Trees not Cubes! Memories of Boris Tsirelson

This post is devoted to a few memories of Boris Tsirelson who passed away at the end of January. I would like to mention that a few days ago graph theorist Robin Thomas passed away after long battle with ALS. I hope  to tell about Robin’s stunning mathematics in a future post.

Boris Tsirelson (1950 – 2020); Boris’ home-page, and Wikipedia. (More links, below.)

The title of the post is taken from the title of a very interesting 1999 paper by Boris Tsirelson and Oded Schramm: Trees, not cubes: hypercontractivity, cosiness, and noise stability

I was very sad and shocked to hear that Boris Tsirelson had passed away. Boris was one of the greatest Israeli mathematicians, and since 1997 or so we established mathematical connections surrounding several matters of common interest.  Here are a few memories.

Love for coding

1) One thing that Boris told me was that he loves to code. Being a “refusnik”, he could not get into Academia and (luckily) he could work as a programmer. And he told me that afterwards deciding what he liked more – programming or doing mathematical research – was no longer a trivial question for him.  Boris chose to go back to mathematical research, but he continued to enjoy programming, and when he needed it in his mathematical research, he could easily program.

Love for quantum

2) Another thing that Boris loved is “quantum”, the mathematics and physics of quantum mechanics and various connections to mathematics. Early on he proved his famous Tsirelson’s bound related to Bell’s inequalities, and later he was enthusiastic about the area of quantum computing. (And he learned it quickly, taught a course about it in 1997, and his 1997 lecture notes are still considered very useful.)

Black Noise and noise sensitivity

3) Perhaps the most significant mathematical connection between us was in the late 90s, and was centered around the theory of noise stability and noise sensitivity by Benjamini, Schramm and myself, which was closely related to a theory initiated by Boris Tsirelson and Anatoly Vershik. The translation between the different languages that we used and that Boris used was awkward, since the analog of Boolean functions that we studied was the “noise” that Boris studied, and the analog of noise sensitive Boolean functions in our language was “black noise” in Boris’s language. In any case, we had email discussions and we also met a few times with Itai and Oded regarding this connection.

Black Noise and noise sensitivity II

4)  Boris developed a very rich theory of black noise with relations to various areas of probability theory and operator algebras. He also found hypercontractivity that we used in our work quite useful to his applications, and also in this theory, he considered both classical and quantum aspects. I know only a little about Boris Tsirelson’s theory and its applications, but as far as tangent points with our Boolean interests are concerned, I can mention that Boris was enthusiastic by the result of Schramm and Stas Smirnov that percolation is a “black noise” and also that, in 1999, Boris and Oded Schramm wrote a paper whose title started with “Trees not cubes!”, presenting a different angle on this theory.

Tsirelson saw white noise (what we call noise stability) as manifesting “linearity” while “black noise” (what we call noise sensitivity) as manifesting “non-linearity”. Over the years, I often asked him to explain this to me.

Tsireslon’s Banach spaces

5) Geometry of Banach spaces is a very strong area in Israel so naturally I heard as a graduate student about “Tsirelson’s space” from 1974 and some subsequent developments in the 80s. Boris Tsirelson constructed a  Banach space that does not contain an imbedding of \ell_p or c_0.

Bible codes

6) My first personal connection with Boris was related to claims regarding a hidden Bible Code, and a 1994 paper claiming a statistical proof of the existence of these Bible codes. For many years my attitude was that these claims should be ignored, but around 1997, I changed my mind and did some work to see what was going on. Now, Boris kept a site linked in his homepage devoted to developments regarding the Bible Code claims. In this site Boris kindly reported about my first 1997 paper on the topic, my observation that the proximity of two reported p-values for the two Bible code experiments was “too good to be true”, and my interpretation that this suggests that the claimed results manifest naïve statistical expectations rather than scientific reality.  A few weeks later, Boris reported about a much stronger evidence (by McKay and Bar Nathan) against the Bible Code claims (they demonstrated codes of similar quality in Tolstoy’s “War and Peace”) and subsequently after some time he lost interest in this topic.

Quantum computing skepticism

6) In 2005 we had some correspondence and meetings regarding my quantum computing skepticism. In his first email he told me that my reference to “decoherence” seemed strange and I realized that I consistently referred to “entanglement” as “decoherence” and to “decoherence” as “entanglement”.

