Here is a web-page by a student in my “Game Theory” course where you can play several two-players variants of John Conway’s game-of-life. (A couple of variants were considered before. See, e.g. this paper.)

I really enjoyed playing Rani’s games and it can certainly cheer you up in difficult times. Questions about the game, remarks, and suggestions for improvements and for new features, are most welcome.

Maya Bar-Hillel (left) and Ester Samuel-Cahn

Here is an interesting 2006 power point presentation entitled ** How to detect lies with statistics** by Maya Bar-Hillel. This was a talk given by Maya at the conference honoring Prof. Ester Samuel- Cahn , Jerusalem, December 18-20, 2006, and it described a planned research project of Maya Bar-Hillel with Yossi Rinott, David Budescu and myself. At the end we did not pursue it, mainly because each of us was involved in various other projects (but also because we were skeptical about some aspects of it.) Ester Samuel-Cahn (1933-2015) was a famous Israeli statistician. (Here is a post by Yosi Levy in Hebrew about the conference and about Ester.)

The lecture starts with “Last year, *Statistical Science* celebrated 50 years for `How to Lie with Statistics’ the book [by Durell Huff] whose title inspired this talk.”

And here are a few other quotes from Maya’s presentation

“We are not sure a general toolkit for detecting lies with statistics can be developed. Perhaps that explains why none yet exists. We have shown just a collection of anecdotes. But they can be classified and categorized. Some do seem generalizeable, at least to some extent.”

and the conclusion

A famous quip by Fred Mosteller: “It is easy to lie with statistics, but easier to lie without them.”

Likewise, we should say: “It is possible to detect (some) lies with statistics, but easier to detect them with other means”.

Peter Keevash, Alexey Pokrovskiy, Benny Sudakov, and Liana Yepremyan’s paper New bounds for Ryser’s conjecture and related problems, describes remarkable progress very old questions regarding transversals in Latin square.

I came across a very interesting paper The topological Tverberg problem beyond prime powers by Florian Frick and Pablo Soberón with new bounds and a new method for topological Tverberg theorem in the non prime-power case.

A year ago I came across this cool facebook post by Rupei Xu

OMG! Just learned that Jaeger’s conjecture is false for every t>=3. An interesting consequence of it is that a specific version of Goddyn’s conjecture on thin spanning trees is false, which shows some negative evidence that the thin spanning tree approaches may fail to lead to a constant factor approximation algorithm for ATSP!

Let me mention two problems I posted 6-7 years ago about Conway’s game of life. Conway’s game of life for random initial position and Does a noisy version of Conway’s game of life support universal computation?

]]>

From left to right: Atle Selberg, Florian Richter, and Paul Erdos

“One of the most fundamental results in mathematics is the Prime Number Theorem, which describes the asymptotic law of the distribution of prime numbers in the integers.

**Prime Number Theorem.** Let π(N) denote the number of primes smaller or equal to a positive integer N. Then

The Prime Number Theorem was conjectured independently by Gauß and Legendre towards the end of the 18th century and was proved independently by Hadamard and de la Vallée Poussin in the year 1896. Their proofs are similar and rely on sophisticated analytic machinery involving complex analysis, which was developed throughout the 19th century by the combined eﬀort of many great mathematicians of this era, including Euler, Chebyshev, and Riemann (to name a few).

Even though it was believed for a long time not to be possible, more than 50 years after the analytic proof of the Prime Number Theorem was discovered, an elementary proof was found by Erdős and Selberg [Erd49; Sel49]. In this context, elementary refers to methods that avoid using complex analysis and instead rely only on rudimentary facts from calculus and basic arithmetic identities and inequalities. Their approach was based on Selberg’s famous “fundamental formula”:

In this paper we provide a new elementary proof of the Prime Number Theorem, using ideas inspired by recent developments surrounding Sarnak’s and Chowla’s conjecture. With the exception of Stirling’s approximation formula, which is used in Section 3 without giving a proof, our proof is self-contained.”

The introduction also cites papers on the history of the analytic method and an abridged version of it by Newman, on several earlier elementary proofs of the PNT, and on recent dynamically inspired way by McNamara of deriving the Prime Number Theorem from Selberg’s fundamental formula.

