- Next Week in Jerusalem: Special Day on Quantum PCP, Quantum Codes, Simplicial Complexes and Locally Testable Codes
- Happy Birthday Ervin, János, Péter, and Zoli!
- My Mathematical Dialogue with Jürgen Eckhoff
- Test Your Intuition (23): How Many Women?
- Happy Birthday Richard Stanley!
- Influence, Threshold, and Noise
- Erdős Lectures 2014 – Dan Spielman
- Answer to Test Your Intuition (22)
- Test your intuition (22): Selling Two Items in a Bundle.
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- Next Week in Jerusalem: Special Day on Quantum PCP, Quantum Codes, Simplicial Complexes and Locally Testable Codes
- Another Forgotten Bet: Is Don Zagier About to Owe Me 1000 Shekels For The Proof of the ABC Conjecture?
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The four princes in summit 200, ten years ago.
(Left to right) Ervin Győri, Zoltán Füredi, Péter Frankl and János Pach
In 2014, Péter Frankl, Zoltán Füredi, Ervin Győri and János Pach are turning 60 and summit 240 is a conference this week in Budapest to celebrate the birthday of those ever-young combinatorics princes. I know the four guys for about 120 years. I first met Peter and Janos (together I think) in Paris in 1979, then Zoli at MIT in 1985 and I met Ervin in the mid late 90s in Budapest. Noga Alon have recently made the observation that younger and younger guys are turning 60 these days and there could not be a more appropriate time to realize the deep truth of this observation than this week.
The mathematical work of our eminent birthday boys was and will be amply represented on this blog. So let me give just one mathematical link to János’s videotaped plenary lecture at Erdős Centennial conference. (Click on the picture to view it. Here is again the link just in case.)
And now, here are a few pictures of our birthday boys.
Jürgen Eckhoff, Ascona 1999
Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first meeting in Oberwolfach in 1982. Here is a description of my mathematical dialogue with Jürgen Eckhoff:
Summary 1) (1980) we found independently two proofs for the same conjecture; 2) (1982) I solved Eckhoff’s Conjecture; 3) Jurgen (1988) solved my conjecture; 4) We made the same conjecture (around 1990) that Andy Frohmader solved in 2007, and finally 5) (Around 2007) We both found (I with Roy Meshulam, and Jürgen with Klaus Peter Nischke) extensions to Amenta’s Helly type theorems that both imply a topological version.
(A 2009 KTH lecture based on this post or vice versa is announced here.)
Let me start from the end:
5. 2007 – Eckhoff and I both find related extensions to Amenta’s theorem.
Nina Amenta proved a remarkable extension of Helly’s theorem. Let be a finite family with the following property:
(a) Every member of is the union of at most r pairwise disjoint compact convex sets.
(b) So is every intersection of members of .
If every r(d+1) members of has a point in common, then all members of have a point in common!
The case r=1 is Helly’s theorem, Grünbaum and Motzkin proposed this theorem as a conjecture and proved the case r=2. David Larman proved the case r=3.
Roy Meshulam and I studied a topological version of the theorem, namely you assume that every member of F is the union of at most r pairwise disjoint contractible compact sets in $R^d$ and that all these sets together form a good cover – every nonempty intersection is either empty or contractible. And we were able to prove it!
Eckhoff and Klaus Peter Nischke looked for a purely combinatorial version of Amenta’s theorem which is given by the old proofs (for r=2,3) but not by Amenta’s proof. An approach towards such a proof was already proposed by Morris in 1968, but it was not clear how to complete Morris’s work. Eckhoff and Nischke were able to do it! And this also implied the topological version for good covers.
The full results of Eckhoff and Nischke and of Roy and me are independent. Roy and I showed that if the nerve of is d-Leray then the nerve of is ((d+1)r-1)-Leray. Eckhoff and showed that if the nerve of has Helly number d, then the nerve of has Helly number (d+1)r-1. Amenta’s argument can be used to show that if the nerve of is d-collapsible then the nerve of F is ((d+1)r-1)-collapsible.
Here, a simplicial comples K is d-Leray if all homology groups vanishes for every and every induced subcomplex L of K.
Roy and I were thinking about a common homological generalization which will include both results but so far could not prove it.
And now let me move to the beginning:
1. 1981 – we give different proofs for the Perles-Katchalski Conjecture
How many women can you find on this poster announcing the 25th Jerusalem School in Economics Theory devoted to Matching and Market Design?
