The Royal Swedish Academy of Sciences decided at its meeting on 24 January 2001 to award the 2001 Crafoord Prize in mathematics to Alain Connes, Professor at IHES and Collège de France, Paris, “for his penetrating work on the theory of operator algebras and for having been a founder of the non-commutative geometry”.

The French mathematician Alain Connes is counted among the world’s foremost mathematicians. He has made pioneering and unique contributions to the theory of *operator algebras and non-commutative geometry. *The latter is a new field of mathematics, in the creation of which Alain Connes has played a decisive part.

## Operator algebras

At the beginning of the 1930s the Hungarian-American mathematician John von Neumann started developing a theory for the algebras of operators in what is termed Hilbert space. He was inspired by developments in quantum mechanics where these algebras played a central part. von Neumann delimited a particular type of such algebras, which mathematicians now term “von Neumann algebras”, together with a special type of building-block of which such algebras are formed, termed factors. Together with F. J. Murray, von Neumann roughly classified these algebras into three types, I, II and III. von Neumann later turned to other interests and it was not until the period between 1966 and 1971 that the development was resumed and many different type-III factors were constructed. It was here that Alain Connes entered the picture, in 1972. While a good deal of preparatory work had been done, during the next ten years Connes totally revolutionised this picture by solving most of the unsolved problems in the area. For this he was awarded the Fields Prize in 1983.

By further developing this theory, Alain Connes soon entered new, untrodden territory. An entirely new area of mathematics began to take shape, the *non-commutative geometry.*

## Non-commutative geometry

Geometry as it has developed from Descartes onwards is based on the notion of points in systems of coordinates. Geometric properties are reflected in algebraic properties of functions where points in space represent variables. The algebras that can be constructed in this way are usually commutative, meaning that the result of an operation is independent of the order in which it is performed. An example is ordinary multiplication: a · b = b · a.

But in the study of the algebras of operators one often encounters *non-commutative *properties. Matrix multiplication is an example of something that is not normally commutative: A · B does not equal B · A. Alain Connes’ idea is, using such a non-commutative algebra as a base, to consider it as an expression of a fictitious “non-commutative” space. Such a space requires a different and more abstract conceptual apparatus than what we are used to from classical geometry. The concept of point, for example, is meaningless in non-commutative geometry.

Alain Connes’ work has also provided powerful new methods useable in theoretical physics for treating e.g. renormalization theory and the standard model of quantum and particle physics. He has also demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann hypothesis of the zeta function, considered the most famous open problem in mathematics.

## Alain Connes

Alain Connes, 53, was born in Draguignan (Var), France on 1 April 1947. He attended the Ecole Normale Supérieure (ENS) in Paris 1966-70. Since 1979 he has held the Léon Motchane Professorship at the Institut des Hautes Études Scientifiques (IHES) at Bures-sur-Yvette outside Paris, and since 1984 also a professorship in analysis and geometry at the Collège de France in Paris. He received the Fields Medal in 1983 (the most highly regarded mathematical prize in the world) and is a member of many scientific academies including Académie des Sciences, Paris, and National Academy of Sciences, USA.

The 2001 Crafoord Prize will be presented by H.M. the King of Sweden on 26 September 2001 at a ceremony at the Royal Swedish Academy of Sciences in Stockholm. The prize consists of a gold medal and 500,000 USD.

**The Anna-Greta and Holger Crafoord Foundation **was established in 1980 for promoting basic research in mathematics, astronomy, the biosciences (particularly ecology), the geosciences and polyarthritis (joint rheumatism). The prize was awarded for the first time in 1982 in mathematics and has since been awarded by subject area in the order given above. The Crafoord Prize consists of an international prize and research grants to Swedish scientists.

**Earlier laureates **in mathematics are Vladimir I. Arnold, Russia and Louis Nirenberg, USA (1982), Pierre Deligne, Belgium and USA and Alexander Grothendieck*, France (1988), and Simon Donaldson, England and Shing-Tung Yau, USA (1994).

* Grothendieck declined the prize