What is the formal distinction between elementary and non-elementary proofs of the PNT? (This was explained to me several times but I don’t remember the details.) The distinctions between “non-explicit” and “explicit” proofs (or constructions) and between “effective” and “non-effective” proofs are quite clear cut. But here I am less certain.

Are there other important distinctions about proofs that you are aware of?

An equivalent formulation of the prime number theorem is the following:

where is the Mobius function. Consider now a Boolean function with variables. Here the variables and the function has values in . Given nonnegative weights and the function $f$ is a weighted majority function if iff . Let be the probability that and let be the top Fourier coefficient of .

Problems:

(1) Find conditions for a weighted majority function that guarantee that

Let be the th prime, , and . The PNT asserts that the Boolean function associated to these parameters satisfies (**). (It is very easy to see that (**) and (*) are essentially the same.) From vague understanding of some fragments of older elementary proofs of the PNT it looks that the proofs could perhaps be described in this way, namely starting with some weaker properties on the weights (namely the primes) perhaps the derivation of (**) apply to general weighted majority functions with these properties. (This is related to PNT for systems of Beurling primes.) Finding conditions for (**) is interesting also for other weighted majority functions where the weights are not related to the growth behavior of the primes. In fact, the following is also interesting:

(2) Find conditions for (general) Boolean functions that guarantee that (**) holds, namely that the top Fourier coefficients is little o of . Signs of low degree polynomials form a very interesting class of Boolean functions, and they are of special interest also here.

**Remark:** The first problem looks very related to PNT for systems of Beurling primes going back to 1937 paper by Arne Beurling. (Based on the inequality 1937 < 1949, I understand that Breuling’s proof was not elementary, and I don’t know if the elementary proofs for PNT gives Breuling’s theorem as well.)

The new paper is related to conjectures by Chowla and Sarnak and to the notion of Mobius randomness that we considered in an earlier post. The basic question is for which functions of , . This is correct if is random (or random-like in a variety of meanings), and it turns out to be correct if (in some sense) if is of low complexity. We also mentioned some analogs of the PNT in this post. Both polymath 4 and polymath5 had some connections to the PNT.

This is another connection to combinatorics, See this post in “In Theory”.

]]>(Update, May18 2020). I failed to explain what WIT is and this may have caused some misunderstanding. Here is a description from the Simons Institute site.

“The Women in Theory (WIT) Workshop is intended for graduate and exceptional undergraduate students in the area of theory of computer science. The workshop will feature technical talks and tutorials by senior and junior women in the field, as well as social events and activities. The motivation for the workshop is twofold. The first goal is to deliver an invigorating educational program; the second is to bring together theory women students from different departments and foster a sense of kinship and camaraderie.”

The original 1978 version is also quite good.

and all so this one

Update: Meeting of the heads of the universities with Israeli postdocs abroad

]]>My short introduction: There is nothing like a new hypercontractivity inequality to cheer you up in difficult times and this post describes an amazing new hypercontractivity inequality. The post describes a recent hypercontractive inequality by Peter Keevash, Noam Lifshitz, Eoin Long and Dor Minzer (KLLM) from their paper: Hypercontractivity for global functions and sharp thresholds. (We reported on this development in this post. By now, there are quite a few important applications.) And for Talagrand’s generic chaining inequality, see this beautiful blog post by Luca Trevisan.

Barry Simon coined the term “hypercontractivity” in the 70s. (We asked about it here and Nick Read was the first to answer.) A few months ago Barry told us about the early history of hypercontractivity inequalities, and, in particular, the very entertaining story on William Beckner’s Ph. D. qualifying exam.

And now to Noam Lifshitz’s guest post.

Analysis of Boolean functions (ABS) is a very rich subject. There are many works whose concern is generalising some of the results on analysis of Boolean functions to other (product) settings, such as functions on the multicube where is very large. However, in some of these cases the fundemental tools of AOBF seem to be false for functions on the multicube However, in the recent work of Keevash, Long, Minzer, and I. We introduce the notion of global functions. These are functions that do not strongly depend on a small set of coordinates. We then show that most of the rich classical theory of AOBF can in fact be generalised to these global functions. Using our machinery we were able to strengthen an isoperimetric stability result of Bourgain, and to make progress on some Erdos-Ko-Rado type open problem.