Please respond to the poll:
This week we are celebrating in Cambridge MA , and elsewhere in the world, Richard Stanley’s birthday. For the last forty years, Richard has been one of the very few leading mathematicians in the area of combinatorics, and he found deep, profound, and fruitful links between combinatorics and other areas of mathematics. His works enriched and
influenced combinatorics as well as other areas of mathematics, and, in my opinion,
combinatorics matured greatly as a mathematical discipline thanks to his work.
Correct or incorrect?
(1) Richard drove cross-country at least 8 times
(2) In his youth, at a wild party, Richard Stanley found a proof of FLT consisting of a few mathematical symbols.
(3) Richard jumped at least once from an airplane
(4) Richard is actively interested in the study of consciousness
(5) Richard found a mathematical way to divide by zero
Seven Early Papers by Richard Stanley That You Must Read.
Richard’s Green book: Combinatorics and Commutative Algebra
Combinatorics and Commutative Algebra
The two seminal papers (1) and (3) (below) showed remarkable and unexpected applications of commutative algebra to combinatorics. In each of these papers a central
conjecture in combinatorics was solved in a completely unexpected way which was the basis for a later remarkable theory. Paper (1) is the starting point for the interrelation between commutative algebra and combinatorics of simplicial complexes and their
topology. In this work Richard Stanley proved the Motzkin-Klee upper bound conjecture for triangulations of spheres. This conjecture asserts that the maximum number
of k-faces for a triangulation of a (d-1)-dimensional sphere with n vertices is attained by the boundary complex of the cyclic d-dimensional polytope with
n vertices. Peter McMullen proved this conjecture for simplicial polytopes and Richard Stanley proved it for arbitrary triangulations of spheres. The key point was that a certain ring (the Stanley-Reisner ring) associated with a simplicial polytope has the Cohen-Macaulay property.
The connection between combinatorics and commutative algebra is
far reaching, and in subsequent works combinatorial problems led to
developments in commutative algebra and techniques from the two areas were
combined. A more recent important paper by Richard on applications of commutative algebra for the study of face numbers is: R. P. Stanley, Subdivisions and local h-vectors. J. Amer. Math. Soc. 5 (1992), no. 4, 805–851.
And here is, a few weeks old important development in this theory: Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums, by Karim A. Adiprasito and Raman Sanyal.
The Cohen-Macaulay property, magic squares and lattice points in polytopes
This paper starts with a theorem about enumeration of certain magic squares. Solving a long-standing open problem, Stanley proved that the generating function for the
number of k by k integer matrices (k- fixed) with nonnegative entries and row sums and column sums equal to nis rational. This is the starting
point of a deep algebraic theory of integral points in polyhedra.
Enters the Hard-Lefshetz Theorem: McMullen’s g-conjecture
The g-conjecture proposes a complete characterization of face numbers of d-dimensional polytopes. One linear equality that holds among face numbers is, of course, the Euler-Poincaré relation. This relation implies additional [d/2] equalities called the Dehn-Sommerville relations. Peter McMullen proposed an additional system of linear and nonlinear inequalities as a complete characterization of face numbers of polytopes. The sufficiency part of this conjecture was proved by Billera and Lee. Richard Stanley’s brilliant proof for McMullen’s inequalities that established the g-conjecture was based on the Hard Lefschetz Theorem from algebraic topology. Starting from a simplicial polytope P (with rational vertices) we associate to it a toric variety T(P). It turned out that the cohomology ring of this variety is closely related to the Stanley-Reisner ring mentioned above. The Hard Lefschetz Theorem implies an algebraic property of the Stanley-Reisner ring from which McMullen inequalities can be deduced by direct combinatorial reasoning. Richard found a number of other combinatorial applications of the Hard Lefschetz theorem (including the solution of the Erdos-Moser conjecture).
Here is the abstract of Lou Billera’s lecture
LOUIS BILLERA (CORNELL)
Even more intriguing, if rather less plausible…
The title is how Peter McMullen described his own conjectured characterization of the f-vectors of simplicial polytopes in his 1971 lecture notes on the upper bound conjecture written with Geoffrey Shephard. Yet by the end of that decade, the so-called g-conjecture would become the g-theorem, and algebraic combinatorics (as practiced at MIT) would have attracted the attention of mainstream mathematics, almost entirely due to the startling proof given by Richard Stanley.
I will briefly describe some of the events leading to this proof and some of its still developing consequences.
Enumeration is Richard’s true mathematical love.