7) In his 1997 lecture notes on quantum computing (that I cannot find on the web, so I count on my memory), Boris addressed the concerns of early quantum computers skeptics like Rolf Landauer. He did not accept the analogy between quantum computing and analog computation, but he also regarded the analogy with digital computation as problematic. Rather, he regarded quantum information based on qubits as something (at least a priori) different from both these examples. (Update: I found one non-broken link to the lecture notes; indeed the subtitle of Chapter 9 is “neither analog nor digital”.)

A joke that I heard from Boris at that time

8) I remember that once when I asked him about some aspects of quantum fault tolerance he told me the following joke: A student is entitled to a special exam, he arrives at the professor’s office, is given three questions to answer and he fails to do so. He request and is granted a make-up exam two weeks later. When the student shows up at the office two weeks later the professor, who forgot all about it, gave him the same three questions. “This is extremely unfair”, said the student “you ask me questions that you already know that I cannot answer.”

Noise sensitivity and high energy physics

9) In 2006 I came up with the idea that noise sensitivity might be a great idea for physics. Knowing very little physics, I wrote a little manifesto with this idea and tried it, among other people, on Boris. As it turned out, Boris had the idea that noise sensitivity could add a useful modeling power to physics (especially high energy physics) well before that time. (And by 2006 he was already a bit skeptical regarding his own idea.) He also told me that one of the motivations of his 1998 paper with Tolya Vershik came from some mathematical ideas related to physics of the big bang. When I asked him if this was written somewhere in the paper itself, he answered: “Of course, not!”

Boris Tsirelson’s lecture at Oded’s memorial school

10) in 2009 we organized a meeting in memory of Oded Schramm and Boris gave a lecture related to the Schramm-Smirnov “percolation is black noise” result with a single theorem. And what was remarkable about it that it was that he presented a classical theorem with a quantum proof. You can find the videotaped lecture here  (And here are the slides. Boris never wrote up this result.) Following this lecture we had a short correspondence with Scott A. and Greg K. about quantum proofs to classical theorems. (Namely theorems that do not mention quantum in the statement).

Tsirelson’s problem

11) Our last correspondence in 2019 was about Thomas Vidick’s  Notices AMS article about Tsirelson’s problem. (This was a couple of months before the announcement of the solution.) Boris was pleased to hear about these developments, as he was regarding earlier developments in this area. He humorously refers to the history of his problem on his homepage and this interview.

12) People who knew Boris regarded him as a genius from a very early age, and former students have fond memories of his classes.


More resources:

Boris’s home page contains  “Museum to my courses” with many useful lecture notes; link to a small page on quantum computation with a link to Boris’ 1997 lecture notes on quantum computing.  Links to comments on some of Tsirelson’s famous papers. Tsirelson’s 1980 bound. Boris published papers, and his “self-published” papers.

Boris was a devoted Wikipedian and his Wikipidea user page is now devoted to his memory; Here is a great interview with Boris; A very nice memorial post on Freeman Dyson and Boris Tsirelson on the Shtetl Optimized; Tim Gowers explains some ideas behind Tsirelson’s space over Twitter; and here in Polymath2.

Below the fold some emails of interest from Boris, mainly where he explained to me various mathematics. (More can be found in this page.)

Some email correspondence

Oct. 2019 Vidick’s paper

Dear Boris
This paper by Thomas Vidick may interest you,
best regards and shana tova Gil

Oh yes, sure!
Thank you.
Shana metuka, Boris

(Remark: “metuka” means “sweet” in Hebrew.)

Dec 2006 (about noise sensitivity and physics)

(Dec 2006) My very first idea in this field (inspired by conversations with Vershik) was
rather physical (that Big Bang could be a natural occurrence of black noise),
and in fact the main example of “Tsirelson and Vershik 1998” follows this line
(not explicitly, of course).

In local (not Big Bang related) physics, I think, nonlinearity could produce
such effect. And then the very idea of `the field operator at a point’ (on
the level of operator-valued Schwartz distributions or something like that)
will fail. However, physicists do not want to consider this possibility
without very serious indications that it really is used by the nature. And
they are right…

2005 quantum computer skepticism

Subject: Re: Noise and more

Dear Gil:
Yes, of course, we can meet and speak.

