Interview with Gilles Pisier, co-director of one of the ICMAT – Severo Ochoa Laboratories

Gilles Pisier is Emeritus Professor at Texas A&M University (USA) and at Sorbonne University (France). His work in functional analysis, probability, and operator algebras has earned him numerous recognitions, including the Salem Prize (1979) and the Ostrowski Prize (1997). His research has had a strong influence in diverse areas such as harmonic analysis, von Neumann algebras, and the geometry of Banach spaces. Pisier obtained his PhD in 1977 under the supervision of the celebrated analyst Laurent Schwartz—Fields Medalist in 1950—at Université Paris VII (France). He is a member of the French Academy of Sciences. Alongside Mikael de la Salle (CNRS and Université de Lyon), he is co-director of one of the new ICMAT–Severo Ochoa Laboratories. We spoke with him during his visit to the ICMAT, where he gave two lectures as part of the center’s Distinguished Lectures series on May 23.

Ágata Timón y Laura M. Iraola (ICMAT-CSIC)

In which field(s) would you place your research?

I mainly work in analysis, but I have always liked probability. My thesis advisor, Laurent Schwartz, encouraged me to explore probability. He was clearly an analyst and did not actually work in probability himself, but he was very drawn to the field. There was a time when probabilists considered me one of their own—I have to say, they were a very open community. That happened when I was working on probability in Banach spaces, a blend of probability and functional analysis, which was a popular subfield in the 1970s. At the time, they even named me a Fellow of the Institute of Mathematical Statistics (IMS), a distinction awarded to leading probabilists.

‘The probabilists considered me one of their own, they were a very open community’

Would you describe yourself more as a problem solver or a theory builder?

I think I fit more in the category of problem solvers, though that distinction is debatable. I enjoy working on problems that have resisted solution for decades, and I -sometimes spend a long time on them without making any progress, trying to find a new angle of attack. Rather than using “brute force”—as some mathematicians I deeply admire do—I prefer to take my time.

You’ve said that a common thread in your work is the interaction of random phenomena in algebraic structures (sets of random vectors, random matrices, random operators, etc.). Where has this perspective taken you?

The best example of this is my work on random Fourier series. Walter Rudin had posed the problem of characterizing Sidon sets of integers as Λ(p)-sets, with the Λ(p)-constant growing like the square root of p as p tends to infinity. I was able to solve this problem quite easily using the so-called metric entropy condition, which characterizes bounded Gaussian processes. The connection came from viewing random Fourier series simply as stationary Gaussian processes. This was a rather unexpected application of probability to harmonic analysis—and it earned me the Salem Prize. In retrospect, it may seem simple, like knowing the right things at the right time, but a few years later I pushed these ideas further to obtain a purely arithmetic characterization of Sidon sets, a deeper problem that, I’m proud to say, caught the attention of Paul Erdős.

‘I like to work on problems that have resisted for decades, and I usually spend a lot of time working on them without moving forward, trying to find a new angle of attack’

What problems interest you today?

For me, the most exciting problems involve estimating the norms of random matrices with operator (or matrix) coefficients. Specifically, I’m thinking of the work by Charles Bordenave and Benoît Collins. I am also interested in the work of Ramón van Handel and his collaborators on the strong convergence of random matrices, as well as Michael Magee’s work on ‘limit groups’ and his collaboration with Mikael de la Salle. Thanks to these contributions, things are progressing very rapidly in this area, which was initiated by Uffe Haagerup and Steen Thorbjørnsen about 20 years ago.

The main focus of the ICMAT Lab you co-direct is noncommutative Lᵖ-spaces. Could you explain what they are and why they are of interest to the research community?

The usual Lᵖ-spaces are ubiquitous in classical analysis, whether functional or harmonic. They go back to Henri Lebesgue (hence the “L”) and are associated with a measure space. They are a fundamental tool in measure theory, for example, in studying the actions of groups on measure spaces.

Naturally, a parallel theory of Lᵖ-spaces adapted to noncommutative groups emerged, and this led John von Neumann to introduce and extensively study (with Francis Murray) the algebras that now bear his name. Although their study lies firmly within pure mathematics, von Neumann’s original motivation was the foundations of quantum mechanics. Von Neumann algebras are now considered noncommutative analogues of measure spaces, where trace functionals replace the classical notion of measure. This naturally leads to the notion of noncommutative Lᵖ-spaces.

What problems will be addressed in the Lab?

It’s hard to identify a single main goal—each of us has our own obsessions. Currently, my favorite problem—my personal obsession—is the Connes embedding problem, and also whether local lifting implies global lifting. This was the subject of my Distinguished Lecture.