Richard’s monumental books EC1 and EC2 (The picture is of EC1 and a young fan)
(4) A baker’s dozen of conjectures concerning plane partitions, in Combinatoire Énumérative (G. Labelle and P. Leroux, eds.), Lecture Notes in Math., no. 1234, Springer-Verlag, Berlin/Heidelberg/New York, 1986, pp. 285-293.
13 beautiful conjectures on counting plane partitions with various forms of symmetry.
(5) Generating functions, in Studies in Combinatorics (G.-C. Rota, ed.), Mathematical Association of America, 1978, pp 100-141.
For me this was the best introduction to generating functions, clear and inspiring. The entire MAA 1978 Rota’s blue little volume on combinatorics is great. Buy it!
(5) Supersolvable lattices, Algebra Universalis 2 (1972), 197-217.
This paper provides a profound link between group theory and the study of
partially ordered sets. It can be seen as a starting point of Stanley’s own work on the Cohen-Macaulay property and it had much influence on later works on combinatorial properties of lattices of subgroups by Quillen and many others, and also on the study of POSETS (=partially ordered sets) arising from arrangements of hyperplanes. The algebraic notion of supersolvable groups is translated to an important combinatorial notion for partially ordered sets. (There is a more detailed paper which I could not find online: R. P. Stanley, Supersolvable semimodular lattices. Mobius algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 80–142. Univ. Waterloo, Waterloo, Ont., 1971.)
Combinatorics and representation theory
(6) On the number of reduced decompositions of elements of Coxeter groups, European J. Combinatorics 5 (1984), 359-372.
This paper gives an important result proved using representation theory. It is one of
many results by Stanley on connections between enumerative combinatorics,
representation theory, and invariant theory. Again, this paper represents an exciting area of research about the connection of enumerative combinatorics and representation theory that I am less familiar with. A very inspiring survey paper is: Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (new series) 1 (1979), 475-511.
Here are abstracts of two lectures from the meeting on some recent developments in combinatorial representation theory and symmetric functions.
GRETA PANOVA (UCLA)
The Kronecker coefficients: an unexpected journey
Kronecker coefficients live at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their journey started 75 years ago, they still haven’t found their explicit positive combinatorial formula, and present a major open problem in algebraic combinatorics. Recently, they were given a new role in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the “”P vs NP”” problem.
We will take the Kronecker coefficients to asymptotics land and bound them. As an unexpected consequence of this trip, we find bounds for the difference of consecutive coefficients in the q-binomial coefficients (as polynomial in q), generalizing Sylvester’s unimodality theorem and connecting with results of Richard Stanley.
Joint work with Igor Pak.
THOMAS LAM (U MICHIGAN)
Truncations of Stanley symmetric functions and amplituhedron cells
Stanley symmetric functions were invented (by Stanley) with applications to the enumeration of reduced words in the symmetric group in mind. Recently, the “amplituhedron” was introduced in the study of scattering amplitudes in N=4 super Yang Mills. I will talk about a formula for the cohomology class of a (tree) amplituhedron variety as the truncation of an affine Stanley symmetric function.
Two combinatorial applications of the Aleksandrov-Fenchel inequalities;
(7) Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combinatorial Theory (A) 31 (1981), 56-65.
In this amazing paper Stanley used inequalities of classical convexity
to settle an important conjecture on probability of events in partially
ordered sets. A special case of the conjecture was settled earlier by Ron
Graham using the FKG inequality. The profound relation between classical
convexity inequalities, combinatorial structures, polytopes, and
probability theory was further studied by many authors including Stanley
himself and there is much more to be done.
I see that I ran out of my seven designated slots. Certainly you should read Richard’s combinatorial constructions of polytopes, like Two poset polytopes, Discrete Comput. Geom. 1 (1986), 9-23, and his papers on arrangements. Let me mention a more recent paper of Stanley in this general area: A polytope related to empirical distributions, plane trees, parking functions, and the associahedron (with J. Pitman), Discrete Comput. Geom., 27 (2002), 603-634.
(Mostly from RS’s homepage.)
Chess and Mathematics
An unusual method for proving the Riemann hypothesis.
Richard Stanley’s Mathoverflow question on MAGIC
Richard the Catalan
(From RS’s homepage)
Excerpt (27 page PDF file) from EC2 on problems related to Catalan numbers (including 66 combinatorial interpretations of these numbers).
Solutions to Catalan number problems from the previous link (23 page PDF file).
