Monthly Archives: June 2014

Test Your Intuition (23): How Many Women?

2014 Economics School poster

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:

Happy Birthday Richard Stanley!

Richard_Stanley 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.

Trivia Quiz

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.

cobcom Richard’s Green book: Combinatorics and Commutative Algebra

Combinatorics and Commutative Algebra

(1) R. P. Stanley, The upper bound conjecture and Cohen-Macaulay rings. Studies in Appl. Math. 54 (1975), no. 2, 135–142. 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

(2) R. P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings. Duke Math. J. 43 (1976), no. 3, 511–531. 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

(3) R. P. Stanley, The number of faces of a simplicial convex polytope. Adv. in Math. 35 (1980), no. 3, 236–238. 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


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.   ec1 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!

Posets everywhere

(6) Supersolvable latticesAlgebra 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

(7) On the number of reduced decompositions of elements of Coxeter groupsEuropean 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 combinatoricsBull. 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.


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.


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 inequalitiesJ. 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 polytopesDiscrete 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

(12 page  PDF file) An excerpt (version of 1 November 1999) from a book Richard is writing with Noam Elkies.

An unusual method for proving the Riemann hypothesis.

Professor Eubanks in Zetaland

Richard Stanley’s Mathoverflow question on MAGIC

Magic trick based on deep mathematics

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 Cn207. The material on Catalan numbers is being collected into a monograph, to be published by Cambridge University Press in late 2014 or early 2015.   bgs

Influence, Threshold, and Noise



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,
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:

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