Left: Nick Read; Right The front page of Nick’s 1990 famous paper with Greg Moore on nonabelions, and below his email to me from March 2005 on topological quantum computation. (click for full view.)

Left: the argument regarding topological QC demonstrated via Harris’ famous cartoon. While not strictly needed I expect the argument to extend from qubits to gates as well. Right: a recent discussion with Nick over Shtetl Optimized (click for full view). **Update**: We are actually not in an agreement as it seems from the above discussion (see the discussion below).

**Update:** A subsequent post by Steve Flammia, Quantum computers can work in principle over The Quantum Pontiff.

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Workshop announcement

The National Academy of Sciences of Armenia together American University of Armenia are organizing a memorial workshop on extremal combinatorics, cryptography and coding theory dedicated to the 60th anniversary of the mathematician Levon Khachatrian. Professor Khachatrian started his academic career at the Institute of Informatics and Automation of National Academy of Sciences. From 1991 until the end of his short life in 2002 he spent at University of Bielefeld, Germany where Khachatrian’s talent flourished working with Professor Rudolf Ahlswede. Professor Khachatrian’s most remarkable results include solutions of problems dating back over 40 years in extremal combinatorics posed by the world famous mathematician Paul Erdos. These problems had attracted the attention of many top people in combinatorics and number theory who were unsuccessfully in their attempts to solve them. At the workshop in Yerevan we look forward to the participation of invited speakers (1 hour presentations), researchers familiar with Khachatrian’s work, as well as contributed papers in all areas of extremal combinatorics, cryptography and coding theory.

The American University of Armenia (www.aua.am) is proud to host the workshop.

Workshop chair: Gurgen Khachatrian

For any inquiries please send E-mail to: gurgenkh@aua.am

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**Smart fluid**

Terry Tao posted a very intriguing post on the Navier-Stokes equation, based on a recently uploaded paper Finite time blowup for an averaged three-dimensional Navier-Stokes equation.

The paper proved a remarkable negative answer for the regularity conjecture for a certain variants of the NS equations, namely (or, perhaps, more precisely) the main theorem demonstrates* finite time blowup for an averaged Navier-Stokes equation.* (This already suffices to show that certain approaches for a positive answer to the real problem are not viable.) The introduction ends with the following words.

“This suggests an ambitious (but not obviously impossible) program (in both senses of

the word) to achieve the same effect for the true Navier-Stokes equations, thus obtaining a negative answer to Conjecture 1.1 (the regularity conjecture for 3D NS equation)… Somewhat analogously to how a quantum computer can be constructed from the laws of quantum mechanics [Here Tao links to Benioff's 1982 paper: "Quantum mechanical Hamiltonian models of Turing machines,"], or a Turing machine can be constructed from cellular automata such as “Conway’s Game of Life” , one could hope to design logic gates entirely out of ideal fluid (perhaps by using suitably shaped vortex sheets to simulate the various types of physical materials one would use in a mechanical computer). If these gates were sufficiently “Turing complete”, and also “noise-tolerant”, one could then hope to combine enough of these gates together to “program” a von Neumann machine consisting of ideal fluid that, when it runs, behaves qualitatively like the blowup solution used to establish Theorem 1.4.[The paper's main theorem] Note that such replicators, as well as the related concept of a universal constructor, have been built within cellular automata such as the “Game of Life.”

Once enough logic gates of ideal fluid are constructed, it seems that the main difficulties in executing the above program are of a **“software engineering”** nature, and would be in principle achievable, even if the details could be extremely complicated in practice. The main mathematical difficulty in executing this “fluid computing” program would thus be to arrive at (and rigorously certify) a design for logical gates of inviscid fluid that has some good noise tolerance properties. In this regard, ideas from quantum computing (which faces a unitarity constraint somewhat analogous to the energy conservation constraint for ideal fluids, albeit with the key difference of having a linear evolution rather than a nonlinear one) may prove to be useful. (Emphasis mine.)

Interesting idea!

And what Tao does go well beyond an idea, he essentially implement this program for a close relative of the NS equation! I am not sure if universal computing is established for these systems but the proofs of the finite-time blow up theorem certainly uses some computational-looking gadget, and also as Terry explains some form of fault-tolerance.

