From Oberwolfach: The Topological Tverberg Conjecture is False

The topological Tverberg conjecture (discussed in this post), a holy grail of topological combinatorics, was refuted! The three-page paper “Counterexamples to the topological Tverberg conjecture” by Florian Frick gives a brilliant proof that the conjecture is false.

The proof is based on two major ingredients. The first is a recent major theory by Issak Mabillard and Uli Wagner which extends fundamental theorems from classical obstruction theory for embeddability to an obstruction theory for r-fold intersection of disjoint faces in maps from simplicial complexes to Euclidean spaces. An extended abstract of this work is Eliminating Tverberg points, I. An analogue of the Whitney trick. The second is a result  by Murad Özaydin’s from his 1987 paper Equivariant maps for the symmetric group, which showed that for the non prime-power case the topological obstruction vanishes.

It was commonly believed that the topological Tverberg conjecture is correct. However, one of the motivations of Mabillard and Wagner for studying elimination of higher order intersection was that this may lead to counterexamples via Özaydin result. Isaak and Uli came close but there was a crucial assumption of large codimension in their theory, which seemed to avoid applying the new theory to this case.  It turned out that a simple combinatorial argument allows to overcome the codimension problem!

Florian’s  combinatorial argument which allows to use Özaydin’s result in Mabillard-Wagner’s theory  is a beautiful example of a powerful combinatorial method with other applications by Pavle Blagojević, Florian Frick and Günter Ziegler.


Both Uli and Florian talked about it here at Oberwolfach on Tuesday. I hope to share some more news items from Oberwolfach and from last week’s Midrasha in future posts.

Midrasha Mathematicae #18: In And Around Combinatorics



Tahl Nowik

photo (4) 17.8.14 midrasha poster 2015 poster

michal-mid mid-irit mid-david nati-mid mid-peter nica-mid   alex-mid2 midjoel mid-sam tami-mid zohar-mid tahl-mid

Update 3 (January 30): The midrasha ended today. Update 2 (January 28): additional videos are linked; Update 1 (January 23): Today we end the first week of the school. David Streurer and Peter Keevash completed their series of lectures and Alex Postnikov started his series.


Today is the third day of our winter school. In this page I will gradually give links to to various lectures and background materials. I am going to update the page through the two weeks of the Midrasha. Here is the web page of the midrasha, and here is the program. I will also present the posters for those who want me to: simply take a picture (or more than one) of the poster and send me. And also – links to additional materials, pictures, or anything else: just email me, or add a comment to this post.

Lecture series and lectures

Irit Dinur: Direct products of games and graphs

mid-irit Continue reading

A Historical Picture Taken by Nimrod Megiddo

Last week I took a bus from Tel Aviv to Jerusalem and I saw (from behind) a person that I immediately recognized. It was Nimrod Megiddo, from IBM Almaden, one of the very first  to relate game theory with complexity theory, one of the pioneers of computational geometry, and one of the leaders in optimization and linear programming, the guy who (with Ehud Kalai) was the first to  invited me to an international conference, and a fresh Laureate of the John von Neumann theory Prize.  I did not see Nimrod more than a year after our last coincidental  meeting at the Berkeley Simons Institute, I called over to him and he was happy to see me as I was happy to see him, and we found a place together at the back of the bus and caught up on things.

Nimrod showed me on the bus a picture he found in his house, taken by him at the  1974  game theory workshop in Bad Salzuflen, Germany. With Nimrod’s kind permission I present the picture here: (The copyright © belongs to Nimrod Megiddo who took the picture).


Here are some of the people left to right, (on some I already told you in other posts):

David Schmeidler (only beard visible)
Guillermo Owen
Lloyd Shapley
Sergiu Hart
Yair Tauman (only back shown)
Robert Aumann
Werner Güth (behind Aumann)
Reinhard Selten
Ehud Kalai (just to the right of Selten looking at the camera)
Gerhard Schwedihauer
Elon Kohlberg (only back shown, looking to the left)
William Lucas (in the back, looking at the wall)
Robert Weber (only hair and jacket visible)
Bezalel Peleg (looking to the right)
Joel Moulen (looking to the left)
Thomas Marschak (sunglasses and beard)
Michael Maschler
Joachim Rosenmüller (with glasses behind Maschler)
Kenjiro Nakamura

Happy new 2015!