Catalan addendum (Postscript or PDF) (version of 25 May 2013; 96 pages). An addendum of new problems (and solutions) related to Catalan numbers. Current number of combinatorial interpretations of Cn: 207.
The material on Catalan numbers is being collected into a monograph, to be published by Cambridge University Press in late 2014 or early 2015.
My dear friend Itai Benjamini told me that he won’t be able to make it to my Tuesday talk on influence, threshold, and noise, and asked if I already have the slides. So it occurred to me that perhaps I can practice the lecture on you, my readers, not just with the slides (here they are) but also roughly what I plan to say, some additional info, and some pedagogical hesitations. Of course, remarks can be very helpful.
I can also briefly report that there are plenty of exciting things happening around that I would love to report about – hopefully later in my travel-free summer. One more thing: while chatting with Yuval Rabani and Daniel Spielman I realized that there are various exciting things happening in algorithms (and not reported so much in blogs). Much progress has been made on basic questions: TSP, Bin Packing, flows & bipartite matching, market equilibria, and k-servers, to mention a few, and also new directions and methods. I am happy to announce that Yuval kindly agreed to write here an algorithmic column from time to time, and Daniel is considering contributing a guest post as well.
The second AMS-IMU meeting
Since the early 70s, I have been a devoted participants in our annual meetings of the Israeli Mathematical Union (IMU), and this year we will have the second joint meeting with the American Mathematical Society (AMS). Here is the program. There are many exciting lectures. Let me mention that Eran Nevo, this year Erdős’ prize winner, will give a lecture about the g-conjecture. Congratulations, Eran! Among the 22 exciting special sessions there are a few related to combinatorics, and even one organized by me on Wednsday and Thursday.
Contact person: Gil Kalai, firstname.lastname@example.org
TAU, Dan David building, Room 103
|Wed, 10:50-11:30||Van H. Vu (Yale University)||Real roots of random polynomials (abstract)|
|Wed, 11:40-12:20||Oriol Serra (Universitat Politecnica de Catalunya, Barcelona)||Arithmetic Removal Lemmas (abstract)|
|Wed, 12:30-13:10||Tali Kaufman (Bar-Ilan University)||Bounded degree high dimensional expanders (abstract)|
|Wed, 16:00-16:40||Rom Pinchasi (Technion)||On the union of arithmetic progressions (abstract)|
|Wed, 16:50-17:30||Isabella Novik (University of Washington, Seattle)||Face numbers of balanced spheres, manifolds, and pseudomanifolds (abstract)|
|Wed, 17:40-18:20||Edward Scheinerman (Johns Hopkins University, Baltimore)||On Vertex, Edge, and Vertex-Edge Random Graphs (abstract)|
|Thu, 9:20-10:00||Yael Tauman Kalai (MSR, New England)||The Evolution of Proofs in Computer Science (abstract)|
|Thu, 10:10-10:50||Irit Dinur (Weitzman Institute)||Lifting locally consistent solutions to global solutions (abstract)|
|Thu, 11:00-11:40||Benny Sudakov (ETH, Zurich)||The minimum number of nonnegative edges in hypergraphs (abstract)|
And now for my own lecture.
Influence, Threshold, and Noise:
Indeed, most people got it right! Bundling sometimes increases revenues, sometimes keeps revenues the same, and sometimes decreases revenues. In fact, this is an interesting issue which was the subject of recent research effort. So here are a few examples as told in the introduction to a recent paper Approximate Revenue Maximization with Multiple Items by Sergiu Hart and Noam Nisan:
Example 1: Consider the distribution taking values 1 and 2, each with probability 1/2.
Let us ﬁrst look at selling a single item optimally: the seller can either choose to price it
at 1, selling always and getting a revenue of 1, or choose to price the item at 2, selling it
with probability 1/2, still obtaining an expected revenue of 1, and so the optimal revenue
for a single item is 1. Now consider the following mechanism for selling both items:
bundle them together, and sell the bundle for price 3. The probability that the sum of
the buyer’s values for the two items is at least 3 is 3/4, and so the revenue is 3·3/4 = 2.25
– larger than 2, which is obtained by selling them separately.
Example 2: For the distribution taking values 0 and 1, each with probability 1/2,
selling the bundle can yield at most a revenue of 3/4, and this is less than twice the
single-item revenue of 1/2.