Somewhat related ideas (unsupported by any results, of course,) appeared in the seventh post “Quantum repetition” of my debate with Aram Harrow on quantum computing. (See, e.g., this remark, and this one, and this one.) The thread also contains interesting links, e.g. to Andy Yao’s paper “Classical physics and the Curch-Turing Thesis.” In addition to the interesting question:

Does the NS-equation in three-dimension supports universal (classical) computation,

we can also ask what about two-dimensions?

Can NS-equations in two dimension be approximated in any scale by bounded depth circuits?

One general question suggested there was the following: “It can be of interest (and perhaps harder compared to the quantum case) to try to describe *classical* evolutions that do not enable/hide fault tolerance and (long) computation.”

Another interesting comment by Arie Israel is: “I was surprised to learn that experimental fluid mechanics people had thought of this analogy before. Apparently the key name is ‘Fluidics’ and those ideas date back at least to the sixties.”

Update: Here is the first paragraph from a nice article by Erica Klarreich entitled A Fluid New Path in Grand Math Challenge on this development in Quanta Magazine:

In Dr. Seuss’s book “The Cat in the Hat Comes Back,” the Cat makes a stain he can’t clean up, so he calls upon the help of Little Cat A, a smaller, perfect replica of the Cat who has been hiding under the Cat’s hat. Little Cat A then calls forth Little Cat B, an even smaller replica hidden under Little Cat A’s hat. Each cat in turn lifts his hat to reveal a smaller cat who possesses all the energy and good cheer of the original Cat, just crammed into a tinier package. Finally, Little Cat Z, who is too small to see, unleashes a VOOM like a giant explosion of energy, and the stain disappears.

And here is a follow up post on Tao’s blog, and a post on Shtetl Optimized.

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A special slide I prepared for my lecture at Gdansk featuring Robert Alicki and I as climber on the mountain of quantum computers “**because it is not there**.”

It has been quite a while since I posted here about quantum computers. Quite a lot happened in the last months regarding this side of my work, and let me devote this post mainly to pictures. So here is a short summary going chronologically backward in time. Last week – Michel Dyakonov visited Jerusalem, and gave here the condensed matter physics seminar on the spin Hall effect. A couple of weeks before in early January we had the very successful Jerusalem physics winter school on Frontier in quantum information. (Here are the recorded lectures.) Last year I gave my evolving-over-time lecture on why quantum computers cannot work in various place and different formats in Beer-Sheva, Seattle, Berkeley, Davis (CA), Gdansk, Paris, Cambridge (US), New-York, and Jerusalem. (The post about the lecture at MIT have led to a long and very interesting discussion mainly with Peter Shor and Aram Harrow.) In August I visited Robert Alicki, the other active QC-skeptic, in Gdansk and last June I took part in organizing a (successful) quantum information conference Qstart in Jerusalem (Here are the recorded lectures.).

And now some pictures in random ordering

With Robert Alicki in Gdynia near Gdansk

With (from left) Connie Sidles, Yuri Gurevich, John Sidles and Rico Picone in Seattle (Victor Klee used to take me to the very same restaurant when I visited Seattle in the 90s and 00s.) **Update:** Here is a very interesting post on GLL entitled “seeing atoms” on John Sidles work.

With Michel Dyakonov (Jerusalem, a few days ago)

With Michal Horodecki in Sopot near Gdansk (Michal was a main lecturer in our physics school a few weeks ago.)

Aram Harrow and me meeting a year ago at MIT.

Sometimes Robert and I look skeptically in the same direction and other times we look skeptically in opposite directions. These pictures are genuine! Our skeptical face impressions are not staged. The pictures were taken by Maria, Robert’s wife. Robert and I are working for many years (Robert since 2000 and I since 2005) in trying to examine skeptically the feasibility of quantum fault-tolerance. Various progress in experimental quantum error-correction and other experimental works give good reasons to believe that our views could be examined in the rather near future.

A slide I prepared for a 5-minute talk at the QSTART rump session referring to the impossibility of quantum fault-tolerance as a mild earthquake with wide impact.

This is a frame from the end-of-shooting of a videotaped lecture on “Why quantum computers cannot work” at the Simons Institute for the Theory of Computing at Berkeley. Producing a videotaped lecture is a very interesting experience! Tselil Schramm (in the picture holding spacial sets of constant width) helped me with this production.