A lecture by Noga


Noga with Uri Feige among various other heroes

A few weeks ago I devoted a post to the 240-summit conference for Péter Frankl, Zoltán Füredi, Ervin Győri and János Pach, and today I will bring you the slides of Noga Alon’s lecture in the meeting. Noga is my genious twin academic brother – we both were graduate students under the supervision of Micha A. Perles in the same years and we both went to MIT as postocs in fall 1983.  The lecture starts with briefly mentioning four results by the birthday boys related to combinatorics and geometry and continues with recent startling results by Alon, Ankur Moitra, and Benny Sudakov. One out of many contributions of Noga over the years is building a large infrastructure of constructions and examples, often very surprising,  in combinatorics, graph theory, information theory,TOC, and related areas. And the new results add to this infrastructure. The slides are very clear. Enjoy!






In And Around Combinatorics: The 18th Midrasha Mathematicae. Jerusalem, JANUARY 18-31

17.8.14 midrasha poster 2015 (1)


The 18th yearly school in mathematics is devoted this year to combinatorics. It will feature lecture series by Irit Dinur, Joel Hass, Peter Keevash, Alexandru Nica, Alexander Postnikov, Wojciech Samotij, and David Streurer and additional activities. As usual grants for local and travel expences are possible.


Media Item from “Haaretz” Today: “For the first time ever…”


Maryam Mirzakhani received the medal from South Korea’s president Park Geun-hye. 

Here is an article from today’s Israeli newspaper Haaretz.  It is based on this article by the Guardian (Thanks, Manor!).  See also this post on Laba’s accidental mathematician and John Baez’ Google+ post.

The ICM 2014 started today in Seoul. The International congress taking place once every four years is an exciting event, celebrated by thousands of mathematicians in Seoul and many others all over the world. The opening ceremonies came with the announcement of  Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani as 2014 Fields medalist, Subhash Khot won the Nevanlinna prize, Stanley Osher won the Gauss prize and Phillip Griffiths is the Chern medalist, and  Adrián Paenza won the Leelavati Prize. Heartly congratulations to Artur, Manjul, Martin, Maryam, Subhash, Stanley, Phillip, and Adrián! This is also the first for Brazil, Austria, Canada, and Iran! More on the Fields medalists works can be found on Terry Tao’s blog. (New) And here is a live bloging (with pictures) on ICM2014 day 1 from Gowers’s blog.   And also here from the ICM site on the work of all prize recipients. And from Quanta Magazine: (More) More on Osher, Griffiths, and Khot on terry Tao’s blog, on Khot on Scott Aaronson’s blog and on a description of the laudations on Gowers blog. 



Next Week in Jerusalem: Special Day on Quantum PCP, Quantum Codes, Simplicial Complexes and Locally Testable Codes

Special Quantum PCP and/or Quantum Codes: Simplicial Complexes and Locally Testable CodesDay

24 Jul 2014 – 09:30 to 17:00

room B-220, 2nd floor, Rothberg B Building

On Thursday, the 24th of July we will host a SC-LTC (simplicial complexes and classical and quantum locally testable codes) at the Hebrew university, Rothberg building B room 202 (second floor) in the Givat Ram campus. Please join us, we are hoping for a fruitful and enjoyable day, with lots of interactions. Coffee and refreshments will be provided throughout the day, as well as free “tickets” for lunch on campus
There is no registration fee, but please email preferably by next Tuesday if there is a reasonable probability that you attend –  so that we have some estimation regarding the number of people, for food planning

Program:SC-LTC day – simplicial complexes and locally testable classical and quantum codes –Rothberg building B202
9:00 gathering: coffee and refreshments

9:30 Irit Dinur: Locally testable codes, a bird’s eye view

10:15: coffee break

10:45 Tali Kaufman, High dimensional expanders and property testing

11:30 15 minutes break

11:45 Dorit Aharonov, quantum codes and local testability

12:30 lunch break

2:00 Alex Lubotzky: Ramanujan complexes

2:50 coffee break

3:15 Lior Eldar: Open questions about quantum locally testable codes and quantum entanglement

3:45 Guy Kindler: direct sum testing and relations to simplicial complexes ( Based on David, Dinur, Goldenberg, Kindler, and Shinkar, 2014)

4:15-5 free discussion, fruit and coffee


Happy Birthday Ervin, János, Péter, and Zoli!


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.



János 72(?)

Continue reading

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