Before attempting to answer what his lesser known"lesser known" results are, one must answer what arelet me give a rundown of his better known"better known" results are. Given the breadth of his workswork, 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 Harmonicharmonic 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 also well known for his competitive spirit. In his memoir "The Way I remember It", 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.
- A 1999 proof of global wellposedness of defocusing critical nonlinear
Schrödinger equation in the radial case. This was a seminal paper in
the field of dispersive PDEs, which lead to an explosion of
subsequent work. An expert in the field once told me that the history
of dispersive PDE is most appropriately segregated into periods
before and after this paper appeared.
- The development of sum-product theory. TerryTao gives an inside account
of the backstory in his remembrances of Jean on his blog. Roughly speaking these are elementary looking
inequalities that state that either the sum set or product set of an
arbitrary set in certain groupsrings is substantially larger than the
original set, unless you’re in certain well-understood / uninteresting situations.
After proving the initial results, Jean realized that it was a key
tool for controlling exponential sums in cases where there were no
existing tools and no non-trivial estimates even known. He then
systematically developed these ideas to make progress on dozens of
problems that were previously out of reach, including improving
longstanding estimates of Mordel and constructing the first explicit
examples of various pseudorandom objects of relevance to computer
science, such as randomness extractors and RIP matrices in compressed
sensing.
- The development (with Demeter) of decoupling theory. This was one of
Jean’s main research foci over the past five years and led the full
resolution of Vinogradov’s Main Conjecture (with Demeter and Guth) a
central problem in analytic number theory. It also led to
improvements to the best exponent towards the Lindelöf hypothesis, a
weakened often substitute for the Riemann Hypothesis and a record
once held by Hardy and Littlewood, as well as the world record on
Gauss’ Circle Problem. It must be emphasized here, that the source of
these improvements were not minor technical refinements, but the
introduction of fundamentally new tools. The decoupling theory also
led to significant advances in dispersive PDEs and the construction
of the first explicit almost $\Lambda(p)$ sets.
Having summarized perhaps a dozen results that one might considered as Bourgain’shis 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:
- Proving the spherical uniqueness of Fourier series. A fundamental
question about Fourier series is the following: if $\sum_{|n|<R} a_n
e(nx) \rightarrow 0$ for almost every $x$ as $R \rightarrow \infty$
must all of the $a_n$’s be zero? The answer is yes, and this is a
result from the nineteenth century of Cantor. The question what
happens in higher dimensions naturally follows. In the 1950’s this
was considered a central question in analysis and a chapter of
Zygmund’s treatise Trigonometric Series is dedicated to it. I also
believe it was the subject of Paul Cohen’s PhD dissertation. This was
resolved in two dimensions in the 1960’s by Cooke, but the proof
techniques break down in higher dimensions. Jean completely solved
this problem in 2000, introducing a fundamentally new approach based
on Brownian motion. The MathSciNet review states:
- Progress towards Kolmogorov’s rearrangement problem. One of the great
results in twentieth century Harmonic analysis is Carleson’s theorem that the Fourier series of an $L^2$ function converges almost everywhere and the slightly stronger results that maximal operator is bounded on $L^2$. Now these are deep results
about characters and therefor rely on careful and deep tools from,
well, Fourier analysis. In the very early 1900’s Kolmogorov asked if
(after possibly reordering it) the result might hold for an arbitrary
orthonormal system. If true this is incredibly deep as: (1) Jean
proved via an ingenious combinatorial argument that this would imply
the Walsh case of Carleson’s theorem and (2) in this generality there
is seems to be no hope of importing any of the tools used in Carlesonthe proof of Carleson's theorem. Despite
this, appealing to deep results from the theory of stochastic
processes Jean proved the result up to a $\log \log$ loss. The
general problem remains open, and might well remain so for the next
100 years. When I first met Jean at the Institute I asked him about
this problem. He told me that prior to the conversation, to the best
of his knowledge, there were only two people on Earth who cared about
the question: him and Alexander Olevskii. He seemed pleased to find a
third in me.
- Construction of explicit randomness extractors. Most readers here
will probably be familiar with the following puzzle from an introductory
probability class: Given two coins of unknown bias, simulate a fair
coin flip. There’s an elegant solution attributed to Von Newmann.
Randomness extractors seek to address a differentrelated problem which
naturally occurs in computer science applications. Given a
multi-sided die with unknown biases, but with some guarantee that no
side is occurs with overwhelmingly biasedlarge bias, find a method for produce a fair coin
flip given using only two roles of the dice. Now there’s a parameter
(referred to as the min-entropy rate) that regulates how biased the
dice can be. The goal is to construct algorithms that permit as much
bias is possible. For many years, ½ was the limitation of known
methods. In 2005, using the sum-product theory mentioned above, Jean
broke the ½ barrier for the first time. This was a substantial
advancement in the field, yet is just one of a dozen or so
applications in a paper titled “More on the sum-product phenomenon in
prime fields and applications”.
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 anything near 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 thisthese things since I was in my 20’s.” He then proceeded to rederivere-derive the proofs of the relevant theory from memorytheorems 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. This came about after the coauthors and I wrote the paper and released it on the arXiv. 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 then 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.