There’s a proposed solution to this problem from five years ago, coming from theoretical computer science and complexity theory, using ideas from a perspective that’s quite foreign to analysis. It comes from a very different field—even from what some pure mathematicians would consider “mathematics.” So far, nobody has been able to fully verify the solution. There are reasons to think it may be correct, but it’s extremely difficult to be sure: it’s 200 pages long and uses tools entirely unrelated to the original formulation. My goal now is to revisit the problem from the analytical point of view and to find another proof. This will be one of the key topics in the Lab.

Another example is a different problem also posed by Alain Connes, which greatly interests both Javier Parcet and Mikael de la Salle. It concerns discrete groups with Kazhdan’s property (T). The central question (Connes’ rigidity problem) is whether the von Neumann algebra of such groups encodes the structure of the group itself. Previous work by Javier and Mikael suggests that various approximation properties of the noncommutative Lp spaces for groups such as SL3(Z) might lead to major progress. During these days, I had the opportunity to get a little update on the ongoing work of Parcet, de la Salle and Tablate. In all, they are trying to obtain substantially new information about Fourier and Schur Lp multipliers over Euclidean spaces and semisimple lattices, with an eye on Connes’ rigidity conjecture.

‘We expect all the leading figures in noncommutative Lᵖ-spaces to take part in our Lab’

What would you say are the Lab’s main strengths in facing these challenges?

The size of the field (noncommutative Lᵖ-spaces) and of the related community is relatively small, so we can hope that all the leading figures in the field will participate in one way or another in our Lab.

How and when did you meet Mikael de la Salle (Université de Lyon, France) and Javier Parcet?

Mikael de la Salle was my PhD student. I gave a lecture on martingales in Banach spaces at the École Normale Supérieure, partly aimed at attracting potential doctoral students, and Mikael must have liked it because he approached me afterward and asked if I would supervise his thesis. That turned out to be very easy—his extraordinary research talent quickly became evident. Since he graduated in 2009, it has been a pleasure to follow his mathematical achievements and occasionally (though not very often) discuss mathematics with him. I’ve learned a great deal from our exchanges, especially on the topic of random matrices.

I met Javier Parcet very early in his career. I believe it was Françoise Piquard who first told me about his work. I invited him to give a seminar in Paris shortly before he completed his PhD. Later, he spent a few months as a postdoctoral researcher at Texas A&M, and we collaborated on a couple of papers—although I was only there part of the time. He then continued his stay in the United States at the University of Illinois at Urbana-Champaign, working with Marius Junge—who will also be a collaborator in the Lab—and together they developed many results in areas that were also of great interest to me. In particular, they significantly expanded the scope of a concept I had introduced: noncommutative vector-valued Lᵖ-spaces. Naturally, we have remained in close contact ever since.

Could you mention a few mathematicians who had a particularly strong influence on your career?

During the first years I was intensively collaborating with Bernard Maurey who was two or three years older than me and I enormously benefitted from all I learnt from him. Later on, I remember Nicolas Varopoulos was very encouraging. I remember him as one of the first high level mathematicians who told me he really liked my work. Perhaps others appreciated it quietly—mathematicians are often very reserved—but I remember Varopoulos’s support was important in helping me gain confidence. Among probabilists, Donald Burkholder supported me as well, and among Banach space theorists, Joram Lindenstrauss and Lior Tzafriri were especially encouraging.

In particular, Lindenstrauss and Tzafriri invited me to visit Jerusalem for two months at the end of 1974. At that time, I was still technically a doctoral student—I defended my thesis in 1977, having started in 1972—but this was under the old French thèse d’État system, later assimilated to the habilitation. I joined the CNRS in 1972 based solely on good grades in my graduate studies, which was not at all unusual at the time. Things have changed! In any case, as soon as I began publishing papers, the established mathematicians I met at conferences treated me as a colleague—and I greatly appreciated that.

By the time you were awarded your doctorate at Paris VII, you had already published 16 papers—something quite rare. What do you think were the key factors behind this early emergence of your mathematical talent?

Actually, it took me five years, which isn’t particularly fast. But because I already held a CNRS position, there was little incentive to rush through the administrative step of defending the thesis.
As I mentioned, it was a thèse d’État, essentially a habilitation thesis—a collection of papers with an introductory, summarizing text.

‘It has been an incredible privilege to have turned my passion into a profession’

Looking back, what would you say has been the most valuable thing that mathematics has brought you, both personally and professionally?

Probably the same answer many of us would give: it has been an incredible privilege to be able to spend almost all my time on my favorite pastime. I’ve turned my passion into a profession.