Jim Geelen, Bert Gerards, and Geoff Whittle Solved Rota’s Conjecture on Matroids

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Gian Carlo Rota

Rota’s conjecture

I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts that for every finite field, the class of matroids representable over the field can be described by a finite list of forbidden minors. This was proved by William Tutte in 1938 for binary matroids (namely those representable over the field of two elements). For binary matroids Tutte found a single forbidden minor.  The ternary case was settled by by Bixby and by Seymour in the late 70s (four forbidden minors).  Geelen, Gerards and Kapoor proved recently that there are seven forbidden minors over a field of four elements.  The notices paper gives an excellent self-contained introduction to the conjecture.

This is a project that started in 1999 and it will probably take a couple more years to complete writing the proof. It relies on ideas from the Robertson-Seymour forbidden minor theorem for graphs. Congratulations to Jim, Bert, and Geoff!

Well, looking around I saw that this was announced in August 22’s post in a very nice group blog devoted by matroids- Matroid Union, with contributions by Dillon Mayhew, Stefan van Zwam, Peter Nelson, and Irene Pivotto. August 22? you may ask, yes! August 22, 2013. I missed the news by almost a year. It was reported also here and here  and here, and here, and here, and here!

This is a good opportunity to mention two additional conjectures by Gian-Carlo Rota. But let me ask you, dear readers, before that a little question.

Rota’s unimodality conjecture and June Huh’s work

Rota’s unimodality conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. This implies that the coefficients are unimodal. A special case of the conjecture is an earlier famous conjecture (by Read) asserting that the coefficients of the chromatic polynomial of a graph are unimodal (and log-concave). This conjecture about matroids was made also around the same time by Heron and Welsh.

June Huh proved Reads’ unimodality conjecture for graphs and the more general Heron-Rota-Welsh conjecture for representable matroids for characteristic 0. Later Huh and  Eric Katz proved the case of  representable matroids for arbitrary characteristics. I already mentioned these startling results earlier and we may come back to them later.

Huh’s path to mathematics was quite amazing. He wanted to be a science-writer and accomponied Hironaka on whom he planned to write. Hironaka introduced him to mathematics in general and to algebraic geometry and this led June to study mathematics and a few years later to use deep connections between algebraic geometry and combinatorics to prove the conjecture.

Rota’s basis conjecture

Rota’s basis conjecture from the late 80’s appears to remain wide open. The problem first appeared in print in a paper by Rosa Huang and Rota. Here is a post about it also from “the matroid union.” It is the first problem in Rota’s article entitled “Ten Mathematics problems I will never solve“. Having access only to page one of the paper I can only guess what the other nine problems might be.

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Rota’s portrait by Fan Chung Graham

 

 

Pictures from Recent Quantum Months

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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

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With Robert Alicki in Gdynia near Gdansk

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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.

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With Michel Dyakonov (Jerusalem, a few days ago)

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

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Aram Harrow and me meeting a year ago at MIT.

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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.

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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.

GTprod2This 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.

Many Short Updates

Things in Berkeley and later here in Jerusalem were very hectic so I did not blog much since mid October. Much have happened so let me give brief and scattered highlights review.

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.

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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.

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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.

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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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NatiFest is Coming

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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.

Nati’s Influence

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.

Influence

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.”

Ben-Or and Linial’s definition of influence

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 f(x_1,x_2,\dots,x_n) 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’. Continue reading

More around Borsuk

Piotr Achinger told me two things abour Karol Borsuk:

 

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

Borsuk trumpet

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.

 Hodowla zwierzątek

(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.