We now discuss some background on the Fourier analysis on functions on the multicube

There are two fundemental types of operators on Boolean functions The first ones are the discrete derivatives, defined by where denotes the we plug in the value for the th coordinate. The other closely related ones are the laplacians defined by where is obtained from by resampling its th coordinate.

The laplacians and the derivatives are closely related. In fact, when we plug in in the th coordinate, we obtain , and when we plug in in it, we obtain

The 2-norm of the th derivative is called the th influence of as it measures the impact of the th coordinate on the value of . It’s usually denoted by .

For functions on the multicube we don’t have a very good notion of a discrete derivative, but it turns out that it will be enough to talk about the laplacians and their restrictions. The Laplacians are again defined via where is obtained from by resampling its th coordinate. It turns out that in the continuous cube it’s not enough to talk about Laplacians of coordinate, and we will also have to concern ourselves with Laplacians of sets. We define the generalised Laplacians of a set by composing the laplacians corresponding to each coordinate in

We now need to convince ourselves that these laplacians have something to do with the impact of on the outcome of In fact, the following notions are equivalent

- For each we have
- For each we have

in the sense that if (1) holds then (2) holds with and conversely if (2) holds, then (1) holds with

The main theme of our work is that one can understand **global** function on the continuous cube, and these are functions that satisfy the above equivalent notions for all small sets .

For the noise operator is given by when is obtained from by independently setting each coordinate to be with probability and resampling it with uniformly out of otherwise. The process which given outputs is called the -noisy process, and we write

The Bonami hypercontractivity theorem, which was then generalised by Gross and Beckner states that the noise operator is a contraction from to i.e.

for any function

One consequence of the hypercontractivity theorem is the small set expansion theorem of KKL. It concerns fixed and a sequence of sets with The small set expansion theorem states that if we choose uniformly and a noisy then will reside outside of almost surely.

The small set expansion theorem and the hypercontractivity theorem both fail for function on the multicube that are of a very local nature. For instance, let be the set of all such that is Then is of size which is if we allow to be a growing function of . However, the -noisy process from the set stays within the set with probability For a similar reason the hypercontractivity theorem fails as is for functions on However we were able to generalise the hypercontractivity theorem by taking the globalness of into consideration.

Our main hypercontractive inequality is the following

**Theorem 1.**

The terms appearing on the right hand side are small whenever has a small dependency on and it turns out that you have the following corrolary of it, which looks a bit more similar to the hypercontractive intequality.

**Corollary 2.**

Let , and uppose that for all sets

Then

Finally, one might ask wonder why this globalness notion appears only when we look at large values of and not when I think the corollary is a good explanation for that as holds trivially for any Boolean function

]]>Aumann himself was born in Frankfurt, Germany, and fled at the age of eight to the United States with his family in, two weeks before the 1938 Kristallnacht.

I plan to mention this story in my game theory class this evening and also to mention five great game theorists named John.

]]>

I will start with sad news. John Horton Conway, an amazing mathematical hero, passed away a few days ago. There are very nice posts on Conway’s work by Scott Aaronson (with many nice memories in the comments section), by Terry Tao, and by Dick Lipton and Ken Regan. And a moving obituary on xkcd with a touch of ingenuity of Conway’s style. (There is also a question on MO “Conway’s less known results,” and two questions on the game of life (I, II).)

Another reading material to cheer you up is my paper: The argument against quantum computers, the quantum laws of nature, and Google’s supremacy claims. It is for *Laws, Rigidity and Dynamics,* Proceedings of the ICA workshops 2018 & 2019 in Singapore and Birmingham. Remarks are most welcome. (Update (May, 25 2020): An interesting discussion on Hacker news.)

**Update:** starting today, the algebraic combinatorics online workshop. Here is the schedule for the first week, and for the second week.

Matching theory by Lovasz and Plummer is probably one of the best mathematics books ever written.

