Extremal problems in additive number theory

Our first lecture dealt with extremal problems for families of sets. In this lecture we will consider extremal problems for sets of real numbers, and for geometric configurations in planar Euclidean geometry. 

Problem I: Given a set A of n real numbers, how small can the set A+A={a+a’: a,a’ A} be?

If A={1,2,…,n} |A+A|=2n-1. Suppose the elements of A are and . Note that   . So we identified 2n-1 distinct elements in A+A. This is the Cauchy-Davenport theorem.  

Problem II: Given a set A of n positive real numbers, how small can be the set A A={a a’: a,a’ A}?

You may protest (again, like in the first lecture,) that I regard problem II a different problem. You can move from problem II to problem I by taking the logarithm of all elements in A, or you can simply use the same proof with the sum replaced by a product. The proof relies only on very basic monotonicity properties of these operations.

OK, lets have another problem II.

Problem II: Given a set A of n real numbers, how small can the quantity max (|A+A|, |A A|) be? 

This problem was asked by Erdös, and in hindsight it is a very good problem. Erdös conjectured that the maximum behaves like , and this is open.  We will see below a wonderful proof by Elekes that the maximum is (up to a multiplicative constant) at least . The exponent 5/4 was improved twice(!!) by Jozsef Solymosi and there is a very nice post about his most recent ingenious proof for a lower bound max (|A+A|, |A A|)  in Izabella Laba’s blog. 

Extremal problems in plane geometry

Problem III:  Given n points in the plane what is the maximum number of pairs among them at distance ‘1′

Problem IV:  Given s points and t lines in the plane what is the maximum number of incidences between them.

An incidence is a pair where  is a point  is a line and .

Problem V: Given a graph G with v vertices and e edges what is the minimum number of crossings in a planar drawing of G.

The second of my extremal combinatorics lectures was devoted to the surprising connections that were found between the above problems. There is also a very nice post about this material on Terry Tao’s blog. To show you something new I will describe a few problems in higher dimensions at the end.

The solution to problem IV is given by:

The Trotter Szemeredi Theorem: The number I of incidence between t points and s lines in the plane is at most .

In particular the number of incidences of n points and n lines in the plane is . This is an Euclidean phenomenon which does not follow from the very basic axioms of geometry about incidences of points and lines: Recall that for a finite projective plane with n lines and n points the number of incidences behaves like .

Elekes: Bounds for product sums via Trotter-Szemeredi

This is an example of an ingenious, simple, and surprising connection between two areas.

Here is how it goes: Let  be a set of n positive real numbers. Consider the planar set . The number  of points in this set is the quantity we would like to estimate. Now, consider the set of  lines of the form , where . Our set of lines contains  lines, . Note that for every line of the form   and every  if we let we get . Thus the point is on the line. We have identified points of Z on every line in our family of lines. In sum, the number of incidence is at least .

Aha! But now we can apply Trotter Szemeredi’s theorem: . This gives so .        

In order to prove the Trotter-Szemeredi theorem we will need the following:

The Crossing Theorem: A graph drawn in the plane with v vertices and e edges, e>5v has at least crossings.

Szekély: Bounds on Incidences via the crossing lemma

This is an example of a surprising ingenious and simple connection in the same area.

Here is how it goes: We can assume that every line is incident to a point. Consider the following graph drawn in the plane. The vertices are the points in our family. The edges correspond to line segments between two consecutive vertices on the same line. Now, this graph has vertices and edges. (A line incident to k points contributes k-1 edges.) The number of crossings is at most because every crossing represents an intersection point between a pair of lines. Therefore either or .  

The bootstrap proof of the crossing lemma

This is an example of a simple new proof that sheds a light on a theorem, yet is it really different from the old proof? (There was an interesting discussion over Gowers’ blog on when two proofs are essentially the same. In this case, I regard the proofs as different while Janos Pach who understand this subject more deeply regards them as the same.)

We start with Euler’s theorem. It implies that a planar graph with v vertices has at most 3v-6 edges. Therefore for a graph G with e edges and v vertices drawn in the plane consider a maximal planar subgraph H and note that every edge not in H crosses an edge in H. So  .

And now we improve this inequality using itself as follows: Start with a graph G with v vertices and e edges, e>5v. Choose at random every vertex with probability p. The resulting random graph H will have v’ verices, e’ edges, and c’ cossings where the expectation of v’ is pv, the expectation of e’ is the expectation of c’ is . It follows that so we can optimize the value of p and this leads to the crossing lemma.   

Problem III about the number of pairs of points at distance one among n points is the plane is a very famous problem (of Erdös, of course) in discrete geometry. Consider a configuration of n points in the plave and the configuration of unit circles around them. The number of pairs of distance one is half the number of points-circles incidences. Trotter and Szemeredi proved an upper bound and Szekely’s argument gives a simple proof of this result as well. It is conjectured that the exponent 4/3 can be pushed down to , for every . The Trotter-Szemeredi theorem itself is sharp - there are examples with matching lower bounds. (Up to the multiplicative constant K.)

High dimensions

Conjecture: Let K be a 2-dimensional simplicial complex with e edges and t triangles. Then for every embedding of K into the number of crossing is at least .

As pointed out by Dey and Pach, this conjecture would follow (using the probabilistic argument above) from the following:

Conjecture (Sarkaria): Let K be a 2-dimensional simplicial complex with  (or even with ) then K cannot be embedded to .

(There are similar conjectures for higher dimensions.)

Sarkaria’s conjecture is closely related to the “g-conjecture” that I plan to devote a special post to.

Problem: What is the maximum number of incidences between s lines and t planes in ?

Unlike the planar case I do not know of a relation between the incidence problem and the crossing problem in high dimensions.

