# Alain Connes

### Quick Info

Draguignan, Var, France

**Alain Connes**is a French mathematician and theoretical physicist who works in operator algebras and noncommutative geometry.

### Biography

**Alain Connes**was born in Draguignan, a town near Cannes in the Provence- Alpes-Côte-d'Azur region of southeast France. He entered the École Normale Supérieure in Paris in 1966, graduating in 1970. After graduating, Connes became a researcher at the Centre National de la Recherche Scientifique. His thesis

*A classification of factors of type III*was on operator algebras, in particular on von Neumann algebras, and the work was supervised by Jacques Dixmier. The thesis was presented to the École Normale Supérieure in 1973.

Connes spent the academic year 1974-75 at Queen's University in Ontario Canada. In 1976 he was appointed a lecturer at the University of Paris VI, then he was promoted to professor. He spent the year 1978-79 at the Institute for Advanced Study at Princeton. He left the University of Paris VI in 1980 but, the previous year, he had been appointed as professor at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette. Connes still holds this professorship.

In 1981 Connes returned to the Centre National de la Recherche Scientifique, this time as its director of research. He held this post for eight years. Another position he was appointed to was professor at the Collège de France at Rue d'Ulm in Paris in 1984. Connes currently holds both the position in the Institut des Hautes Études Scientifiques and the one in the Collège de France.

Connes's work is on operator algebras and it is put in context by Moore [5]:-

To place Alain Connes's fundamental and pioneering contributions to operator algebras in context, recall that von Neumann and Murray in the 1930s and 1940s were led by, among other things, the spectral theory of operators on Hilbert space, and by considerations of constructing mathematical models for quantum mechanical systems, to introduce what they called rings of operators - since renamed von Neumann algebras. ... One of the main problems has been and remains the classification of these algebras as intrinsic algebraic and topological objects.In [5] Moore describes Connes' thesis as:-

... already a major, stunning breakthrough in the classification problem.One of the first major international distinctions for Connes was an invitation to give one of the invited lectures at the International Congress in Helsinki in 1978. Four years later the award of a Fields Medal to Connes was announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. The Medal was not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Connes which led to his receiving the Medal were made at the 1983 International Congress. Araki, described Connes' contributions, see [2]:-

His most remarkable contributions areAfter describing these contributions, Araki also notes other work of Connes such as his applications of operator algebras to differential geometry and his work on non-commutative integration theory which he published in 1979.

(1) general classification and a structure theorem for factors of type III, obtained in his thesis

(2) classification of automorphisms of the hyperfinite factor, which served as a preparation for the next contribution

(3) classification of injective factors, and

(4) application of the theory of C*-algebras to foliations and differential geometry in general.

Moore, in [5], sums up some of Connes' work:-

Taken altogether Connes's work in the last decade on operator algebras and its applications has transformed the subject, and opened up entire new areas of research.Connes' recent work has been on noncommutative geometry and he published a major text on the topic in 1994. He has studied applications to theoretical physics and his work is of major importance showing, in the words of Araki in [2]:-

... the incredible power of Alain Connes and the richness of his contributions.Connes has received many awards for his work. In addition to the Fields Medal he was awarded the Prix Aimeé Berthé in 1975, the Prix Pecot-Vimont in 1976, the gold medal of the Centre National de la Recherche Scientifique in 1977, the Prix Ampère from the Académie des Sciences in Paris in 1980 and in 1981 the Prix de Electricité de France.

He was elected to the Académie des Sciences in 1982 being at that time only one of thirteen mathematicians in the Academy. He has been elected as a foreign member of the Royal Danish Academy of Sciences (1980), the Norwegian Academy of Sciences (1993) and the Canadian Academy of Sciences (1995).

Connes continues to hold the Léon Motchane Chair at the Institut des Hautes Études Scientifiques and the Chair of Analysis and Geometry in the Collège de France. In addition, in 2003 he was appointed Distinguished Professor at Vanderbilt University, Tennessee, in the United States. He has continued to receive major honours such as the Clay Research Award, presented on 24 May 2000 in Paris:-

... for revolutionizing the field of operator algebras, for inventing modern non-commutative geometry, and for discovering that these ideas appear everywhere, including the foundations of theoretical physics.On 24 January 2001 the Royal Swedish Academy of Sciences decided to award Connes their highly prestigious Crafoord Prize:-

... for his penetrating work on the theory of operator algebras and for having been a founder of the non-commutative geometry.In 2004 he received the Gold Medal from the Centre National de la Recherche Scientifique. The announcement reads:-

In naming Connes the 2004 Gold Medallist, the CNRS called him "one of the greatest mathematicians of our time." Throughout his career, Connes has applied himself to solving mathematical problems arising from quantum physics and the theory of relativity. He revolutionized the theory of operator algebras and was a primary founder of a new branch of mathematics - noncommutative geometry.In addition to these awards we note that Connes has been given honorary doctorates by Queen's University, Kingston, Ontario, Canada (1979), the University of Rome Tor Vergata, Italy (1997), and the University of Oslo, Norway (1999). In addition to the academies mentioned above, he has been elected to the American Academy of Arts and Sciences (1990), the United States National Academy of Sciences (1997), and the Russian Academy of Sciences (2003).

### References (show)

- Biography in
*Encyclopaedia Britannica.*http://www.britannica.com/biography/Alain-Connes - H Araki, The work of Alain Connes,
*Proceedings of the International Congress of Mathematicians, Warsaw, 1983***1**(Warsaw, 1984), 3-10. - H Araki, Profiles of 1982 Fields Medal winners (Japanese),
*Sugaku***35**(1) (1983), 70-77. - J Mi, The work of mathematicians awarded the Fields Medal in 1983 and 1986 (Chinese),
*J. Northwest Univ.***19**(1) (1989), 103-104. - C C Moore, The work of Alain Connes,
*Notices Amer. Math. Soc.***29**(1982), 499-501. - A K Seda, One aspect of the work of Alain Connes,
*Irish Math. Soc. Newslett. No.***11**(1984), 38-48.

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Written by J J O'Connor and E F Robertson

Last Update September 2009

Last Update September 2009