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Joel David Hamkins’ profile over MathOverflow reads: “My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the set-theoretic multiverse, engaging with the emerging field known as the philosophy of set theory.”

Joel is a wonderful MO contributor, one of those distinguished mathematicians whose arrays of MO answers in their areas of interest draw coherent deep pictures for these areas that you probably cannot find anywhere else. And Joel is also a very highly decorated and prolific MO contributor, whose 999th answer appeared today!!

Here is a very short selection of Joel’s answers. To (MO founder) Anton Geraschenko’s question What are some reasonable-sounding statements that are independent of ZFC? Joel answered; “If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y.” Joel gave a very thorough answer to my question on Solutions to the Continuum Hypothesis; His 999th answer is on the question Can an ultraproduct be infinite countable? (the answer is yes! but this is a large cardinal assumption.) **Update**: Joel’s 1000th answer on a question about logic in mathematics and philosophy was just posted.

Joel also wrote a short assay, the use and value of MathOverflow over his blog. Here it is:

The principal draw of mathoverflow for me is the unending supply of extremely interesting mathematics, an eternal fountain of fascinating questions and answers. The mathematics here is simply compelling.

I feel that mathoverflow has enlarged me as a mathematician. I have learned a huge amount here in the past few years, particularly concerning how my subject relates to other parts of mathematics. I’ve read some really great answers that opened up new perspectives for me. But just as importantly, I’ve learned a lot when coming up with my own answers. It often happens that someone asks a question in another part of mathematics that I can see at bottom has to do with how something I know about relates to their area, and so in order to answer, I must learn enough about this other subject in order to see the connection through. How fulfilling it is when a question that is originally opaque to me, because I hadn’t known enough about this other topic, becomes clear enough for me to have an answer. Meanwhile, mathoverflow has also helped me to solidify my knowledge of my own research area, often through the exercise of writing up a clear summary account of a familiar mathematical issue or by thinking about issues arising in a question concerning confusing or difficult aspects of a familiar tool or method.

Mathoverflow has also taught me a lot about good mathematical exposition, both by the example of other’s high quality writing and by the immediate feedback we all get on our posts. This feedback reveals what kind of mathematical explanation is valued by the general mathematical community, in a direct way that one does not usually get so well when writing a paper or giving a conference talk. This kind of knowledge has helped me to improve my mathematical writing in general.

So, thanks very much mathoverflow! I am grateful.

Thanks very much, Joel, for your wonderful mathoverflow answers and questions!

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Here is one of the central and oldest problems in combinatorics:

**Problem:** Can you find a collection S of *q*-subsets from an *n*-element set X set so that every *r*-subset of X is included in precisely λ sets in the collection?

A collection S of this kind are called a **design **of parameters (*n,q,r*, λ)**, **a special interest is the case λ=1, and in this case S is called a **Steiner system.**

For such an S to exist n should be **admissible** namely should divide for every .

There are only few examples of designs when* r>2.* It was even boldly conjectured that for every *q r* and λ if *n* is sufficiently large than a design of parameters (*n,q,r*, λ) exists but the known constructions came very very far from this. … until last week. Last week, Peter Keevash gave a twenty minute talk at **Oberwolfach** where he announced the proof of the bold existence conjecture. Today his preprint** the existence of designs**, have become available on the arxive.

The existence of designs and Steiner systems is one of the oldest and most important problems in combinatorics.

1837-1853 – The existence of designs and Steiner systems was asked by Plücker(1835), Kirkman (1846) and Steiner (1853).

1972-1975 – For* r=2* which was of special interests, Rick Wilson proved their existence for large enough admissible values of *n*.

1985 -Rödl proved the existence of approximate objects (the property holds for (1-o(1)) *r*-subsets of *X*) , thus answering a conjecture by Erdös and Hanani.

1987 – Teirlink proved their existence for infinitely many values of *n* when *r* and* q* are arbitrary and λ is a certain large number depending on *q* and *r* but not on n. (His construction also does not have repeated blocks.)

2014 – Keevash’s proved the existence of Steiner systems for all but finitely many admissible values of *n* for every *q* and* r. *He uses a new method referred to as **Randomised Algebraic Constructions.**

**Update:** Just 2 weeks before Peter Keevash announced his result I mentioned the problem in my lecture in “Natifest” in a segment of the lecture devoted to the analysis of Nati’s dreams. 35:38-37:09.