After the wonderful “being a cosmonaut” taxi-story, I am happy to present you:

 An Ambassador in Singapore

A story by Michal Linial

It was a long trip, 4 days in Singapore. I visited the tallest artificial waterfall in the world, and went on a night safari with artificial moon light. Well, time to go back home to reality. Before leaving my hotel, I made sure that I had enough money for a taxi ride to the airport. Due to last minute extra taxes, I found myself with only 15 Singaporean Dollars (~ 10 $ US). In the hotel, they promised me that it is more than enough. I entered the Taxi and immediately mentioned that I have 15$.

In the next 30 minutes my taxi driver explained that he actually does not want these 15$ at all. At the matter of fact, he is honored to take me to the airport as a treat of the Singaporean people to their distinguished guests. He said: ”I want your memories from Singapore to be enjoyable so you will tell everyone to visit us, and so you come back as well”. I was impressed. He continued: “my real satisfaction is to be the ambassador of my country”. He even went into some psychological arguments: “Can you remember the last time you were in Paris, can you tell me who took you to the airport?” I admitted that I did not remember, and then he continued: “Can you remember who took you to the airport from Jerusalem only 5 days ago ?” I had to admit that I did not remember any of this…and then he made his last point ”You will always remember me and you will want to come again to Singapore just because you met people like me”. Indeed, I was so impressed and vaguely remembered the Israeli slogan “HASBER PANECHA LATAYAR”, which is no longer relevant… I promised him that I would cherish these memories forever.

We arrived at the airport, my ‘ambassador’ helped me carry my heavy luggage, he shook my hand and immediately said, “I can take whatever you want, 50 US Dollars, or 50 Euro and I do not even mind 50 UK pounds”. I did a quick calculation to sort all these attractive alternatives and humbly offered him my 50 US $. My taxi driver added “But this does not include your heavy luggage.” I replied ”I can give you the 15 Singaporean Dollars.” My ambassador answered without hesitation: ”No way, I want you to have good memories and save them as souvenirs for your kids”.

Well, as you see I have not forgotten him, and he was right, this week I fly back to Singapore.

A poster with the slogan: “HASBER PANECHA LATAYAR” (This means “Be nice to the tourist.” The poster also hints at the literal translation: “explain your face to the tourist.”)

The annual Summer School in Economics at HU was directed until last year by Kenneth Arrow, along with Eyal Winter. Arrow decided this year to step down as a director and Eric Maskin is replacing him. The 2008 Summer School was devoted to Arrow’s economics. The list of speakers was quite impressive, with six lecturers who are already Nobel Laureates. (Our local Institute for Advanced Study runs five schools every year, in Physics, Economics, Life Sciences, Jewish Studies, and Mathematics.)    

Economic puzzles told by Arrow

Let me tell you about three economics puzzles mentioned by Arrow in an earlier summer school. I doublechecked some details with Arrow himself; still, if my description contains errors I will be happy to be corrected. (Arrow spent a considerable amount of time talking with the workshop students. Another remarkable thing about him: he takes lecture notes! Is it a good idea to take detailed lecture notes at lectures? Let’s return to this question sometime.)

Puzzle 1: Why is there unemployment?

Why is this even a puzzle? Because the economics teaching that “the market will clear” means that all people who can work will. A person who can work and is not working represents inefficiency, which is not supposed to exist in a competitive economy. Part of the issue is referred to as “friction” and accounts for economics processes being slow rather than instantaneous. But it appears to be true that there is more to unemployment than that. What can explain the 30% unemployment that was witnessed in the US in the 1930s?

Is this puzzle a scientific problem? You bet it is! And it is a fairly clear-cut scientific problem. I suppose there are several answers to this puzzle in the literature but we are far from a definite understanding of the issue.

Puzzle 2: What is the reason for high volatility of prices in markets, say in stock markets?

The price of a stock, according to economics theory, represents the long-term value of the company. What accounts for the fact that the overall value of the entire stock market may fluctuate by more than 1% on a typical day? What accounts for fluctuations (more often drops) of 3-5% in one day? (Such fluctuations are not rare.) A famous question is to explain the one-day drop of 20% in October 1987.

Arrow mentioned in this context the Milgrom-Stokey “no trade theorem” which asserts that under certain assumptions markets in equilibrium will exhibit no trade (even if traders have private information).

Private companies conduct a lot of research on stock market behavior, probably much much more so than universities. I asked Arrow whether we should expect some progress toward understanding the fundamental issues regarding stock-market behavior to be achieved there. Arrow was quite skeptical about it. 

In my opinion, stock market behavior is an example where scholarly research is important even in areas where much research is taking place outside academia. (It is also an important and delicate matter to ensure that the external research and activity not vitiate academic goals and integrity.) 

A relevant blog post concerning financial mathematics is Tao’s recent description of the Black-Scholes formula. Explaining the systematic difference between the Black-Scholes formula and the actual behavior of options prices is another interesting question.

Puzzle 3: What accounts for the huge futures trading in foreign currencies?

Another puzzle that Arrow mentioned is this: futures trading in foreign currencies can be explained by agents involved in international trade wanting to reduce their risk. This suggests that the volume of currency futures trading  will be below the volume of international trade. Yet currency futures trading is 300 times larger. What can explain this phenomenon?

Following are a few lectures that I would like to tell you about, in some detail, in a later post. Maskin’s lectures on social choice and “the robustness of majority rule” were perhaps the lectures closest to my own research interests (and related to Arrow’s theorem about voting). Roger Myerson in his lecture asked: “Is capitalism better than socialism?” He was referring to the Soviet Union-type of socialism and the way firms operate under such a system. John Geanakoplos talked about models of general equilibrium theory with collateral and gave a hilarious account of Shakespeare as economist. His model provides some insight into the current subprime crisis in the US.  And Herb Scarf returned to cooperative game theory and proved his theorem on the nonemptyness of the core for balanced games. A link to all lectures is here. 

