# Vladimir Drinfeld

Wolf Prize Laureate in Mathematics 2018

**Vladimir Drinfeld**

#### Affiliation at the time of the award:

**University of Chicago, USA**

#### Award citation:

**“for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in presentation theory, a field which helps to understand complex algebraic structures”.**

#### Prize Share:

**Vladimir Drinfeld**

**Alexander Beilinson **

An “algebraic structure” is a set of objects, including the actions that can be performed on those objectas, that obey certain axioms. One of the roles of modern algebra is to research, in the most general and abstract way possible, the properties of various algebraic structures (including their objects), many of which are amazingly complicated.

Vladimir Drinfeld, born in Kharkov, Ukraine (1954), represented the USSR at the International Mathematical Olympiad at the age of 15 and won a gold medal. In the same year he also began his studies at the University of Moscow. Since the eighties he has been considered one of the world’s leading mathematicians. In 1990 he won the prestigious Fields Medal and in 2008 was elected to the National Academy of Sciences (USA). Drinfeld has contributed greatly to various branches of pure mathematics, mainly algebraic geometry, arithmetic geometry and the theory of representation – as well as mathematical physics. The mathematical objects named after him – the “Drinfeld Modules”, the “Drinfeld Chtoucas”, the “Drinfeld Upper Half Plane”, the “Drinfeld Associator”, and so many others that one of his endorsers jokingly said, “one could think that “Drinfeld” was an adjective, not the name of a person”.

In the seventies, Drinfeld began his work on the aforementioned “Langlands Program”, the ambitious program that aimed at unifying the fields of mathematics. This program was proposed by the American-Canadian mathematician Robert Langlands (winner of the Wolf Prize in 1996) and discovered for the first-time tight and direct links between different branches of mathematics. Numbers theory (the field based on arithmetic, “sums”), algebraic representation theory and another field called “automorphic forms”, (which is related to harmonic analysis and assists, for example, in the physical study of waves and frequencies). By means of a new geometrical object he developed, which is now called “Drinfeld Chtoucas”, Drinfeld succeeded in proving some of the connections that had been indicated by the Langlands Program. In the eighties he invented the concept of algebraic “Quantum Group”, which led to a profusion of developments and innovations not only in pure mathematics but also in mathematical physics (for example in statistical mechanics).

Drinfeld and Beilinson, together created a geometric model of algebraic theory that plays a key role in both field theory and physical string theory, thereby further strengthening the connections between abstract modern mathematics and physics. In 2004 they jointly published their work in a book that describes important algebraic structures used in quantum field theory, which is the theoretical basis for the particle physics of today. This publication has since become the basic reference book on this complex subject.