I have just learned that Jean Bourgain passed away last week in Belgium, aged 64, after a prolonged battle with cancer. He and Eli Stein were the two mathematicians who most influenced my early career; it is something of a shock to find out that they are now both gone, having died within a few days of each other.

Like Eli, Jean remained highly active mathematically, even after his cancer diagnosis. Here is a video profile of him by National Geographic, on the occasion of his 2017 Breakthrough Prize in Mathematics, doing a surprisingly good job of describing in lay terms the sort of mathematical work he did:

When I was a graduate student in Princeton, Tom Wolff came and gave a course on recent progress on the restriction and Kakeya conjectures, starting from the breakthrough work of Jean Bourgain in a now famous 1991 paper in Geom. Func. Anal.. I struggled with that paper for many months; it was by far the most difficult paper I had to read as a graduate student, as Jean would focus on the most essential components of an argument, treating more secondary details (such as rigorously formalising the uncertainty principle) in very brief sentences. This image of my own annotated photocopy of this article may help convey some of the frustration I had when first going through it:

Eventually, though, and with the help of Eli Stein and Tom Wolff, I managed to decode the steps which had mystified me – and my impression of the paper reversed completely. I began to realise that Jean had a certain collection of tools, heuristics, and principles that he regarded as “basic”, such as dyadic decomposition and the uncertainty principle, and by working “modulo” these tools (that is, by regarding any step consisting solely of application of these tools as trivial), one could proceed much more rapidly and efficiently. By reading through Jean’s papers, I was able to add these tools to my own “basic” toolkit, which then became a fundamental starting point for much of my own research. Indeed, a large fraction of my early work could be summarised as “take one of Jean’s papers, understand the techniques used there, and try to improve upon the final results a bit”. In time, I started looking forward to reading the latest paper of Jean. I remember being particularly impressed by his 1999 JAMS paper on global solutions of the energy-critical nonlinear Schrodinger equation for spherically symmetric data. It’s hard to describe (especially in lay terms) the experience of reading through (and finally absorbing) the sections of this paper one by one; the best analogy I can come up with would be watching an expert video game player nimbly navigate his or her way through increasingly difficult levels of some video game, with the end of each level (or section) culminating in a fight with a huge “boss” that was eventually dispatched using an array of special weapons that the player happened to have at hand. (I would eventually end up spending two years with four other coauthors trying to remove that spherical symmetry assumption; we did eventually succeed, but it was and still is one of the most difficult projects I have been involved in.)

While I was a graduate student at Princeton, Jean worked at the Institute for Advanced Study which was just a mile away. But I never actually had the courage to set up an appointment with him (which, back then, would be more likely done in person or by phone rather than by email). I remember once actually walking to the Institute and standing outside his office door, wondering if I dared knock on it to introduce myself. (In the end I lost my nerve and walked back to the University.)

I think eventually Tom Wolff introduced the two of us to each other during one of Jean’s visits to Tom at Caltech (though I had previously seen Jean give a number of lectures at various places). I had heard that in his younger years Jean had quite the competitive streak; however, when I met him, he was extremely generous with his ideas, and he had a way of condensing even the most difficult arguments to a few extremely information-dense sentences that captured the essence of the matter, which I invariably found to be particularly insightful (once I had finally managed to understand it). He still retained a certain amount of cocky self-confidence though. I remember posing to him (some time in early 2002, I think) a problem Tom Wolff had once shared with me about trying to prove what is now known as a sum-product estimate for subsets of a finite field of prime order, and telling him that Nets Katz and I would be able to use this estimate for several applications to Kakeya-type problems. His initial reaction was to say that this estimate should easily follow from a Fourier analytic method, and promised me a proof the following morning. The next day he came up to me and admitted that the problem was more interesting than he had initially expected, and that he would continue to think about it. That was all I heard from him for several months; but one day I received a two-page fax from Jean with a beautiful hand-written proof of the sum-product estimate, which eventually become our joint paper with Nets on the subject (and the only paper I ended up writing with Jean). Sadly, the actual fax itself has been lost despite several attempts from various parties to retrieve a copy, but a LaTeX version of the fax, typed up by Jean’s tireless assistant Elly Gustafsson, can be seen here.

About three years ago, Jean was diagnosed with cancer and began a fairly aggressive treatment. Nevertheless he remained extraordinarily productive mathematically, authoring over thirty papers in the last three years, including such breakthrough results as his solution of the Vinogradov conjecture with Guth and Demeter, or his short note on the Schrodinger maximal function and his paper with Mirek, Stein, and Wróbel on dimension-free estimates for the Hardy-Littlewood maximal function, both of which made progress on problems that had been stuck for over a decade. In May of 2016 I helped organise, and then attended, a conference at the IAS celebrating Jean’s work and impact; by then Jean was not able to easily travel to attend, but he gave a superb special lecture, not announced on the original schedule, via videoconference that was certainly one of the highlights of the meeting. (I unfortunately do not have records on the precise topic of the talk; perhaps it pertained to applications of decoupling theorems.)

I last met Jean in person in November of 2016, at the award ceremony for his Breakthrough Prize, though we had some email and phone conversations after that date. Here he is with me and Richard Taylor at that event (demonstrating, among other things, that he wears a tuxedo much better than I do):

Jean was a truly remarkable person and mathematician. Certainly the world of analysis is poorer with his passing.