Apropos of the comparison between capitalism and socialism, Myerson’s work does not deal with a comparison between the US system and the slightly more socialist West European version. (The difference does not lie in the way firms operate but in governmental redistribution of resources.) Personally, I like the West European economic free-market system and even the most “socialist,” Scandinavian version of  it, and this once almost got me in trouble. When I visited Stockholm, in the late 80s, I was sitting next to a local Swedish person in a Chinese restaurant on Nybrogatan Street, and I was telling him at some length how highly I thought of the Swedish system. The guy listened carefully to what I said and at the end he was so angered by it that I thought he would kill me. (In Sweden, at that time, the punishment for murder was 10 years in prison, of which only five had to be served; yet murder rates were low.) It appears that one should be careful about giving compliments almost as much as about criticism.

Erdos and Turan asked in 1936: What is the largest subset of {1,2,…,n} without a 3-term arithmetic progression?

In 1946 Behrend found an example with 

Now, sixty years later, Michael Elkin pushed the the factor from the denominator to the enumerator, and found a set with   !

Here is a description of Behrend’s construction and its improvment as told by Michael himself:

“The construction of Behrend employs the observation that a sphere in any dimension is convexly independent, and thus cannot contain three vectors  such that one of them is the arithmetic average of the two other. The new construction replaces the sphere by a thin annulus. Intuitively, one can produce larger progression-free sets because an annulus of non-zero width contains more integer points than a sphere of the same radius does. However, unlike in a sphere, the set of integer points in an annulus is not necessarily convexly independent. To counter this difficulty I show that as long as the annulus is sufficiently thin, the set U of its integer points contains a convexly independent subset W whose size is at least a constant fraction of the size of U. The subset W is, in fact, the exterior set Ext(U) of the set U.

The set U above is the set of integer points of the the intersection of a very thin annulus with a cube. The (minimum) dimension k of the space   in which this body has non-zero volume is not constant, but rather it tends to infinity logarithmically with the radius of the annulus. Consequently, it becomes not trivial to estimate the volume of this body, leaving alone the the number of integer points that it contains.  In addition, most known estimates for the discrepancy  between the number of integer points and the volume assume that the dimension is fixed, and thus these estimates are inapplicable in this case.  Moreover, since the annulus is very thin, its volume is not much larger than its surface area, and thus crude estimates of the discrepancy between the number of integer points and the volume do not suffice. Showing more precise estimates involves a rather delicate analysis.”

I decided to split long part III into two parts. This (truly) last part of this series deals with probabilistic problems and with combinatorial questions regarding higher Laplacians. 

21. Higher Laplacians and their meanings

Our high dimensional extension to Cayley’s theorem reads:

The right hand side of our formula corresponds to the eigenvalues of a higher Laplacian of a complete d-dimensional complex with n vertices. In several general cases there are nice expressions for these eigenvalues - for matroidal complexes, for shifted complexes, for complete skeleta of the cubes and in other cases. There are also nice general high-dimensional matrix-tree theorems.

If is the space of k cycles and we identify it via a certain inner product with the space of k-cocycles, then we can talk about the Laplacian defined by latex where is the boundary operator from to and is the coboundary operator from to . 

Spectra of graphs’ Laplacians are very important, for understanding random walks on graphs, for expansion properties and they are also related to many graph parameters like the diameter. The recent series of posts on Luca Trevisan’s blog give a detailed description of these connections (See also this post on James Lee’s blog.) What about higher Laplacians? (Those do not correspond to connectivity but to higher homology groups.) 

What is the analog of the random walk interpretation of the spectral gap?

What is the analog of the relation between the spectral gap and expansion properties?

What is the analog of the diameter?   

22. Probabilistic questions

 Consider the class G of all d-dimensional simplicial complexes on n labelled vertices with d-faces and a complete (d-1)-skeleton. Is there a substantial probability, for d>1,  that a random complex in G with the property that every (d-1)-face is contained in a d-face (no isolated (d-1)-faces) is Q-acyclic? This is not the case for graphs (d=1). The probability for a graph with n vertices and n-1 edges to be a tree tends to 0 even if it has no isolated vertices. This question has a similar flavour to results regarding singularity of random matrices with 0,1 entries.

Here are other natural probabilistic questions that go back to my old paper.

Inside G consider  

A) The class of collapsible complexes

B) The class of contractible complexes

C) The class of Z-acyclic complexes

D(p)) The class of Z/p-acyclic complexes

E) The class of Q-acyclic complexes 

For d=1 (graphs) all classes A-E are the same. For d>2 the class C equals the class D.

We can ask if for d >1 as n tends to infinity, a random complex in E is almost surely not in D(p), and a random complex in D(p) is almost surely not in C, and a random complex in C is, for d=2, almost surely not in B, and a random complex in B is almost surely not in A.  

Let me also mention that there are several recent results about random simplicial complexes by Linial and Meshulam and by Babson Hoffman and Kahle.

23. Russell Lyons problem: Generating random hypertrees 

Find a way to generate random Q-acyclic spanning subcomplexes L of a d-dimensional simplicial complex K (with the distribution given by ). One way to do it is to choose a d-face based on the ratio between the weighted number of hypertrees containing and not containing this face. But we want to do something else -  mimicking the beautiful ways of Broder-Aldous and of Wilson to generate a random spanning tree.

This question was raised by Russell Lyons in the context of studying determinental probability measures.

24. Diversion: the amazing algorithms of Aldous-Broder and of Wilson to generate random spanning trees.

The Aldous Broder algorithm  goes as follows: Start a random walk on the edges of the graph and add to your tree any edge visited which does not create a cycle, the distribution on the resulting spanning trees is uniform.

The Wilson algorithmgoes as follows: Mark a vertex as a root. (at a later stage the root will be a subtree.) Choose an arbitrary vertex not in the root and make a random walk until hitting the root. Delete all cycles created in this walk and add the remaining path to the root. Returning this process also leads to a uniform random spanning tree.