**Update:** Some other blog post on this achievement: Van Vu, Jordan Ellenberg, The aperiodical . A related post from Cameron’s blog Subsets and partitions.

**Update**: Danny Calegary pointed out a bird-eye similarity between Keevash’s strategy and the strategy of the recent Kahn-Markovic proof of the Ehrenpreis conjecture http://arxiv.org/abs/1101.1330 , a strategy used again by Danny and Alden Walker to show that random groups contain fundamental groups of closed surfaces http://arxiv.org/abs/1304.2188 .

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**Two “real analysis” workshops at the Simons Institute** – The first in early October was on Functional Inequalities in Discrete Spaces with Applications and the second in early December was on Neo-classical methods in discrete analysis. Many exciting lectures! The links lead to the videotaped lectures. There were many other activities at the Simons Institute also in the parallel program on “big data” and also many interesting talks at the math department in Berkeley, the CS department and MSRI.

To celebrate the workshop on inequalities, there were special shows in local movie theaters

**My course at Berkeley on analysis of Boolean functions** - The course went very nicely. I stopped blogging about it at weak 7. Just before a lecture on MRRW upper bounds for binary codes, a general introductory lecture on social choice, and then several lectures by Guy Kindler (while I was visiting home) on the invariance principle and majority is stablest theorem. The second half of the course covered sharp threshold theorems, applications for random graphs, noise sensitivity and stability, a little more on percolation and a discussion of some open problems.

**Back to snowy Jerusalem, Midrasha, Natifest, and Archimedes.** I landed in Israel on Friday toward the end of the heaviest snow storm in Jerusalem. So I spent the weekend with my 90-years old father in law before reaching Jerusalem by train. While everything at HU was closed there were still three during-snow mathematics activities at HU. There was a very successful winter school (midrasha) on analytic number theory which took place in the heaviest storm days. Natifest was a very successful conference and I plan to devote to it a special post, but meanwhile, here is a link to the videotaped lectures and a picture of Nati with Michal, Anna and Shafi. We also had a special cozy afternoon event joint between the mathematics department and the department for classic studies where Reviel Nets talked about the **Archimedes Palimpses**.

The story behind Reviel’s name is quite amazing. When he was born, his older sister tried to read what was written in a pack of cigarettes. It should have been “royal” but she read “reviel” and Reviel’s parents adopted it for his name.

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Eran Nevo and Stedman Wilson have constructed triangulations with *n* vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 -sphere have?

1) Since the number of facets in an *n*-vertex triangulation of a 3-sphere is at most quadratic in* n*, an upper bound for the number of triangulations of the 3-sphere with *n* vertices is . For certain classes of triangulations, Dey removed in 1992 the logarithmic factor in the exponent for the upper bound.

2) Goodman and Pollack showed in 1986 that the number of simplicial 4-polytopes with *n* vertices is much much smaller . This upper bound applies to simplicial polytopes of every dimension *d*, and Alon extended it to general polytopes.

3) Before the new paper the world record was the 2004 lower bound by Pfeifle and Ziegler -

4) In 1988 I constructed triangulations of the *d*-spheres with *n* vertices. The new construction gives hope to improve it in any odd dimension by replacing* [d/2]* by *[(d+1)/2]* (which match up to log*n* the exponent in the upper bound). [**Update** (Dec 19) : this has now been achieved by Paco Santos (based on a different construction) and Nevo and Wilson (based on extensions of their 3-D constructions). More detailed to come.]

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The conference Poster as designed by **Rotem Linial**

A conference celebrating Nati Linial’s 60th birthday will take place in Jerusalem December 16-18. Here is the conference’s web-page. To celebrate the event, I will reblog my very early 2008 post “Nati’s influence” which was also the title of my lecture in the workshop celebrating Nati’s 50th birthday.

When do we say that one event causes another? Causality is a topic of great interest in statistics, physics, philosophy, law, economics, and many other places. Now, if **causality** is not complicated enough, we can ask what is the **influence** one event has on another one. Michael Ben-Or and Nati Linial wrote a paper in 1985 where they studied the notion of influence in the context of **collective coin flipping**. The title of the post refers also to Nati’s influence on my work since he got me and Jeff Kahn interested in a conjecture from this paper.

