Jean Bourgain passed away on December 22, 2018.
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions, such as the proof of Vinogradov's conjecture with Guth and Demeter. He was of course very prolific and his work spanned many areas. There are a few questions at mathoverflow (e.g., here and here) regarding his technical contributions.
I am sure there are some of his contributions that may not be as widely known, but deserve to be so. Perhaps focusing on results with unexpected aplications may be a good start, but any answers are welcome.
I was unable to find the list of talks at the conference in his honour at the IAS mentioned and linked to in Tao's post, but that may be just me.
Thanks to Josiah Park for pointing out the link to the IAS Bourgain conference, titled 'Analysis and Beyond'. In fact, the videos of the talks are also there.
Edit: Revisiting this question, I read again all the wonderful and informative answers. Mathoverflow is a greeat community.
Edit 2: Sept 2020: There is an upcoming collection of papers on Bourgain's work to be published by the Bulletin of the American Mathematical Society. Terry Tao has blogged about it here, as well as uploading his contribution to arXiv .
Best Answer
There are many accounts of the truly exceptional breadth, depth and ingenuity of Jean's work and the quickness of his mind. These accounts are not exaggerations or embellishments. Jean collected nearly every honor and prize possible, including the Fields medal and Breakthrough prize. Even considering this, it is hard not to think that in some ways the weight of his contributions has still somehow gone underappreciated within the larger mathematical and scientific community.
Before attempting to answer what his "lesser known" results are, let me give a rundown of his "better known" results are. Given the breadth of his work, the answer to this first question is likely a function of the mathematical expertise and taste of the answerer. One place to start is his 1994 Fields medal citation which discusses the following:
What's remarkable about the 1994 Fields medal citation is that it manly focuses on his then-recent contributions to harmonic analysis from the late 1980's and early 1990's and completely skips over the equally impactful and significant contributions to functional analysis and Banach space theory that rose Jean to prominence in the early 1980’s.
The first story about Bourgain I ever heard was during my first year of graduate school in which Ted Odell—an leading expert in Banach space theory himself—recounted being at a conference and having a renown colleague explain a difficult problem about Banach spaces he had been working on for several years. Anyone remotely familiar with the Jean’s legend can already fill in the ending. Jean joins the conversation, listens to the problem, and emerges the next morning with a complete solution. Indeed there are dozens of similar stories.
As one might glean from the above story, Jean was well known for his competitive spirit. In his memoir, Walter Rudin recounts that Jean told him that his 1988 solution to the $\Lambda(p)$ problem, a question in Harmonic Analysis which Jean reduced to deep statements about the geometry of stochastic processes and then solved, was the most difficult problem he had ever solved and that he was disappointed that that wasn't mentioned in his Fields medal citation. Rudin also recounts there that at his retirement conference he offered a prize for solving a problem that had eluded him for many years about the radial variation of bounded analytic functions, a topic Rudin was perhaps the world expert on. Of course, Jean solved the problem and claimed the prize in short order. This is perhaps one of the many examples of results that would stand out as exceptional on any bibliography other than Jean's.
While we are still on the better known results, we should speak of his banner results subsequent to 1994:
Having summarized perhaps a dozen results that one might considered his better known work, let me turn the question of what are some of his lesser known works. Here are three examples that reflect my own personal tastes than anything else:
There are many other contributions that could have been career making results for others, such as: improving bounds on Roth’s theorem; finding a deep connection between the Kakeya conjecture and analytic number theory; obtaining the first super-logarithmic bound on the cosine problem (this had been worked on by Selberg, Cohen, and Roth, among others); obtaining polynomial improvements to the hyperplane slicing problem, development of the Radon-Nikodym property in functional analysis, development of the Ribe program, progress on Falconer’s conjecture, disproving a conjecture of Montgomery about Dirichlet series, disproving a conjecture of Hormander on oscillatory integrals, obtaining (what was for a long time) the best partial progress on the Kadison-Singer conjecture, constructing explicit ultra-flat polynomials, work on invariant Gibbs measure, etc.
Not that it belongs on Jean’s highlight reel, but I’d like to share two stories from my own collaboration with Jean. The first came about when I ran into Jean on a weekend in Princeton during my time at IAS. I told Jean I was thinking about a problem about Sidon sets. This is a topic I knew he extensively worked on in the early to mid-1980’s and had since became dormant. He listened to my problem and ideas carefully and then said “well I haven’t thought about these things since I was in my 20’s.” He then proceeded to re-derive the proofs of the relevant theorems in careful and precise writing on a blackboard from scratch. Over the course of the next few hours the first result of our joint paper was obtained. We eventually broke for the day, but I receive an email the next morning that he had made further progress that evening.
Jean also wrote an appendix to a paper I wrote with two coauthors. In the paper we raised two problems related to our work that we couldn’t settle. Shortly after posting the preprint on the arXiv, I received, out of the blue, an email from Jean with a solution to one of the problems. Knowing Jean’s competitive spirit, I thanked him for sharing the development but pointed out that we had raised two problems in the paper. I received a solution to the second the next morning.
Bell famously wrote that mathematics was set back 50 years because Gauss didn’t publish all his results. I’m certain that many areas of mathematics are 50 or more years ahead of where they would have been without Jean’s contributions.