The paths obtained by Wilson algorithm are called loop erased random walk; we can regard them as a certain random 1-cycle of whose boundary is a presecribed 0-cycle. Something analogous in higher dimension is quite desirable.

Professor Sarah Stroumsa

On Wednesday, the Senate of the Hebrew University of Jerusalem elected Professor Sarah Stroumsa (homepage) as the next Rector (provost) of the Hebrew University. For the first time since its establishment, the Hebrew University has elected a woman to its highest post of academic leadership.

The situation of the Israeli higher education is complicated and constantly on the verge of a crisis. But this election is a reason for celebration for the HU, and is part of an ongoing change in the Israeli society as a whole.

Professor Leah Goldberg

 On this occasion, I would like to dedicate to Sarah and to the HU community a poem entitled “Rainbow”, written by the poet Leah Goldberg.  Leah Goldberg was an important scholar and a professor at the Hebrew University, and she is most famous as one of the greatest Hebrew writers of modern times. It is a poem for children, which I also regard as a metaphor of academic and other quests. To further celebrate the event (and in the tradition of Ehud Friedgut’s translation contest of a limerick on Konigsberg’s bridges and Euler), I am announcing an open contest for translating the poem into English. Please email me your translations or post them directly here. (You can find a very rough literal translation below.)

בשמיים ראיתי קשת,
היא עמדה ממש מול הרחוב
ורציתי אליה לגשת
ולראות אותה מקרוב.
והתחלתי ללכת, ללכת,
ועליתי על ראש גיבעה.
אך ככל שהרחקתי לכת
הקשת לא קרבה:

 עופי, עופי, יפת כנפיים!
עופי, עופי אל המרום,
אל הקשת אשר בשמיים
ומיסרי לה בירכת שלום.

 היא עמדה במרום רקיע
כמצויירת על עננה,
אז ידעתי שלא אגיע
ושלחתי אליה יונה.
בשמיים יונה וקשת,
בשמיים הדרך פתוחה
וטיפות ראשונות של גשם
מבשרות שנת שלום וברכה

עופי, עופי, יפת כנפיים…

To get started here is a literal translation:

In the sky I saw a rainbow.

She stood just across the street

and I wanted to go to her(*)

and see her up close.

And I started going and going,

and climbed to the top of a hill,

but the further I went

The rainbow did not get any closer.

Fly, fly, you with the beautiful wings!

fly fly to the sky,

to the rainbow in the sky and

give her my blessing.

She stood there on the top of the sky

as if painted on a cloud,

and then I knew I will not reach her

and I sent her a dove.

In the sky - a dove and a rainbow,

in the sky the road is open,

and first drops of rain

signal a year of blessing and peace.

Fly, fly, you with the beautiful wings! …

(*) In Hebrew “to her” (”el-lea”), also sounds as ”to Leah”.

A rainbow picture on Izabella Laba’s blog.

This is the third and last part of the journey from a Helly type conjecture of Katchalski and Perles to a Cayley’s type formula for “hypertrees”.  (On second thought I decided to divide it into two devoting the second to probabilistic questions.) This part will include several diversions, open problems, and speculations.  

11. How to make it work - the matrix tree theorem

Our high dimensional extension to Cayley’s theorem reads:

where the sum is over all d-dimensional simplicial complexes K on n labelled vertices, with a complete (d-1)-dimensional skeleton, and which are Q-acyclic, namely all their (reduced) homology groups with rational coefficients vanish.  

Looking at the various proofs of Cayley’s formula (there are many many many beautiful proofs), the proof that I know to apply is the one based on the matrix tree theorem.

Consider the signed incidence matrix A’ between all (d+1)-subsets and all d-subsets of {1,2,…,n} that represents the boundary operator of simplicial homology. The rank of this matrix is , and just like in the ordinary matrix tree theorem you delete rows to be left with linearly independent rows. Here you delete all rows corresponding to sets containing ‘n’ and you are left with a matrix A. Now we compute the determinant of  directly, and compare the result to a computation based on the Cauchy-Binet Formula.

The eigenvalues of  are the eigenvalues of the d-th Laplacian of the complete d-dimensional simplicial complex with n vertices. It is easy to inspect what they are and the determinant of is indeed .

The many square determinants correspond to d-dimensional simplicial complexes K on our labelled set of vertices, which satisfy and . Now if K has non-vanishing d-th homology, the determinant is zero. If K is a Q-acyclic simplicial complex (i.e., its (reduced) homology groups with rational coefficients are trivial) then it has a non zero determinant. So far, it is like trees, but next comes a surprise. The contribution of K is the square of the number of elements in , the (d-1)th homology group of K. This torsion group is finite, but for d >1 it need not be trivial.

Emily Peters presents the matrix-tree theorem. From “The Sarong Theorem Archive” - an electronic archive of images of people proving theorems while wearing sarongs.

12. An even simpler use of Cauchy-Binet worth knowing

Consider the matrix A, whose columns are all +1 -1 vectors of length n. Computing , via Cauchy-Binet Formula (or by other easy methods) asserts that the expected value  for all   +1/-1 matrices behaves roughly like . This was observed by Turan and Szekeres who also found a formula for the sum of the fourth powers of all 0-1 n by n matrices. See a leter paper by Turan. (I am not aware of a similar formula for the fourth power of the size of the homology groups for hypertrees.) Much is known about the determinant and related properties of random 0-1 matrices and the analogy between torsion in the homology groups of random complexes and determinants of random matrices looks like a good analogy.

13. Torsion

One consequence of the formula compared to the total number of available simplicial complexes is that the torsion group is typically huge. (For d>1, the expected value of for Q-acyclic complexes counted by the formula, is asymptotically larger than ; I am not aware of explicit examples with such a huge torsion group.)

