9 Jan 2023

Yuri Ivanovich Manin, emeritus director of the Max Planck Institute for Mathematics in Bonn, passed away on Saturday, 7 January.

Yuri Manin made fundamental contributions to number theory, algebraic geometry, and mathematical physics, contributions that are themselves just the tip of the iceberg as concerns his incredibly deep and broad insight into mathematics, physics, and other domains of scholarly endeavour. In his mid-twenties, Manin initiated a systematic study of formal groups via the Dieudonne-Manin classification, which set up one of the key themes in arithmetic geometry for 60 years via p-adic Hodge theory, Shimura varieties, and the Langlands programme. Furthermore, he was the originator of numerous streams of research in Diophantine geometry, such as the theory of the Brauer-Manin obstruction or the distributional study of rational points of bounded height on higher-dimensional varieties. His most famous work in number theory was possibly the first proof of the function field Mordell conjecture, in the course of which he introduced the Gauss-Manin connection, a differential equation satisfied by periods of families of algebraic varieties. This simple yet remarkable innovation illustrates well the unity of vision he brought to different areas of mathematics, since the Gauss-Manin connection became an absolutely fundamental tool in number theory, algebraic geometry, and the rich interaction between physics and geometry over the last three decades that has come to be known as ‘mirror symmetry’. Also in mathematical physics, together with Atiyah, Drinfeld, and Hitchin, he gave an algebro-geometric construction of instantons, solutions to the self-dual Yang-Mills equations, which had an enormous impact on the work of Donaldson, Witten, and many others. It has sometimes been referred to as the first application of algebraic geometry to physics.

As a teacher, Manin supervised an army of the most productive and original thinkers in the history of Soviet (and world) mathematics, many of whom went on to become great gurus themselves, such as the Fields medalist Vladimir Drinfeld or the Wolf prize winner Alexander Beilinson. Probably no other single figure has commanded the level of respect he received from the former Soviet mathematical community, both before and after the diaspora.

However, if I were to name the most distinctive characteristic of Yuri Manin as a scholar, I would say that he was the one of the most authentic philosophers among mathematicians of the last hundred years. In his research papers, his expository lectures, and his numerous books, one finds hardly a single superfluous sentence, most of them being steeped in wisdom expressed with inimitable clarity. Many mathematicians, in addition to proving deep theorems, secretly or openly aspire to have a philosophical overview of what mathematics is truly about. There are few indeed who achieved this kind of perspective to the level that Yuri Manin did.

His far-reaching mathematical ideas, his inspiring expositions, and his unadulterated wisdom will be greatly missed by the ICMS community as well as by the world.

Minhyong Kim, Director of the ICMS