Example 3 (and 4): In some other cases neither selling separately nor bundling
is optimal. For the distribution that takes values 0, 1 and 2, each with probability 1/3,
the unique optimal auction turns out to oﬀer to the buyer the choice between any single
item at price 2, and the bundle of both items at a “discount” price of 3. This auction
gets revenue of 13/9 revenue, which is larger than the revenue of 4/3 obtained from
either selling the two items separately, or from selling them as a single bundle. (A similar situation happens for the uniform distribution on [0, 1], for which neither bundling nor selling separately is optimal (Alejandro M. Manelli and Daniel R. Vincent ).
Example 5: In yet other cases the optimal mechanism is not even deterministic and must oﬀer lotteries for the items. This happens in the following example from a 2011 paper “Revenue Maximization in Two Dimensions” by Sergiu Hart and Phil Reny: Let F be the distribution which takes values 1, 2 and 4, with probabilities 1/6, 1/2, 1/3, respectively. It turns out that the unique optimal mechanism oﬀers the buyer the choice between buying any one good with probability 1/2 for a price of 1, and buying the bundle of both goods (surely) for a price of 4; any deterministic mechanism has a strictly lower revenue. See also Hart’s presentation “Two (!) good to be true” Update: See also this paper by Hart and Nisan: How Good Are Simple Mechanisms for Selling Multiple Goods?
Update: See also Andy Yao’s recent paper An n-to-1 Bidder Reduction for Multi-item Auctions and its Applications. The paper is relevant both to the issue of bundling and to the issue of using randomized mechanisms for auctions. (Test your intuition (21).)
You have one item to sell and you need to post a price for it. There is a single potential buyer and the value of the item for the buyer is distributed according to a known probability distribution.
It is quite easy to compute which posted price will maximize your revenues. You need to maximize the price multiplied by the probability that the value of the item is greater or equal to that price.
1) When the value of the item for the buyer is 10 with probability 1/2 and 15 with probability 1/2. The optimal price is 10 and the expected revenue is 10.
2) When the value of the item for the buyer is 10 with probability 1/2 and 40 with probability 1/2. The optimal price is 40 and the expected revenue is 20.
Now you have two items to sell and as before a single potential buyer. For each of the items, the buyer’s value behaves according to a known probability distribution. And these distributions are statistically independent. The value for the buyer of having the two items is simply the sum of the individual values.
Now we allow the seller to post a price for the bundle of two items and he posts the price that maximizes his revenues.
In summary: The values are additive, the distributions are independent.
Test your intuition:
1) Can the revenues of a seller for selling the two items be larger than the sum of the revenues when they are sold separately?
2) Can the revenues of a seller for selling the two items be smaller than the sum of the revenues when they are sold separately?
PS: there is a new post by Tim Gowers on the cost of Elseviers journals in England. Elsevier (and other publishers) are famous (or infamous) for their bundling policy. The movement towards cheaper journal prices, and open access to scientific papers that Gowers largly initialted two years ago is now referred to as the “academic spring.”
Here are links to a videotaped lecture in two parts entitled “why quantum computers cannot work” recorded at the Simons Institute for the Theory of Computing on December 2013 and two additional videos: a short talk on topological quantum computers and a twelve minute overview. Here are the links: Overview, Part I, Part II, Topological QC.
Why Quantum Computers Cannot Work:
Overview and Vision.
Why Quantum Computers Cannot Work I:
From the “Trivial Flow” to Smoothed Lindblad Evolutions
Why Quantum Computers Cannot Work II:
Debate, Reasons to Disbelieve, and Experimentation
Why Topological Quantum Computers Cannot Work
The Geometry of Spacetime is Enabled by the Failure of Quantum Fault-Tolerance
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).
Added later (April 18): Let me quote from what Steve wrote about the videos: The surprising part is the superior production value relative to your typical videotaped lecture (at least for the first overview video). Producing the videos was an interesting and demanding experience and I was certainly happy to read Steve’s description of the production value. (Of course, the main purpose of Steve’s post was to express his disagreement with the content of the videos. See either the post or Co-’s comment below.)
Also there are two earlier versions of my lecture (in 1-hour format) that were videotaped. The first was taken by Jesus De Loera in Davis. Very interesting shooting-angle and interesting comments by Greg Kuperberg, Bruno Nachtergaele and other participants. The second was taken in Seattle in a UW-PIMS colloquium lecture. Again interesting questions by several participants including James Lee and John Sidles.
(July:) The Simons Institite (almost identical) versions of the movies are now linked from the web-page of my November 15 lecture at SI.