How should we think about torsion in homology? It seems that thinking about the size of the torsion as a behaving like the determinant of a random matrix, may give a good intuition for many cases.

14. Extending other proofs for Cayley’s theorem?     

Cayley’s counting trees theorem has many wonderful proofs.  Can any other proof extend to the case of Q-acyclic simplicial complexes? For example, one proof relies on the exponential theorem that relates the exponential generating functions for connected and general graphs with a certain property P. (Followed by the Lagrange inversion formula.) Is there an analog of the exponential formula when connectivity is replaced by higher homology? Is there any analog of Prüfer sequences? I am not aware of any other proof that works.

15. Weights to the rescue of other conjectures? 

Can we use subtle weights to save other promising but false enumerative conjectures? The farthest reaching fantasy in this direction would be to try to save MacMahon’s conjecture regarding space partitions of the number n. This conjecture is about enumerating spacial arrays of numbers that sum up to n. The conjecture is true for small values of n but fails for larger values. Can subtle weights come to the rescue?  (MacMahon’s conjecture extends the formulas for ordinary partitions and for plane partitions.) 

16. Incidence matrices

Perles’ observation in the beginning of this story was about the rank of the incidence matrices modulo 2 of k-subsets versus (k+1) subsets of an n element set. This was the starting point for a work by Linial and Rothschild. They asked:  What is the rank of the incidence matrix of  versus modulo p? and gave a complete answer for p=2. Richard Wilson gave a complete answer for general values of p and came quite close  to presenting the “Smith form” of these matrices. Frumkin and Yakir gave representation-theoretic interpretation of Wilson’s result and proved “q-analogs”. Namely they replaced k-subsets of an n-element set by k-dimensional linear (and in a later work affine) subspaces of an n dimensional linear space over a field with q elements. They gave a complete formula when p and q are prime.   

17. Duality and Self-Dual trees

 Here is a very nice notion of duality that occurs in many places. Start with a simplicial complex K on a set X of vertices. Take the family F of all the complements to all sets in K. (This is not a simplicial complex, it is closed under supersets and not under subsets.) Now, take the family K* of all  subsets of X not in F. Formally, {}.

K* is a simplicial complex again. It is the Alexander dual of algebraic topology, and the “blocker” of polyhedral combinatorics.

If n=2d+2 the duals of our hypertrees are also d-dimensional.  Molly Maxwell counted self-dual hypertrees with the same weights we used, and for odd dimensions the count gives precisely the square root of . She deduced it from a more general theorem on matroids duality. For even values of d this is not the case but something may still work.

For d=1 the number of self-dual trees (simply stars) on 4 vertices is four the square root of 16 the total number of labelled trees. There is a theorem of Tutte extending this to self dual trees inside self dual planar maps. For d=3 and 8 vertices, the weighted number of all hypertrees is and by Maxwell’s theorem the weighted number of self dual ones is . For d=2, n=6 - the weighted number of hypertrees is and if we exclude the triangulations of the real projective plane we get . For d=4 the total weighted sum of hypertrees with 10 vertices is , and somehow, a clever weighted sum of the self dual ones should give you .

18. The Perles-Katchalski conjecture and associated eumeration problem

The assertion of the Perles-Katchalski conjecture holds for general classes of simplicial complexes described by homological properties, and we can ask again if the extremal examples enumerate nicely.

Let  be the class of (d+r)-dimensional simplicial complexes with the properties that

(L) For every induced subcomplex K’,

Now, the homological extension of the Perles-Katchalski Theorem asserts that a (d+r)-dimensional simplicial complex K with n vertices satisfying condition (L) has at most   d-dimensional faces.(For r=0 we need not worry about induced subcomplexes since, in this case, non trivial d-th homology for a subcomplex immediately extends to the whole complex. 

We can try to “enumerate” simplicial complexes with n labelled vertices satisfying property (L) with precisely d-dimensional faces. (All these complexes will have the same number of i-faces for every i.)

We can expect that an appropriate enumeration of these objects (probably those containing a specific r-face), will give us a formula of the form , where m is the number of r-dimensional faces of such a simplicial complex K. For the case d=1, no weights are needed and this speculation reduces to a formula of Beineke and Pippert for counting “k-trees”. (We may even expect finer enumeration formulas according to degree-sequences of r-faces; This is known for “k-trees”.)

19. Adin’s colorful extension.

Ron Adin extended the weighted enumeration of hypertrees to “colored complexes”, thus confirming (with extra weights added) another conjecture of Bolker.

20. Gelfand’s question.

Ron Adin gave a lecture about his work in a Stockholm ‘89 meeting in combinatorics which was one of the earliest meetings with many participans from Russia, among them Gelfand, Vershik, Zelevinskii, Serganova, and others. Gelfand was excited about combinatorics (or what he regarded as combinatorics) at the time and was quite interested in Adin’s result. One question he asked me was: why is it that in combinatorics there is so much emphasis on graphs compared to higher dimensional objects.

I personally like the combinatorics of high dimensional objects but I could think of three answers. (Gelfand was quite satisfied with them).

a) For many purposes moving from sets to graphs represents a major conceptual jump, more than moving up from graphs to higher dimensional objects.

b) Higher dimensional objects can often be represented by graphs.

c) Many of the miracles of graph theory fail at higher dimensions.

Another memory from the 89 conference is this: Israel Gelfand has a somewhat wide-spanned competative nature. Gelfand looked at Ron Adin and asked me: “He is orthodox isn’t he?”, “yes” I replied. Gelfand thought a little and then said: “But not as orthodox as my Dima.”

