# Jean Georges Gilles Pisier

### Quick Info

Nouméa, New Caledonia

**Gilles Pisier**is a French mathematician, born in New Caledonia, whose main research field is functional analysis.

### Biography

**Gilles Pisier**was born in Nouméa, the capital of the French overseas country of New Caledonia in the southwest Pacific Ocean, situated about 1500 km east of Australia. His father, who was colonial governor of New Caledonia, had previously served as colonial governor of Vietnam. Gilles has an older sister, Marie-France, born in 1944 while their father was serving in Dalat, Vietnam; she is now a well-known French actress. In 1956, while Gilles was approaching his sixth birthday, the family returned to Paris when Gilles was educated. He attended the Lycée Buffon in Paris from 1966 to 1967, being awarded his Baccalauréat in 1967. He then entered the Lycée Louis-le-Grand in Paris where he prepared for university entrance. After spending the years 1967-69 at the Lycée Louis-le-Grand, Pisier entered l'École Polytechnique in Paris.

Pisier spent the years 1969-72 at l'École Polytechnique. He was awarded his Master's Degree in Mathematics from the Université Denis Diderot, Paris VII, in 1971 and a D.E.A. in Pure Mathematics (with distinction) from the Université Pierre et Marie Curie, Paris VI, in 1972. The D.E.A. (diplôme d'études approfondies) was a postgraduate degree which has now been replaced by the M.A.S. (Master of Advanced Studies). Then in October 1972, as a Stagiaire de Recherche (trainee researcher) at the Centre national de la recherche scientifique (C.N.R.S.) (National Centre of Scientific Research), he began research for his doctorate with Laurent Schwartz as his thesis advisor [2]:-

I started being advised by Laurent Schwartz, and I really profited from the extraordinarily challenging atmosphere of his laboratory at the École Polytechnique. Very early on I worked with Bernard Maurey who was my elder by three years and greatly influenced me. I really owe him a lot.In October 1974 Pisier became an Attaché de Recherche having already published a whole range of papers including:

*Bases suites lacunaires dans les espaces Lp d'après Kadec et Pelczynski*Ⓣ (1973); (with Bernard Maurey)

*Un théorème d'extrapolation et ses conséquences*Ⓣ (1973);

*"Type" des espaces normés*Ⓣ (1973);

*Sur les espaces de Banach qui ne contiennent pas uniformément de*$l^{1}_{n}$ Ⓣ (1973) and (with Bernard Maurey)

*Charactérisation d'une classe d'espaces de Banach par des propriétés de séries aléatoires vectorielles*Ⓣ (1973).

On 10 November 1977 Piser was awarded his doctorate from the Université Denis Diderot, Paris VII. He had already published 16 papers between 1973 and 1976, a remarkable achievement. He was appointed as a Chargé de recherche (Research associate) at the C.N.R.S. in October 1979 and, in October 1981, he was appointed Professor of Mathematics at the Université Pierre et Marie Curie, Paris VI. While continuing to hold this position, he was also appointed as a Distinguished Professor at the Texas A&M University in the United States from 1985, being appointed to the A G and M E Owen Chair of mathematics. At Paris VI he became Professeur, classe exceptionnelle (Distinguished Professor) in February 1991.

In [2] he gives the following overview of his work:-

My main research field is functional analysis, taken in a broad sense, ranging from the geometry of Banach spaces to the theory of star algebras or von Neumann algebras, through single operator theory on a Hilbert space. But, in fact, a hallmark of my work is allowing me to make significant breakthroughs in fields previously thought distinct, in probability and harmonic analysis. ... Perhaps the common thread that reappears every time is the interaction of random phenomena in algebraic structures (sets of random vectors, random matrices and random operators, etc. ...).Let us give a few more quotes from [2] in which Pisier gives a fascinating account of his contributions up to 2001. He explains that the main inspiration for his work on harmonic analysis was the book:-

... by J P Kahane ( "Random series of functions", 1968) based on the remarkable work of Paley, Salem and Zygmund and the work of his student P Billiard on random Fourier series. My American colleague Michael Marcus and me have been, from the beginning, substantially influenced by this book. We were really keen to solve the problems posed in this book, and it is with immense joy and pride that we have seen "return" in the second edition (1985) containing several of our results!Pisier's main contributions to random Fourier series are contained in two works, both with Michael Marcus. First there is the book

*Random Fourier series with applications to harmonic analysis*(1981). Jack Cuzick begins a review as follows:-

The authors generalize to the framework of locally compact abelian groups some results which were first established for Rademacher series and later for more general random Fourier series and for Gaussian processes. Most of the results are also obtained for compact nonabelian groups. While the results are stated in this general setting, the intuition developed in the concrete case of random Fourier series on a finite interval of the line pervades the presentation. The fundamental property in question is the a.s. uniform convergence of a random Fourier series and most other results are dependent on the existence (or not) of this property.The second of his major works with Michael Marcus on random Fourier series is the paper

*Characterization of almost surely continuous p-stable random Fourier series and strongly stationary processes*(1984).

