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:
- Proving the first restriction estimates beyond Stein-Tomas and
related contributions towards the Kakeya conjecture;
- Proof of the boundedness of the circular maximal function in two
dimensions;
- Proof of dimension free estimates for maximal functions associated to
convex bodies;
- Proof of the pointwise ergodic theorem for arithmetic sets;
- Development of the global well-posedeness and uniqueness theory for
the NLS with periodic initial data;
- Proof that harmonic measure on a domain does not have full Hausdorff
dimension; and
- Proof with Milman of Mahler's reverse-Santalo conjecture in convex
geometry.
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:
- 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. Tao 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 rings 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 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:
- 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 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:
This masterful paper solves what had been the most important open problem in this area of harmonic analysis. The special case when d=1 was done in 1870 by Georg Cantor... V. L. Shapiro [Ann. of Math. (2) 66 (1957), 467–480; MR0090700] had solved the d=2 case, subject to a side condition that was later shown by R. Cooke to follow from the original hypothesis [Proc. Amer. Math. Soc. 30 (1971), 547–550; MR0282134].
...The power and originality used here prompted Victor Shapiro to say to me that he had casually mentioned this 40-year-old problem to Jean Bourgain, but if he had not it might very well have gone unsolved for another century.
- 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 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 the 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.
- 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 related 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 large bias, find a method for produce a fair coin
flip 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 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.
Thanks to @reuns for the answer in the comments. I've asked him to post as an answer, and I will accept it if he does. Meanwhile, his comments encouraged me to look again, and here is another approach, quite easy (Lemma 4.11 in Equivalents of the Riemann Hypothesis vol. II)
Let $\Xi(s)=\xi(1/2+is)$, so $\Xi(-s)=\Xi(s)$. Let $\gamma_n$ be the zeros with positive real part, so the Hadamard product looks like
$$
\Xi(s)=\xi(1/2)\prod_{n=1}^\infty\left(1-\frac{s^2}{\gamma_n^2}\right)
$$
Expanding $\Xi(s)$ as a power series about $s=0$
$$
\sum_{k=0}^\infty \frac{(-1)^k\xi^{(2k)}(1/2)}{2k!}s^{2k}=\Xi(s)=\xi(1/2)\left(1-s^2\left(\sum_n \gamma_n^{-2}\right)+s^4\left(\sum_{m,n}\gamma_m^{-2}\gamma_n^{-2}\right)-\ldots\right)
$$
Equating coefficients and the Newton Identities gives the moments as polynomials in the $\xi^{(2k)}(1/2)$, for example
$$
\sum_n\frac{1}{\gamma_n^6}=\frac{\xi^{(6)}(1/2)}{240\,\xi(1/2)}-\frac{\xi^{(2)}(1/2)\xi^{(4)}(1/2)}{16\,\xi(1/2)}+\frac{\xi^{(2)}(1/2)^3}{8\,\xi(1/2)}.
$$
Best Answer
With respect to the recent breakthrough, Bourgain states in this preprint:
This is also discussed in Ford's papers Vinogradov's Integral and Bounds for the Riemann Zeta Function, Zero-Free Regions for the Riemann Zeta Function, where he states:
Here Theorem $1$ is the inequality $$\zeta(\sigma+it)\leq A t^{B(1-\sigma)^{3/2}}\log^{2/3}t\ \ \ \ \ \ (t\geq 3,\ \frac{1}{2}\leq \sigma\leq 1)$$ which Ford proves with $A=76.2$ and $B=4.45$.
[For additional discussion on the implications of the Bourgain-Demeter-Guth breakthrough see, this recent preprint of Heath-Brown A New $k^{th}$ Derivative Estimate for Exponential Sums via Vinogradov’s Mean Value]