Ladies and gentelmen, I am very happy to present to you:

Being a Cosmonaut

A story by Michal Linial

I am back from the airport… not in the best mood for a long discussion but quite open to hear refreshing political statements. 50 minutes taxi ride that is all what it takes… This time, the communication took its own turns… I start with my regular question: “What is new this week”. This time, my Russian Taxi driver ignored my question and asked back: “How many planets are there?” I told him that to the best of my knowledge there are 9 planets that orbit around the sun. Really, he shouted: “my wife tells me there are 8 and she tells me she is a cosmonaut” and he continues, she knows nothing! I said that there might be some debate, so 8 is a good number and probably I got confused. My Taxi driver got really mad now: “You are not a cosmonaut and you know better”.
Two minutes later, he asks me again “how old is the earth”. This time I was a bit less decisive and told my taxi driver that I am not sure, but the accepted number is 4-5 billion years. To make sure this time there will be no conflicts with his marriage, I immediately added, the universe is much older, also, many people believe that the earth is much younger. My taxi driver now is very excited and answers me without hesitation: “You see, I can not even ask my wife this question, she says she is a cosmonaut, but I am sure that if I ask her, she will not know”… I had a creative suggestion to solve this unsolvable catch and quietly whispered: “maybe you should try to ask her about it”. It was a big mistake… He told me, “if I will ask her she may say the right answer but it is only because she is guessing, how should she know what is the age of earth?? Who can know the age of the earth?. What does she think of herself, is she god?? Did she create the earth??” Just before we got to Jerusalem, he told me “I do not believe her that she is a cosmonaut, and even if she is, they know nothing there, I am telling you”… 

Michal Linial is a HU Professor of Biology and the Director of The Sudarsky Center for Computational Biology .

Christine Björner’s words at the Stockholm Festive Combinatorics are now available to all our readers. What makes this moving and interesting, beyond the intimate context of the conference, is our (mathematician’s) struggle (and usually repeated failures) to explain to others what we are doing and why we are doing it.

The Golden Room and the Golden Mountain

Christine Björner

I want to tell you a story about Anders. Actually two stories. The story of The Golden Room. And the story of The Golden Mountain.

I’ll begin with the Golden Room.

Once when I had seen Anders, night after night, week after week, working at his desk, totally immersed in a world of his own, I asked him this question: Can you explain to me what you are doing? And he answered: Christine, I have found the most beautiful room. The whole room is a mosaic of gold, dazzling in its splendour. What I am trying to do is make this room visible to others.

I am not a mathematician, but this metaphor gave me an insight into the magical world that all of you who have come to the Festive Combinatorics conference share. I want to honour today, in Anders, and all of you, not only your beautiful minds, but your intuition, your persistence, and your passion for truth.

A papyrus showing Queen Ankhes-Tut, wife of King Tut, with her husband in the golden room. The Egyptian Museum, Cairo.

Now I will tell you the second story. The story of The Golden Mountain.

The Golden Mountain is a rocky peninsula on an island in the Stockholm archipelago. Before I lived there I saw it in a vision. So I know that it belongs in the realm of the sacred and the magical. What I didn’t know was that someday it would be mine. And that it would be Anders who would unlock the door to this magical realm. Just like the Golden Room that he visits, the Golden Mountain for me is dazzling in its simplicity. And I don’t need any math theorems to show it to you. All I need is a boat, which Svante has so kindly arranged.

Thank you. I know how much these past three days have meant to Anders. I know that I speak for both of us when I say that you are very welcome to join us on the deck of our little cottage on Golden Mountain for afternoon coffee and tea.

La Torre del Oro - The Golden Tower in Sevilla

Ladies and Gentelmen: Amir Ban (right, in the picture above) the guest blogger, was an Israeli Olympiad math champion in the early 70s, with Shay Bushinsky he wrote Deep Junior, and he is also one of the inventors of the “disc on key”. This post is about computer chess. 

Let me introduce myself: I’m Amir Ban, and I wrote the computer chess program Junior, also known as Deep Junior. When Gil invited me to guest-blog for him on the subject of computer chess, I was honored and pleased, but what can I write to introduce the subject to the uninitiated? Well, as luck has it, I participated last week in a unique event in Barcelona, Spain: a man with machine vs. machine match! The “man” was veteran grandmaster and Spanish champion Miguel Illescas. The “machine” he assisted himself by was my own Junior 10 (a commercially available product) on his Toshiba notebook. On the other side of the board was my latest Deep Junior 11, due to be released later this summer, running on an ordinary core-duo Dell desktop.

Susan Polgar, a former women’s world chess champion, and the eldest of the three Hungarian-Jewish Polgar sisters, describes the event on her popular blog. The comments to the blog entry show some difference of opinion:

Comment 1: “That’s not fair to the computer at all.”
 
Comment 2: “Big deal, Junior 10 vs. Junior 11 with a grandmaster moving the mouse….”

  
Visualization of Deep Junior Bxh2 sacrifice in the 5th game, New York, 2003.
Hmm… We need some perspective here. For that, let us take a few quick flashbacks, starting ca. 60 years ago with the pioneering efforts of Alan Turing, and especially Claude Shannon, who in 1950 wrote “Programming a Computer For Playing Chess“. In the article, Shannon lays out the foundations of computer chess, still practiced to this day: Given the current position where the computer must play a move, it will generate the tree of all hypothetical continuations: All moves playable at the position, then all possible replies to each of these moves, then all possible replies to the replies, and so on. Theoretically this process may continue indefinitely until a terminal position is reached, i.e. a checkmate, or a draw by the rules of chess. Unfortunately (or rather, fortunately) this possibility is merely theoretical, as the typical number of legal moves per position is around 40, and a chess game may last a hundred or so moves, the exponential explosion of possibilities forces a limit to the practical depth of the moves tree.
 

The Shannon trophy

Shannon proposed stopping the tree generation at some practical depth, and attaching an evaluation to the position at the leaf.