Simons@UCBerkeley

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Raghu Meka talking at the workshop 

I spend the semester in Berkeley at the newly founded Simons Institute for the Theory of Computing. The first two programs demonstrate well the scope of the center and why it is needed. One program on real analysis in computer science seems to demonstrate very theoretical aspects of the theory of computing and its relations with pure mathematics. The second program is on big data analysis, a hot topic in statistics, machine learning and many areas of science, technology, and beyond. And one surprise is that there is a lot in common to these two areas. Next semester, there will be one special program on quantum Hamiltonian complexity which again may seem in the very far-theoretic side of TOC as it largely deals with computational classes between NP and QMA, (far beyond what we seem relevant to real-life), and yet this study is related to classical efficient algorithms in condensed matter physics, with connections to real-life physics and technology. The other special program in Spring 2014 is on evolution (evolutionary biology and TOC)!

The program I am mainly involved with is on real analysis in computer science. It started with a very successful and interesting workshop Real Analysis in Testing, Learning and Inapproximability. This week (Spet 9-13) there is a “Boot camp on real analysis” with three mini-courses. The amount of activity in the center and around it is too vast to allow any sort of live-blogging but I do hope to share with you a few things over the semester. Two things for now: Kalvin Lab, the building where the institute is located is beautiful! The video coverage of talks, (both the live streaming and the video archives) is of very high quality (reflecting both the excellent equipment and the people that operate it).

It is always a special feeling to witness the first days of the academic year whether it is in Israel or in the US. Many students returning to school, many first-year students, at times, accompanied by their proud parents, and quite a few colleagues who share this excitement and are just as surprised on how younger and younger these students are becoming (and some other colleagues, who this year are proud parents themselves). This time, I spent a few days in Yale and saw the excitement there, and then continued with the same “high” mood on the first days of school here at Berkeley.

The Kadison-Singer Conjecture has beed Proved by Adam Marcus, Dan Spielman, and Nikhil Srivastava

…while we keep discussing why mathematics is possible…

The news

Adam Marcus, Dan Spielman, and Nikhil Srivastava posted a paper entitled “Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem,” where they prove the 1959 Kadison-Singer conjecture.

(We discussed part I of “interlacing families” in this post about new Ramanujan graphs.  Looks like a nice series.)

The Kadison-Singer Conjecture

The Kadison-Singer conjecture refers to a positive answer to a question posed by Kadison and Singer: “They asked ‘whether or not each pure state of \cal B is the extension of some pure state of some maximal abelian algebra’ (where \cal B is the collection of bounded linear transformations on a Hilbert space.”) I heard about this question in a different formulation known as the “Bourgain-Tzafriri conjecture” (I will state it below) and the paper addresses a related well known discrepancy formulation by Weaver. (See also Weaver’s comment on the appropriate “quantum” formulation of the conjecture.)

Updates: A very nice post on the relation of the Kadison-Singer Conjecture  to foundation of quantum mechanics is in this post in  Bryan Roberts‘ blog Soul Physics. Here is a very nice post on the mathematics of the conjecture with ten interesting comments on the proof by Orr Shalit, and another nice post on Yemon Choi’s blog and how could I miss the very nice post on James Lee’s blog.. Nov 4, 2013: A new post with essentially the whole proof appeared on Terry tao’s blog, Real stable polynomials and the Kadison Singer Problem.

Update: A very nice blog post on the new result was written by  Nikhil Srivastava on “Windows on theory.” It emphasizes the discrapancy-theoretic nature of the new result, and explains the application for partitioning graphs into expanders.

The Bourgain-Tzafriri theorem and conjecture

Let me tell again (see this post about Lior, Michael, and Aryeh where I told it first) about a theorem of Bourgain and Tzafriri related to the Kadison-Singer conjecture.

Jean Bourgain and Lior Tzafriri considered the following scenario: Let a > 0 be a real number. Let A be a n \times n matrix with norm 1 and with zeroes on the diagonal. An s by s principal minor M is “good” if the norm of M is less than a.

Consider the following hypergraph F:

The vertices correspond to indices {}[n]=\{1,2,\dots,n\}. A set S \subset [n] belongs to F if the S \times S sub-matrix of M is good.