The word “influence” (dating back, according to Merriam-Webster dictionary, to the 14th century) is close to the word “fluid”. The original definition of influence is: “an ethereal fluid held to flow from the stars and to affect the actions of humans.” The modern meaning (according to Wictionary) is: “The power to affect, control or manipulate something or someone.”

Collective coin flipping refers to a situation where n processors or agents wish to agree on a common random bit. Ben-Or and Linial considered very general protocols to reach a single random bit, and also studied the simple case where the collective random bit is described by a Boolean function of n bits, one contributed by every agent. If all agents act appropriately the collective bit will be ’1′ with probability 1/2. The purpose of collective coin flipping is to create a random bit R which is immune as much as possible against attempts of one or more agents to bias it towards ’1′ or ’0′.

Given such a protocol, the influence of a set S of agents towards ’0′ is the probability that R=0 if the agents in S try to tilt the outcomes of the coin flipping towards ’0′ as much as possible. The influence towards ’1′ is defined in the same way. And the influence of S is the sum of these two quantities. To make the definition clearer we should explain what the agent in S can do. Here we assume that in case of simultaneous action by all agents, the “bad guys” can wait to the contributions of all other agents before making their move. The bad guys can only change their inputs to the procedure.

**Notations**: When the protocol is denoted by , for a set of processors, denote by their influence toward ’1′, by their influence toward ’0′, and let . The influence of a single processor is denoted by and the sum is called the **total influence** of . (In the Boolean case we refer to a “processor” or “agent” simply as a variable, and talk about influence of a variable, and influence of a set of variables.)

Boolean functions can be regarded as “voting rules”. A Boolean function describes a way to move from the votes of n voters between two candidates to the collective decision of the society. Thinking of Boolean functions as voting rules provides nice names for special kinds of Boolean functions. “Dictatorship” refers to functions of the form . For the “majority function” (when is odd) the value of is ’1′ if and only if for more than half the variables . For monotone Boolean functions, the influence of the kth variable coincides with the “Banzhaf power index” defined in game theory. Another related important notion of power is the Shapley-Shubik power index.

Picture: Muli Safra

After many months of working on the conjecture, Jeff, Nati, and I managed to prove it. Let be a Boolean function.

**Theorem 1 (KKL): **If the then there exists a variable k so that .

A repeated application of this theorem shows that:

**Theorem 2 (KKL): **If the then there exists a set S of variables so that .

**Majority**

For the majority function with n variables the influence of every variable is proportional to .

**The tribes example**

This is the basic example of Ben-Or and Linial for Boolean functions with low influence. The society is divided into a large number of tribes, each having members. The value of f is one if and only if there exists a tribe whose members all vote ’1′. For this example the influence of every variable is .

The next two examples represent more complicated protocols for collective coin flipping. (Not just Boolean functions.)

**Mike Saks’ and Shafi Goldwasser’s “passing the baton” example**

We start with some voter who holds the baton. This voter passes the baton to another random voter. Every voter who gets the baton passes it to a random voter who did not yet hold it. The last voter to hold the baton chooses the random collective bit. In this example, bad agents cannot tilt the outcome significantly. (The best strategy for the “bad guys” is to pass the baton to a “good guy”.)

**Uri Feige’s “two rooms” example**

Every agent enters at random one out of two rooms. The room with fewer agents is selected and every agent in this room enters at random one out of two rooms. This process is continued (more or less) and at the end, as before, the last remaining agent contributes the collective random bit.

This process is immune against a constant number of “bad guys”. (The first such example was found by Noga Alon and Moni Naor.) The number of rounds in this protocol (appropriately optimized) goes to infinity extremely slowly. It is not known whether there is a protocol with similar properties with a bounded number of rounds.

Consider a monotone Boolean function . Let be the product probability space where for every bit The definition of influence extends without change to the setting of biased product distribution. The influence of the th variable on the Boolean function f with respect to is denoted by . The probability that is a monotone function in and **Russo’s lemma **asserts that the derivative of with respect to is precisely the total influence . Therefore, large influence is related to “sharp threshold behavior”. Namely, to a very short interval between the value of where is very close to 0, and the value of where is very close to 1. Simple consequences of KKL’s theorem to the study of threshold behavior were noted by Ehud Friedgut and me, and Friedgut found an important theorem giving conditions for sharp threshold behavior when p itself is a function of n. Muli Safra and I wrote a survey paper on influences and threshold behavior.