Bipartite Perfect Matching as a Real Polynomial, by Gal Beniamini and Noam Nisan

**Abstract:** We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.

And here is how the paper starts

Every Boolean function can be represented in a unique way as a Real multilinear polynomial. This representation and related ones (e.g. using the {1,−1} basis rather than {0,1}– the “Fourier transform” over the hypercube, or approximation variants) have many applications for various complexity and algorithmic purposes. See, e.g., [O’D14] for a recent textbook. In this paper we derive the representation of the bipartite-perfect-matching decision problem as a Real polynomial.

**Deﬁnition.** The Boolean function is deﬁned to be 1 if and only if the bipartite graph whose edges are has a perfect matching, and 0 otherwise.

And here are the two main theorems regarding this polynomial and the polynomial for the dual representation:

(For the second theorem you need the notion of totally ordered bipartite graphs.)

And here is a nice picture!

A very interesting open problem is:

**Problem:** Can the Beniamini-Nisan results be extended to general (non-bipartite) graphs

This reminds me of an old great problem:

**Problem:** Does Lovasz’ randomized algorithm for matching extend to the non-bipartite case?

For both problems methods of algebraic combinatorics may be helpful.

A Real Polynomial for Bipartite Graph Minimum Weight Perfect Matchings, Thorben Tröbst, Vijay V. Vazirani

**Abstract:**

In a recent paper, Beniamini and Nisan gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph has a perfect matching, together with an efficient algorithm for computing the coefficients of the monomials of this polynomial. We give the following generalization: Given an arbitrary non-negative weight function w on the edges of , consider its set of minimum weight perfect matchings. We give the real multilinear polynomial for the Boolean function which determines if a graph contains one of these minimum weight perfect matchings.

Three more VVV remarks: in the Tel Aviv theory ~~fast~~ fest three months ago (it seems like ages ago) Vijay Vazirani gave a lecture about matching. Here is the link to Vijay’s lecture, and to all plenary lectures. At the end, I asked him how he explains that matching theory is such inexhaustable gold mine and Vijay mentioned the fact that a polynomial-time algorithm for the assignment problem (which is closely related to matching) was already found by Jacobi in 1890. (Unfortunately VJ’s inspiring answer was not recorded). A few years ago Vijay published a simplified proof of a fantastic famous result he first proved with Silvio Micaly 34 years earlier. And here is a most amazing story: a few years ago I went to the beach in Tel Aviv and I discovered Vijay swimming just next to me. We were quite happy to see each other and Vijay told me a few things about matching, economics and biology. This sounds now like a truly surrealistic story, and perhaps we even shook hands.

]]>

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.

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.

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.)

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.

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.

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 or .

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.

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.”

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!”

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).

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.

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.)

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) 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…

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.

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.

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

more).

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

answer?

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

examples.

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.

]]>

Eight weeks ago I wrote that my heart goes out to the people of Wuhan and China, and these days my heart goes out to people in Italy, Spain, the US, Iran, France, the United Kingdom, Germany, Netherland and many other countries all over the world. Of course, I am especially worried about the situation here in my beloved country Israel, and let me tell you a little about it.

The pandemic started here late but it hit us pretty hard with 5,358 identified cases yesterday. Severe measures of social distancing were gradually introduced, and right now it is too early to tell if the pandemic is under control.

My part in this struggle is to stay at home. (Many Israeli scientists are making various endeavors and proposing ideas of various kind for fighting the disease and I salute them all for their efforts.) Like all of us I am very thankful to medical and other essential workers who are in the front-lines. As a scientist, I am especially impressed by the people from the Ministry of Health who manage the crisis and communicate with the public. They represent the very best we can offer in terms of science and medicine, decision making, gathering information, communicating with the public, and managing the crisis. In the picture below you can see three of the leaders – Moshe Bar Siman Tov (middle) Prof. Itamar Grotto (right) and Professor Sigal Sadetzki (left).

We had a tradition of sharing entertaining taxi-and-more stories and this post belongs to this category. We note that our highest quality story teller Michal Linial, a prominent Israeli biologist, is now involved in various aspects of the struggle against the disease. Our post today is part of a report by Michal Feldman and me on our experience from the ICA3 conferences in Singapore and Birmingham.