Bourgain and Tzafriri showed that for every a > 0 there is C(a) > 0 so that for every matrix A we can find S \in F so that |S| \ge C(a)n.

Moreover, they showed that for every nonnegative weights w_1,w_2,\dots w_n there is S \in F so that the sum of the weights in S is at least C(a) times the total weight. In other words, (by LP duality,) the vertices of the hypergraph can be fractionally covered by C(a) edges.

The “big question” is if there a real number C'(a) > 0 so that for every matrix M, {}[n] can be covered by C'(a) good sets. Or, in other words, if the vertices of F can be covered by C'(a) edges. This question is known to be equivalent to an old conjecture by Kadison and Singer (it is also known as the “paving conjecture”). In view of what was already proved by Bourgain and Tzafriri what is needed is to show that the covering number is bounded from above by a function of the fractional covering number. So if you wish, the Kadison-Singer conjecture had become a statement about bounded integrality gap. Before proving the full result, Marcus, Spielman and Srivastava gave a new proof of the Bourgain-Tzafriti theorem.

Additional references:

KADISON-SINGER MEETS BOURGAIN-TZAFRIRI by PETER G. CASAZZA, ROMAN VERSHYNIN,  The Kadison-Singer Problem in Mathematics and Engineering: A Detailed Account pdf, and many other recent publications by Pete Casazza.

Polymath8: Bounded Gaps Between Primes

Yitang Zhang’s very recent shocking paper demonstrated that bounded gaps between primes occur infinitely often, with the explicit upper bound of 70,000,000 given for this gap. Polymath8 was launched for the dual purpose of learning Zhang’s proof and improving the upper bound for the gaps. Here are links for three posts (I, II, III) on Terry Tao’s blog, a post on the secret blogging seminar,  and for a post on the polymath blog. And here is the table for the world records so far.

Updates: Record for June 16 – 60,744

Why is mathematics possible?

Spectacular advances in number theory

Last weeks we heard about two spectacular results in number theory.  As announced in Nature, Yitang Zhang proved that there are infinitely many pairs of consecutive primes (p_n, p_{n+1}) which are at most 70 million apart! This is a sensational achievement. Pushing 70 million to 2 will settle the ancient conjecture on twin primes, but this is already an extremely amazing breakthrough.  An earlier breakthrough came in 2005 when Daniel Goldston, János Pintz, and Cem Yıldırım proved that the gaps between consecutive primes p_{n+1}-p_n is infinitely often smaller than \sqrt {\log p_n} \log \log ^2 p_n.

Update: A description of Zhang’s work and a link to the paper can be found on Emmanuel Kowalski’s bloog Further update: A description of Zhang’s work and related questions and results can be found now in Terry Tao’s blog. Terry Tao also proposed a new polymath project aimed to reading Zhang’s paper and attempting to improve the bounds.

Harald Helfgott proved that every integer is the sum of three primes.  Here the story starts with Vinogradov who proved it for sufficiently large integers, but pushing down what “sufficiently large” is, and pushing up the computerized methods needed to take care of “small” integers required much work and ingenuity.

Why is Mathematics possible?

The recent news, and a little exchange of views I had with Boaz Barak, bring us back to the question: “Why is mathematics possible?” This is an old question that David Kazhdan outlined in a lovely 1999 essay “Reflection on the development of mathematics in the twentieth century.” The point (from modern view) is this: We know that mathematical statements can, in general, be undecidable.  We also know that a proof for a short mathematical statement can be extremely long. And we also know that even if a mathematical statement admits a short proof, finding such a proof can be computationally intractable. Given all that, what are the reasons that mathematics is at all possible?

It is popular to associate “human creativity” with an answer. The problem with incorrect (or, at least, incomplete) answers is that they often represent missed opportunities for better answers. I think that for the question “why is mathematics possible” there are opportunities (even using computational complexity thinking) to offer better answers.

Please offer your answers.