Four sentences about the connection with Game Theory: The sharp threshold phenomenon is called in economics “asymptotically complete aggregation of information”. This property goes back to an old theorem from the theory of voting called “Condorcet’s Jury Theorem“. The Shapley-Shubik power index, mentioned above, can be defined as the integral . (This is not the original axiomatic definition but a later Theorem by Owen.) It turns out that for a sequence of monotone Boolean functions, “sharp threshold phenomenon” is equivalent to “diminishing individual Shapley-Shubik power indices”. (But the quantitative aspects of this result are not satisfactory.)

A major conceptual challenge is to understand the concept of influence (and related notions of “sharp threshold phenomenon”) for distributions which are not product distributions, namely when the probabilities for individual bits to be ’1′ are **not** statistically independent. Does an observer in a committee meeting have an influence? Can there be a negative influence? Moving away from statistical independence is often very difficult and yet very important for most applications. This issue is addressed in the paper of Graham and Grimmett, and that of Haggstrom, Mossel, and myself.

There are quite a few problems regarding influences which remained unsolved. I will mention only two related conjectures both dealing with the Boolean case.

**Conjecture 1:** **(Benny Chor): **Suppose that there is , such that there is a set with .

**Conjecture 2**: Suppose that . Then there is a set , so that .

KKL theorem gives a weak form of Conjecture 1 where is replaced by and a weak form of Conjecture 2 where is replaced by . The main difficulty here (and in various other problems in extremal combinatorics) is that arguing about influences of single variables is the only known method toward influences of large sets. Both these conjectures (as KKL’s theorem itself) have natural formulations in terms of traces of families of sets related to the Sauer-Shelah Theorem.

**Update** (February 2013): Both these conjectures have now been disproved in a paper with Jeff Kahn: Functions without influential coalitions.

The theory of poetic influence was developed by Yale’s literary critic Harold Bloom. His book “The Anxiety of Influence” deals with the process of influence as well as with the psychology of influence in literature. Philosopher Avishai Margalit studied influence in Philosophy in his paper on Wittgenstein. The paper will appear in a Festschrift for Peter Hacker by Blackwell Oxford, 2009. Section 3 opens with a distinction between influence and power: “Naked power is for anyone to see. Influence is not. It works its wonders in ways not readily observable. Influence is inferred from its effects. This is a major reason for the elusive nature of influence.”

Michael Ben-Or

What do Michael Ben-Or, Uzi Segal (whose calibration theorem was mentioned in the post on the controversy around expected utility theory), Mike Werman, Ehud Lehrer, Yehuda Agnon, myself, and quite a few others, all have in common.

Hint:

Mike Saks made valuable comments regarding this post. The first protocol for collective coin-flipping immune to a positive proportion of processors by protocol with rounds was by Alexander Russell and David Zuckerman. If each processor contributes only one bit per round there is a matching lower bound by Russell, Saks, and Zuckerman. The open problem Section from RSZ’s paper is very interesting and, as far as we know, no further progress on the problems presented there was made. The Baton passing method was suggested also by Shafi Goldwasser.

If we allow every processor to contribute many bits in every round then the situation is not clear even for one round. This is related to conjectures discussed by Ehud Friedgut.

Related problems are of great interest for quantum computation. Carlos Mochon have recently solved one of the most important longstanding open problems in quantum information theory, and found a protocol for weak coin flipping with arbitrarily small bias, using quantum bits. His work strengthens earlier work by Kitaev.

Related post: The Entropy Influence Conjecture. (Terry Tao’s blog)

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From Wikipedea: Dunce hat Folding. The blue hole is only for better view

is another name for the contractible non-collapsible space commonly called also the “dunce hat“. (See also this post.) For a birthday conference of Borsuk, a cake of this shape was baked and served.

(polish, in English: Animal Husbandry) is (from Wikipedea, here is the link) a dice game invented and published by Karol Borsuk at his own expense in 1943, during the German occupation of Warsaw. Sales of the game were a way for Borsuk to support his family after he lost his job following the closure by the German occupation authorities of Warsaw University. The original sets were produced by hand by Borsuk’s wife, Zofia. The author of the drawings of animals was Janina Borsuk née Śliwicka. The game was one of the first in the world to feature a 12-sided dice.

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