Ytdyklly

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.

Bourgain has made so many contributions to analysis. I will describe just a few of my favorites in harmonic analysis, and mention a few in number theory. First to cover my bases, I will name what I think he is best known for in harmonic analysis and are so significant that they do not fit your criteria:

• Progress on Stein's restriction conjecture and on the Kakeya conjecture.


• Introducing discrete restriction theory and the decoupling method with Ciprian Demeter.

I also include closely related problems to the above such as $Lambda(p)$ sets, Roth's theorem, the study of solutions to certain PDEs(especially the Schroedinger equation), problems in additive combinatorics, etc.

Bourgain also made numerous contributions to number theory aside from his work with Demeter and Guth. These include:

• Disproving Montgomery's conjecture via the Kakeya problem


• The affine sieve with Gamburd and Sarnak and related work on expansion with Kontorovich


• Sarnak's conjecture on Mobius orthogonality


• Quantitative versions of Oppenheimer's conjecture and related works (e.g. with Lindenstrauss, Michel and Venkatesh)


• Statistics of eigenfunctions and of lattice points on spheres with Rudnick and Sarnak


• Additive combinatorics - He did so much here that I am not sure where to begin, maybe Mordell's exponential sums revisited?

Here are three of Bourgain's results in harmonic analysis that I think were big when he proved them but are not as well known now:

1. 
   $L^p$-boundedness of the circular maximal function


2. Dimension-free bounds for maximal functions over convex sets


3. Pointwise ergodic theorems for arithmetic sets

The first two may not be known outside of harmonic analysis. The last one is known to ergodic theorists though they seem less interested in it than harmonic analysts. In succession, I will justify why I think of these works were important.

1. Bourgain proved that the circular maximal function is bounded on $L^p(mathbb{R}^2)$ for $p>2$. The circular maximal function is unbounded on $L^2$. That suggests that purely Fourier analytic methods are insufficient to solve the problem. This is contrast to higher dimensions where Fourier analysis suffices which was done by Stein about 10 years prior to Bourgain's result. Bourgain's result was considered quite an achievement at the time. To this date, no one has adequately explained Bourgain's proof to me. (There are other, "better" proofs nowadays.) More importantly I think Bourgain's solution foreshadowed the direction he would take harmonic analysis towards. Bourgain's proof combined Fourier analysis of the operator with incidence geometry and combinatorics. Later Bourgain used this perspective on the restriction problem mentioned above. These approaches currently dominate approaches to related problems in harmonic analysis. (Bourgain even joked during his talk at Stein's 80th birthday conference that harmonic analysts need to stop doing Fourier analysis and start doing combinatorics.)




2. Bourgain's work on dimension free inequalities began roughly at the same time as his work on the circular maximal function. I believe that this was also considered a big problem at the time. Here, Bourgain generalized a result of Stein and Strömberg for balls to convex sets with a bound on their geometry. His expertise of and intuition from functional analysis is on display in this work. Various lemmas in these works are still of use today. In particular he uses a discretized form of the classical Littlewood--Paley inequality to derive certain bounds which are much more intuitive than Stein's g-functions. (Bourgain did not introduce these LP decompositions - I would like to know who the first to do so was.) Recently these works were adapted to variational operators and discrete operators by Bourgain--Mirek--Stein--Wrobel. For instance, one may readily prove results for variational operators by combining Bourgain's approach for maximal functions with Jones--Seeger--Wright's work on variations.




3. Bourgain's work on pointwise ergodic theorems for arithmetic sets solved a then-outstanding problem about sparse averages in ergodic theory. (I think the question was posed by Furstenberg.) Bourgain combined methods from harmonic analysis and number theory to attack this problem. In these works, Bourgain introduced discrete operators, the circle method, transference principles and variational operators to harmonic analysts. This is a field that Bourgain initiated which is still active. One important contribution here is the paradigm that Bourgain demonstrated which is that the circle method is analogous to Littlewood--Paley theory in a sense. This paradigm was later used by Bourgain's seminal work on discrete restriction theory and periodic non-linear Schroedinger equation, and recently superseded by decoupling for the same problem.

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, one must answer what are his better known results are. Given the breadth of his works, 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 expert in Banach space himself—recounted being at a conference and having a renown colleague explain a difficult problem 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.

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. Terry 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 groups is substantially larger than the
  original set, unless you’re in certain 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’s 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:

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,
  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 no hope of importing any of the tools used in Carleson. 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 different problem which
  naturally occurs in computer science applications. Given a
  multi-sided die with unknown biases, but with some guarantee that no
  side is overwhelmingly biased 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, 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 this since my 20’s.” He then proceeded to rederive the relevant theory from memory in careful and precise writing on a blackboard. 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. 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.

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.

It follows from results in Bourgain's Acta 1984 paper (but I have the impression these results were announced some years ealier?) that the dual of the disc algebra $A({bf D})$ has cotype 2; and every bounded linear map from $A({bf D})$ to a Banach space of cotype 2 is 2-summing. As a corollary, the canonical injection $A({bf D}) hatotimes A({bf D}) to C({bf T})hatotimes C({bf T})$ has closed range. (See this MO question for my attempt to find out if this holds for any other examples of uniform algebras.)

These results are sometimes packaged together under the description "Grothendieck's inequality holds for the disc algebra", the point being that the Grothendieck's inequality can be interpreted as a statement about bounded bilinear maps on $C(K)$-spaces, and one doesn't expect proper uniform algebras to have all the nice Banach-space properties that $C(K)$-spaces do.

His work with Dilworth, Ford, Konyagin, and Kutzarova solved an open problem in compressed image sensing. Thanks to this blog post of Tao from 2007, we can get a feel of the general mentality before this work. One of their key insights was to incorporate the sum-product phenomenon.

One body of works which I found mind Boggling is related to the question: "Can you hear the degree of a map". I think the general theme is about the connection between Fourier analysis and algebraic topology. The beautiful videotaped lecture by Haim Brezis is about this topic. On the negative side, Bourgain and Kozma's paper One cannot hear the winding number "constructed an example of two continuous maps $f$ and $g$ of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis." On the positive side (going back to Cauchy) under further restrictions on $f$, the degree (and other topological invariants) can be "heard" (and is equal to $sum_{n=-infty}^{infty}|hat f(n)|^2n$), and Bourgain mainly with Brezis and other coauthors have made various important contributions.