The evaluation would be a number on a scale whose two extremes would be “white wins” and “black wins”. It may be computed based on, say, the balance of chess pieces material, for a start, and on identifiable strengths or weaknesses in the position, according to some formula. Having attached an evaluation to each of the leaves, the minimax algorithm may be applied to derive an evaluation for all non-leaf positions, including the root position. The minimax algorithm is simple to describe: At each node with white to play, the value is that of the successor node with maximum evaluation. At each node with black to play, the value is that of the successor node with minimum evaluation. Finally, the move to be played is one that achieves the root position value.
 
Shannon proposed two strategies: Type A, which envisions searching to a fixed depth, is now commonly known as brute-force. Type B, which proposes searching variations to variable depths dependent on their supposed game-specific relevance, is now commonly known as selective search. Neither approach seemed promising for a long time: The exponential explosion (and the weakness of computing machinery of the 50’s & 60’s) made attainable depths very modest, with or without selectivity. The selectivity idea, lucrative on paper, turned out to be difficult to implement without harming playing strength. Besides, even in brute-force mode programs suffered from a hilarious effect that became known as the horizon effect, which sometimes made them commit suicide with no objective provocation.

 
Well into the 70’s, the common reaction of chess experts to the efforts of chess programs was a patronizing smile. Skepticism of Shannon’s framework was rampant, and many abandoned it for dead. Douglas Hofstaedter, of Godel, Escher, Bach fame, speculated that only a program equipped with human-like faculties of reasoning, perceptions and feelings would suffice to play chess at the expert level (and so, according to Hofstaedter, if invited to play a game, may refuse and say that it would rather read poetry …). Hans Berliner, an international chess master and professor of Computer Science at Carnegie-Mellon, opined that the horizon effect places a limit on the strength of Shannon-type programs.
 
Yet, progress was being made, slowly. In the 60’s & 70’s McCarthy, Newell, Simon & Knuth discovered and refined the alpha-beta algorithm, a risk-free improvement to the minimax algorithm, which, under best conditions could nearly double search depths. In 1967 Greenblatt entered his program MacHackSix into a tournament where it managed to earn a chess rank. Other advances at the time produced some irrational optimism: In 1968, John McCarthy of Stanford University and Donald Michie of Edinburgh University told British International Master David Levy that in ten years there would be a program strong enough to beat him. Levy was so scandalized by this statement that he placed a 1000 pounds bet to prove them wrong. Ten years later, in 1978, Levy played against Slate & Atkin’s Chess 4.7, the strongest program of the day, in a 6-game match to settle the bet.
 
Chess 4.7 was not a pushover. Slate and Atkin revisited the brute-force approach, in a sophisticated manner, and were able to make progress. Their program’s rating was around 2000 (candidate master level). It was not good enough against David Levy, however. At Toronto, he beat it 3.5-1.5 to win the “Levy bet”.
 

International master, and president of the International Computer Games Association, David Levy.
 

Optimism and progress returned in the 80’s. Following Slate and Atkin’s example, it was the age of brute-force, and dedicated chess hardware. Hans Berliner, whom we met above as a skeptic, admitted his error and jumped on the bandwagon. He built the dedicated chess machine Hitech, which became world champion in the mid 80’s. Unrelated to him, two undergraduates at his university, Carnegie-Mellon, Feng-Hsiung Hsu and Murray Campbell, were doing even better: They designed and built very ambitious dedicated chess integrated circuits, which they named Deep Thought. Deep Thought’s calculating ability surpassed by a mile anything seen until then, and scored sensational achievements against strong players (and in a nostalgic meeting, gave David Levy a thrashing). For a few years in the late 80’s and early 90’s, they thoroughly dominated the competition. IBM Corporation took interest, offering Hsu and Campbell contracts. Soon Deep Thought was renamed to Deep Blue, which became IBM’s flagship project. They set their aim at defeating world champion Garry Kasparov.
 
But meanwhile, things were changing again. In the 90’s PC’s were coming of age, and good programming tools became available. Computer chess was no longer the exclusive domain of academia. Many independent developers appeared, with their innovation. The null-move pruning heuristic, popularized by Donninger, and my own half-ply heuristic were very effective depth enhancers. Soon, chess programs gained reputations of tactical monsters, being able to out-calculate humans in forced variations. In long-term strategy and positional understanding, however, they were not as proficient, and sometimes fared very poorly. So in man vs. machine games a pattern emerged where the programs fared well in “blitz”, i.e. a fast game where all moves must be completed in several minutes, because the human opponent could not keep up with the tactical complications, while in slow games (of 2-3 hours per game) the human master, if he could control the game and exploit the computer’s strategic errors, had an edge.
 
At the 1995 world championship at Hong-Kong, the first in which I participated, it was becoming apparent that the age of big hardware was drawing to a close: IBM’s Deep Blue came to play, expecting to take a stroll. However, they were surprisingly defeated by the Dutch program Fritz, who went on to win the world title, while running on a measly Pentium III 90 MHz! Unfazed, IBM announced at the closing ceremony that their 1996 Philadelphia match against world champion Garry Kasparov.
 
Kasparov won 4-2 in Philadelphia, rather easily, so IBM asked and got a rematch the following year, 1997, in New York City. The Deep Blue team worked feverishly throughout the year to improve, and introduced a new machine design, dubbed Deeper Blue. It was reputed to be able to calculate an incredible 200 million positions per second. Kasparov won the first game, but lost the second game in a way that left him psychologically crushed for the rest of the match: Deep Blue played a fine game, in which at some point, rather than seizing on an easy advantage, it played a solid move that left Kasparov with little counter-play. This was so uncharacteristic of the way computers play chess (or so Kasparov thought) that Kasparov became suspicious that IBM had cheated with illegal human help during the game. Hinting at such a possibility soured the atmosphere of the match, and ultimately boomeranged against Kasparov himself. Even more amazingly, hours after Kasparov resigned the 2nd game as lost, chess fans discovered that he could force a draw in that position, and that therefore he resigned a game that was not lost, an unprecedented occurrence (I believe I was a party to the original discovery, where chess programmer Bruce Moreland, myself, and others on the Internet Chess Club discussed and analyzed the game minutes after Kasparov’s resignation. Goaded on by Moreland’s repeated observation “I don’t see why this is lost”, we found a save).
 