For now, I am not much bothered. I am not an expert in quantum error
correction, but anyway, my feeling is that all physically reasonable
“attacks” of Nature are repelled. Especially, your three-qubit attack
looks to me not dangerous. And, “der Herr Gott is raffiniert, aber
boschaft ist er nicht”; Nature never attacks like an enemy.

Yours, Boris.

Quick 2000 comments on a (sloppy) draft of my survey paper

Dear Gil:

Thank you for the text; I am reading it.
For now, only a trivial remark: “Tsilerson” should be “Tsirelson” in
[133] and [134]; and in [132] “” should be “Tsirelson”…

Shabat shalom,
Yours, Boris.

November 1998: Noise sensitivity and black noise

Dear Gil,

I am reading your (with Itai and Oded) paper. Thanks.

Moreover, I am thinking about changing the title of my future talk in
Vien accordingly: from “The five noises” to “The six noises” (or even
To this end, however, I need to answer the following question.

Is there a mesh refinement limit for the percolation?

That is, take the lattice with a small pitch \eps. Choose two
“electrodes”, say, two vertical intervals on two parallel vertical
lines, and ask about the probability that they are connected (via the
bond percolation on the whole band between the two vertical lines).
Let the electrodes be macroscopic; that is, they do not depend on
\eps. Does the probability of the event have a limit for \eps \to 0 ?
If it does, then one more question: what about the joint distribution
for a finite collection of such events? That is, I want to see a weak
limit of these “discrete” random processes. It seems to me, the
question is well-known and was discussed. However, do you know the

Yours, Boris.

Dear Itai, Oded, and Gil,

Thank you for the information. I see that for now we have a
conditional result: if there exists “the noise of percolation”, then
it is not a white noise.
Yours, Boris.

Dear Gil:
> I dont understand yet the concept of “noise” precisely

One of ways is this. A noise is a scaling limit for coin tossing.
You choose a class of “macroscopic observables” and look, whether
their joint distribution converges, when n\to\infty. If it does, you
get a noise. (For percolation we do not know, does it or not.)
Now, if all “macroscopic observables” are noise insensitive, it means
that the noise is white. For a white noise, there is only one
invariant, its dimension (or multiplicity).
If all “macroscopic observables” are noise sensitive, the noise is
black. Probably, there are a lot of black noises, but for now we have
only two examples, without knowing, whether they are isomorphic, or
not. Spectra may be used for classifying black noises. Say, it may
happen that for each “macroscopic observable”, its spectrum is
concentrated on sets having Hausdorf dimension less than something.
If some “macroscopic observables” are noise sensitive but some others
are not (except for constants, of course), then the noise is neither
white nor black. It may happen that it is a direct sum of a white
noise and a black noise. However, it may happen that it is not. We
have for now two such examples: “noise if splitting” and “noise of
stickiness” (they are probably non-isomorphic); both are found by Jonathan
Warren. I am trying to understand, whether your matter can give more


August 1998: Bible code story

Dear Gil,

Thanks for the text.
As for me, it is already an `overkill’, since for me the WRR94
is basically dead. But maybe for others…

Yours, Boris.


This entry was posted in Combinatorics, Obituary, Probability, Quantum and tagged . Bookmark the permalink.

4 Responses to Trees not Cubes! Memories of Boris Tsirelson

  1. Jon Awbrey says:

    Cacti 🌵 ¬ Trees 🌲🌳

  2. Leonid Vishnevsky says:

    Regarding (7), Tsirelson’s lecture notes from the 1997 quantum computing course can be found on his website under “other courses”.
    (the links to the pdf files are not clickable, but one can copy the url’s – they work).
    GK: Thanks, Leonid. The link to the home page is already in the post. (I uploaded the ps files and may add later pdf files.)

  3. Peter Tobler says:

    This comment by Leonid Vishnevsky is infected with a virus – Blacklist – don’t click on the link

  4. I did not know Boris personally, but he was the first to use some of the ideas from my book on diffeology in his paper “Brownian local minima, random dense countable sets and random equivalence classes”. The book was still in draft form, but the fact that it was cited in such an article encouraged me to continue writing, and I have cited Boris’s article in several reports. I always wanted to thank him, but I didn’t. It seems that the distance between Jerusalem and Tel Aviv was infinite at that time. Now it is too late, I regret not having known him more. Thank you Gil for this page which gives me the opportunity to pay tribute to him, even belatedly.

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