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 finally 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.)