complex of sheaves on the stack of rank n vector bundles on X, which is related to E in a certain way. In the 1990’s Beilinson and I developed such a construction in the de Rham context for a certain class of local systems assuming that k has characteristic 0; the main idea was to quantize a certain Hamiltonian integrable systems discovered by Hitchin in 1987.
In 1999 I moved with my family to the USA. Since then I have been working at the University of Chicago.
In 2012 I proved that for any smooth variety X over a finite field, the set of isomorphism classes of irreducible l-adic local systems whose determinant has finite order does not depend on the prime number l. In the case of curves, this had been deduced by L Lafforgue from the Langlands conjecture proved by him; my contribution was to deduce the general case from his result. It is still unknown whether the smoothness assumption can be replaced by normality (as conjectured by P Deligne).
Since 2018 I have been working on prismatic cohomology. This is a new cohomology theory for p-adic formal schemes introduced by B Bhatt and P Scholze and developed further by Bhatt and J Lurie.
12 November 2023 Hong Kong
