
Jean Bourgain - furcyd
https://terrytao.wordpress.com/2018/12/29/jean-bourgain/
======
aportnoy
Tangential, but Tao is such a great teacher of mathematics and person. A rare
combination of skill and modesty. Highly recommend his analysis and measure
theory textbooks.

~~~
weinzierl
> [..] Tao is such a great teacher of mathematics and person.

And a good writer too. I could read his blog posts all day long. In this post
I particularly like how he describes his feelings when reading through one of
Jean's papers.

>> _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._

> A rare combination of skill and modesty.

That's what I was thinking when reading the following paragraph. He is one of
the greatest mathematicians of our times but what he writes is so relatable.

>> _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._

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GlenTheMachine
Fascinating video. I use Fourier analysis all the time. It's a foundational
tool of controls engineering, which is my field. But I never knew it was
invented to describe thermal dynamics problems.

~~~
hprotagonist
Depending on where you draw the line (heh), Lagrange's vibrating string work
predates the heat equation work by a few years.

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cosmic_ape
I learned so much from Bourgain's work. While most of his work was in Fourier
analysis, and later in number theory, he really was a very versatile analyst.
Just two of his early papers, on embeddings of metric spaces, and on
concentration for covariance matrices started whole fields later. They are
very useful now in theoretical machine learning.

~~~
heinrichf
Interesting! Could you share examples of works in theoretical ML using these ?

~~~
cosmic_ape
Well, lets try. For the embeddings, you'll have to search the internet for
details, and this may be not a "proper" ML. But the general idea is that if
you have some nice euclidean algorithm, like k-means, and you want to cluster
some data w.r.t a different metric, you could try putting the data in the
euclidean space without distorting the distances too much and then apply the
algo. there. For k-means in particular the nearest neighbor search in
Euclidean space can be made efficient, I think. This didn't take off much
because most of the results were unfortunately negative. Although there were
some recent positive results, even with discussion here [1]. Bourgain had some
early results on the embeddability of finite metric spaces into classical
spaces.

The concentration of covariance matrices is fundamental for most "spectral"
methods, in particular spectral clustering. See for instance [2]. People
proved and reproved these concentration results a lot of times, but Bourgain's
was the earliest with N\log N sample complexity for log concave measures,
improving on a result of Lovazs et al. (!!!) More importantly, the method with
which he proved this, controlling the concentration by decomposing into
bounded and "rarely non-zero" parts is now all around in statistical learning
theory. So each time people want to prove uniform law of large numbers and
nothing standard works, people try to find a good decomposition.

[1]
[https://news.ycombinator.com/item?id=17752836](https://news.ycombinator.com/item?id=17752836)
[2] Clustering with Spectral Norm and the k-means Algorithm, Amit Kumar and
Ravindran Kannan (Section 6...)

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sn41
Bourgain's paper on monotone expanders is the most difficult paper I have
tried to read, since it has some implications on open questions in
computational complexity. Spoiler alert: I have still not understood it.

~~~
throwawaymath
Is this the paper you're referencing?
[https://arxiv.org/abs/1108.2555](https://arxiv.org/abs/1108.2555)

~~~
sn41
Yes, that's the one.

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southerndrift
> 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.

Does anybody know what those "basic" tools and heuristics are? Has anybody
created a list of them somewhere for others to understand the papers more
easily?

~~~
southerndrift
The answer is in the blog comments:

>Dear Terry,

>Have you ever considered writing a blog post on “Bourgain’s toolbox”
summarizing the kinds of techniques used continually (without explanation) in
his papers?

>[...]

Tao's final conclusion, besides other hints like
[http://www.tricki.org/](http://www.tricki.org/) is:

> I think that ultimately the best way to acquire these tools is to constantly
> be exposed to interesting new mathematics, whether it is through reading a
> challenging paper from an expert, or listening to a talk on a topic outside
> your own specialty, or by working (particularly with collaborators) on a
> problem outside of your own comfort zone. Bourgain is not the only author
> whose writings happen to be particularly worth investing the effort to
> properly read; I for instance found reading Szemeredi’s work or Perelman’s
> work to be a broadly similar experience, and I am sure that there are
> several other mathematicians with this property also.

~~~
southerndrift
Though:

>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.

If he hasn't managed to do this on his own, how should anybody else acquire
those tools just by reading challenging papers?

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raphlinus
That is a beautiful memorial. Thank you Terry for writing it, and helping me
appreciate Jean's life and work better.

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yesenadam
Also off-topic: Why are 6 of the 8 'comments' on that page autocomments merely
saying that it's a top story on HN? That seems a bit excessive.

edit: Ah, downvoted within seconds. This is a Displeasing Question apparently.
Sorry for whyever that is. But I'm curious. ..and won't be bullied into
deleting the question.

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aportnoy
Upvoted you, I asked myself the same question. Looks like a bot that annoys
bloggers by notifying them that their writing is trending on various
aggregators.

~~~
samsolomon
The "pingbacks" are common on WordPress blog using WordPress comments. As you
mentioned they simply appear when another website provides a backlink.

This is default behavior, but most disable it.