After hearing about him for many years, it was a great pleasure for both Michal Feldman and myself to finally meet Partha Dasgupta in person and to listen to his lecture. Partha who is the Frank Ramsey Professor of Economics at Cambridge was introduced by a person, who entered the room directly from an intercontinental flight, whom we did not know but who made a strong impression on us. He devoted part of his introduction to Frank Ramsey who was a mathematician, philosopher and economist, and who had made fundamental contributions to algebra and had developed the canonical model of saving in economic growth, before he died at the young age of 26. (And yes! also Ramsey’s theorem!)

Seeing the introducer, Robin Mason, three words came into our minds (more precisely two words, one repeated twice): “Bond, James Bond.”

Indeed, this has led to the following sequence of profound ideas:

1) Robin Mason is a perfect choice for a new generation James Bond.

2) The name “James Bond” is overused. “Robin Mason” is a perfect name to replace the name “James Bond”.

3) Espionage is a little obsolete and it lost much of its prestige and charm. Science and academia is the new thing! An international interdisciplinary academics is the perfect profession which, at present, deserves the prestige formely associated with espionage.

In summary, we came a full circle. Robin Mason is the perfect new choice for James Bond, “Robin Mason” is the perfect new name to replace the name “James Bond,” and Mason’s academic activities and title of Pro-Vice-Chancellor (International) are the perfect replacement for Bond’s activities and the title ‘007’.

(The option of Mason playing his role on the movies rather than in real life should be considered. ‘Q’ could be handy for science as well. )

Clique here for Robin’s introduction and Partha’s lectur

]]>Starting Tuesday March 31, I am giving an on-line course (in Hebrew) on Game theory at IDC, Herzliya (IDC English site; IDC Chinese site).

In addition to the IDC moodle (course site) that allows IDC students to listen to recorded lectures, submit solutions to problem sets , etc., there will be a page here on the blog devoted to the course. Zoom link for the first meeting. https://idc-il.zoom.us/j/726950787

A small memory: In 1970 there was a strike in Israelis’ high schools and I took a few classes at the university. One of these classes was Game theory and it was taught by Michael Maschler. (I also took that trimester a course on art taught by Ziva Meisels.) Our department at HUJI is very strong in game theory, but once all the “professional” game theorists retired, I gave twice a game theory course which I enjoyed a lot and it was well accepted by students. In term of the number of registered students, it seems that this year’s course at IDC is quite popular and I hope it will be successful.

(Click to enlarge)

Game Theory 2020, games, decisions, competition, strategies, mechanisms, cooperation

The course deals with basic notions, central mathematical results, and important examples in non-cooperative game theory and in cooperative game theory, and with connections of game theory with computer science, economics and other areas.

What we will learn

1. Full information zero-sum games. The value of a game. Combinatorial games.

2. Zero-sum games with incomplete information. Mixed strategies, the Minmax Theorem and the value of the game.

3. Non cooperative games, the prisoner dilemma, Nash equilibrium, Nash’s theorem on the existence of equilibrium.

4. Cooperative games, the core and the Shapley value. Nash bargaining problem, voting rules and social choice.

Background material:

Game theory alive by Anna Karlin and Yuval Peres (available on-line).

In addition I may use material from several books in Hebrew by Maschler, Solan, Zamir, by Hefetz, and by Megiddo (based on lectures by Peleg). (If only I will manage to unite with my books that are not here.) We will also use a site by Ariel Rubinstein for playing games and some material from the book by Osborne and Rubinstein.

Requirement and challenges:

**Play, play, play games**, in Ariel Rubinshtein site and various other games.- Solve 10 short theoretical problem set.
- Final assignment, including some programming project that can be carried out during the semester.
- Of course, we will experience on-line study which is a huge challenge for us all.

Games and computers

- Computer games
- Algorithms for playing games
- algorithmic game theory:
- Mechanism design
- Analyzing games in tools of computer science
- Electronic commerce

- Games, logic and automata: there will be a parallel course by Prof. Udi Boker

I still have some difficulty with the virtual background in ZOOM.

]]>Zero-knowledge answers please.

Comments regarding this view itself and on “what is mathematics” are also welcome.

(Here are other posts on “What is mathematics.”)

PS. The last facetious sentence was omitted in the Journal version of the paper. (Indeed it was a good decision to take it out.) PPS Yannai Gonczarowski pointed out the the journal formulation is also rather condescending (perhaps even more so) towards non-mathematicians.

]]>