Other major achievements by Pisier include solving a problem posed by Walter Rudin in 1960 on Sidon sets and doing fundamental work on probability in Banach spaces [2]:-

In this direction, my results are on the classical limit theorems: the law of large numbers, the central limit theorem, the law of iterated logarithm.Other areas of Pisier's research appear in the books

*Factorization of linear operators and geometry of Banach spaces*(1986) and

*The volume of convex bodies and Banach space geometry*(1989). The first of these came about because of his fascination with Grothendieck's famous article

*Resumé de la théorie métriques des produits tensoriels topologiques*Ⓣ (1956). Pisier writes [2]:-

The article ends with a list of six open problems which are a real research agenda for the next generation.In fact the 1986 book:-

... describes the development centred around the six problems formulated and discussed at the end of the Résumé, and presents various results which led to their solutions.The 1989 book mentioned above was reviewed by Mikhail Ostrovskii who explains that it:-

... gives an almost self-contained presentation of a number of recent results which relate the volume of convex bodies symmetric with respect to 0 in $n$-dimensional Euclidean space ... and the geometry of the corresponding finite-dimensional normed spaces. ... This book makes more accessible many important recent results from the theory of convex sets and the local theory of Banach spaces.Pisier has received many awards for his outstanding contributions. These include the Salem Prize (1979), the Cours Peccot at the Collège de France (1981), the Prix Carrière from the Académie des Sciences in Paris (1982), the Grand Prix from the Académie des Sciences in Paris: Prix Fondé par l'Etat (1992), the Faculty Distinguished Achievement Award in Research from Texas A&M University (1993) and the 1997 Ostrowski Prize. The Ostrowski Prize, presented by the Ostrowski Foundation created by Alexander Markowich Ostrowski and funded by his entire estate, was presented to Pisier on 25 April 1998 at the University of Leiden. The citation reads [1]:-

Pisier has obtained many fundamental results in various parts of analysis. In recent years he concentrated his efforts on the area of operator spaces. He transformed this area into a deep research area. In the framework of his research on this area Pisier solved in the last three years two extremely long-standing open problems. In C* theory he solved, jointly with Junge, the problem of uniqueness of C* norms on the tensor product of two copies of B(H), the algebra of all bounded operators on Hilbert space. Pisier and Junge were able to produce two such tensor norms that are nonequivalent. In operator theory Pisier solved (again negatively) the problem of whether an operator which satisfies the von Neumann inequality (with a constant) is similar to a contraction. Both results are based on elegant constructions, and the verifications of the examples are ingenious. Both of these results had a considerable impact and have already led to important further research.In June 2001 Pisier was awarded the Stefan Banach Medal by the Polish Academy of Sciences. He was invited to address the International Congress of Mathematicians in Warsaw in 1983 when he gave the lecture

*Finite rank projections on Banach spaces and a conjecture of Grothendieck*, and to give the plenary address

*Operator spaces and similarity problems*at the International Congress of Mathematicians in Berlin in 1998. Twice Pisier has been invited to give plenary addresses to the British Mathematical Colloquium. The first was at the meeting at Heriot-Watt University, Edinburgh, Scotland, in 1995, when he gave the lecture

*Operator spaces and group representations,*and the second was in 2004 at Belfast University when he spoke on

*Factorization of completely bounded maps on "exact" C*-algebras and operator spaces*. He has also been elected to many academies including the Institute of Mathematical Statistics (1989), the Académie des Sciences (corresponding member in 1984, full member 2002), the Real Academia de Zaragoza (2002) and the Polish Academy of Sciences (2005).

Finally let us mention two more recent books by Pisier. First these is the important

*Similarity problems and completely bounded maps*(1996). Christian Le Merdy begins his detailed review as follows:-

This book is devoted to three similarity problems, which we explain below, concerning unitary group representations, contractions on Hilbert space, and *-representations on C*-algebras. For each of them, the solutions that were known (at the time of the writing of the book) are given with complete proofs and the necessary background. Furthermore, many important related questions are discussed throughout the book. As the title suggests, the author emphasizes the link between these problems and the theory of completely bounded maps on operator algebras (or operator spaces). The book is truly remarkable for at least three reasons. First, it gathers in one volume almost all the important results on similarity problems and is therefore a valuable tool for anybody working on similarity problems. Second, the book contains several new or simplified proofs which shed some new light on several aspects of the problems. Third, the reader can find here several new results and complements not available elsewhere. It will undoubtedly become a reference book on the subject.Next we mention

*Introduction to operator space theory*(2003). Marius Junge writes:-

The recent theory of operator spaces combines operator algebra theory and the theory of Banach spaces. Pisier's introduction to the subject starts with the basic concepts in this theory and eventually leads the reader to top-notch results. ... Pisier's unique and clear way of presenting the material might even surprise researchers in the field: complicated results look very natural and simple in Pisier's presentation.

### References (show)

- Nesterenko and Pisier Share Ostrowski Prize,
*Notices Amer. Math. Soc.***45**(7) (1998), 879. - G Pisier, Notice sur les Travaux Scientifiques de Gilles Pisier, Académie des sciences (21 September 2001, Paris). http://www.academie-sciences.fr/membres/P/Pisier_Gilles_notice.htm

### Additional Resources (show)

Other websites about Gilles Pisier:

### Honours (show)

Honours awarded to Gilles Pisier

Written by J J O'Connor and E F Robertson

Last Update February 2010

Last Update February 2010