The 3rd, 4th and 5th games were drawn, and on the 6th game Kasparov played weakly and lost, to lose the match 2.5-3.5. This was a historic result, which however was marred by the controversy created by his accusations (almost certainly unfounded), and by the fact that Kasparov was clearly out-psyched. It did not help, too, that IBM immediately disbanded the Deep Blue project and never played again. I and others in the computer chess community felt particularly betrayed by that disappearance act. We noted that the greatest achievement in computer chess history was received by a machine with a mere 12-game public career which was on top of that hardly convincing.
 
At this time Junior’s own career was taking off. Later that year (1997) Junior won the computer chess world championship in Paris, and followed through with four more world titles (Maastricht 2001, Maastricht 2002, Ramat-Gan 2004 and Turin 2006). The winner of the world champion gets to keep the Shannon trophy, a beautifully carved horse’s head named in honor of Claude Shannon, and nicknamed Shanny. Shanny graced my living room for many years. Junior is my joint project with Shay Bushinsky, and Deep Junior is how we call the version of the program that runs on several processors.
 
Kasparov, bruised by the defeat to Deep Blue, stayed away from man vs. machine competitions for several years, but came back in answer to a $1 million challenge by FIDE President Kirsan Ilyumzhinov to play against … Deep Junior. This 6 game match took place in the winter of 2003 in New York City, and was drawn 3-3. It was a high-class match and a wonderful achievement for Deep Junior, but it became immortal due to Deep Junior’s 11th move in the 5th game. The out-of-the-blue bishop sacrifice created a sensation when it was played, justified itself over the board by holding Kasparov to a draw (with black), and its correctness or incorrectness will perhaps be forever debated. This was no mere technical feat of calculation, one whose result is worked out in advance, but a plunge into darkness played against conventional wisdom and on general merits: True creativity. You can read all about it here http://www.chessbase.com/newsdetail.asp?newsid=777.
 
At present the point of parity between the strongest computers and the strongest masters may have already been passed. In 2004 and again in 2005, an event in Bilbao, Spain pitted three former world champions against three programs (Junior, Fritz and Hydra). In both cases the programs won by a wide margin. In 2007, Fritz beat world champion Vladimir Kramnik 4-2 in a match at Bonn.
 
What does all this mean? I don’t know, but obviously appearances may mislead: No, computers do not have to have higher intelligence to play at the highest level, and yes, they can be highly creative. An approach to a problem that at a time seems hopeless can turn out to be so successful that no one bothers to consider others. And, no, the computers’ ascendancy did not kill the interest in the game.
 
Returning to the Barcelona event, the result was 1-1, two draws in two games… The games were unspectacular, but high-level, and according to GM Illescas, free of errors. An excellent crowd of around 100 people at the auditorium of the CosmoCaixa science museum was pleased and stayed on for a discussion panel lasting late into evening, in which, predictably, brute-force, creativity, the human spirit and Deep Junior’s sacrifice were mentioned. For the interested, the game scores are below:
 
 [Date "2008.06.05"]
[White "Illescas, Miquel"]
[Black "Deep Junior 11"]

1. Nf3 d5 2. d4 Nc6 3. c4 Bg4 4. cxd5 Bxf3 5. gxf3 Qxd5 6. e3 e6 7. Nc3 Qh5 8.
f4 Qxd1+ 9. Kxd1 O-O-O 10. Bg2 f5 11. Ke2 Nf6 12. Bd2 Ne7 13. Rac1 Ned5 14.
Nxd5 Nxd5 15. h4 c6 16. Rhg1 Rg8 17. h5 Be7 18. Bf3 Kd7 19. a3 a6 20. Kd3 Ra8
21. Bd1 a5 22. f3 Bd6 23. Bb3 a4 24. Ba2 Raf8 25. Rg2 Rf7 26. Rcg1 Be7 27. Ba5
Ra8 28. Be1 Rg8 29. Bd2 Bd6 30. Rg3 Be7 31. R3g2 Bd6 32. Rg3 Be7 33. R3g2
1/2-1/2
 
 [Date "2008.06.05"]
[White "Deep Junior 11"]
[Black "Illescas , Miquel"]

1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. d3 Bc5 5. c3 O-O 6. O-O
d5 7. exd5 Qxd5 8. Bc4 Qd6 9. b4 Bb6 10. Qe2 Bg4 11. Nbd2 Rfe8 12. Ne4 Nxe4 13.
dxe4 Be6 14. a4 a6 15. Ra2 Bxc4 16. Qxc4 Qe6 17. Qxe6 Rxe6 18. Rd1 Ree8 19. a5
Ba7 20. Rd7 Rad8 21. Rad2 Rxd7 22. Rxd7 Rc8 23. Kf1 f6 24. Ke2 Nb8 25. Rd3 Kf7
26. Nd2 Ke6 27. Nc4 Nc6 28. Rh3 Rh8 29. g4 Ne7 30. f4 exf4 31. Bxf4 Ng6 32. Bg3
h6 33. Rh5 Rd8 34. e5 fxe5 35. Nxe5 Nxe5 36. Rxe5+ Kf7 37. h4 c6 38. h5 Re8 39.
Rxe8 Kxe8 40. Be5 Kf7 41. c4 Bg1 42. Bc3 Bh2 43. Ke3 Bd6 44. Ke4 g6 45. Ke3
1/2-1/2

Banner of Barcelona event