# The Simplex, the Cyclic polytope, the Positroidron, the Amplituhedron, and Beyond

A quick schematic road-map to these new geometric objects. The  positroidron can be seen as a cellular structure on the nonnegative Grassmanian – the part of the real Grassmanian G(m,n) which corresponds to m by n matrices with all m by m minors non-negative. The cells in the cellular structure of the positroidron correspond to those matrices with the same (+,0) pattern for m by m minors. When m=1 we get a (spherical) simplex. When we project the positroidron using an n by k totally positive matrix we get for m=1 the cyclic polytope, and for general m the  amplituhedron. When we project using general matrices we obtain general polytopes for m=1, and an interesting extension of polytopes proposed by Thomas Lam for general m.

Alex Postnikov’s recent lectures series in our Midrasha was an opportunity to understand slightly better some remarkable combinatorial objects that drew much attention recently. Continue reading

# My Mathematical Dialogue with Jürgen Eckhoff

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

Nina Amenta proved a remarkable extension of Helly’s theorem. Let $\cal F$ be a finite family with the following property:

(a) Every member of $\cal F$ is the union of at most r pairwise disjoint compact convex sets.

(b) So is every intersection of members of $\cal F$.

(c) $|{\cal F}| > r(d+1)$.

If every r(d+1) members of $\cal F$ has a point in common, then all members of $\cal F$ 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

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 $\cal G$ is d-Leray then the nerve of $\cal F$ is ((d+1)r-1)-Leray. Eckhoff and showed that if the nerve of $\cal G$ has Helly number d, then the nerve of $\cal F$ has Helly number (d+1)r-1. Amenta’s argument can be used to show that if the nerve of $\cal G$ 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 $H_i(L)$ vanishes for every $i \ge d$ 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.

# Many triangulated three-spheres!

### The news

Eran Nevo and Stedman Wilson have constructed $\exp (K n^2)$ triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 -sphere have?

### Quick remarks:

1) Since the number of facets in an n-vertex triangulation of a 3-sphere is at most quadratic in n, an upper bound for the number of triangulations of the 3-sphere with n vertices is $\exp(n^2 \log n)$. For certain classes of triangulations, Dey removed in 1992  the logarithmic factor in the exponent for the upper bound.

2) Goodman and Pollack showed in 1986 that the number of simplicial 4-polytopes with n vertices is much much smaller $\exp (O(n\log n))$. This upper bound applies to simplicial polytopes of every dimension d, and Alon extended it to general polytopes.

3) Before the new paper the world record was the 2004 lower bound by Pfeifle and Ziegler – $\exp (Kn^{5/4}).$

4) In 1988 I constructed $\exp (K n^{[d/2]})$ triangulations of the d-spheres with n vertices.  The new construction gives hope to improve it in any odd dimension by replacing [d/2] by [(d+1)/2] (which match up to logn the exponent in the upper bound). [Update (Dec 19) : this has now been achieved by Paco Santos (based on a different construction) and Nevo and Wilson (based on extensions of their 3-D constructions). More detailed to come.]

# Richard Stanley: How the Proof of the Upper Bound Theorem (for spheres) was Found

The upper bound theorem asserts that among all d-dimensional polytopes with n vertices, the cyclic polytope maximizes the number of facets (and k-faces for every k). It was proved by McMullen for polytopes in 1970, and by Stanley for general triangulations of spheres in 1975. This theorem is related to a lot of mathematics (and also computational geometry) and there are many interesting extensions and related conjectures.

Richard Stanley posted (on his  homepage which is full with interesting things) an article describing How the proof of the upper bound theorem for triangulations of spheres was found.

It is interesting how, for Richard, the work oמ face-numbers of polytopes started with his work on integer points in polytopes and especially the Anand-Dumir-Gupta” conjecture on enumeration of “magic squares.” (See this survey article by Winfried Bruns.)  Integer points in polytopes, and face numbers represent the two initial chapters of Richard’s “green book” on Commutative algebra and combinatorics. Both these topics are related to commutative algebra and to the algebraic geometry of toric varieties.

See also these relevant posts “(Eran Nevo) The g-conjecture II: The commutative algebra connection,), (Eran Nevo) The g-conjecture I, and How the g-conjecture came about. Various results and problems related to the upper bound theorem can be found in Section 2 of my paper Combinatorics with a Geometric Flavor.

# F ≤ 4E

## 1. E ≤ 3V

Let G be a simple planar graph with V vertices and E edges. It follows from Euler’s theorem that

## E ≤ 3V

In fact, we have (when V is at least 3,) that E 3V – 6.

To see this,  denote by F the number of regions or faces determined by G (in other words, the number of connected components in the complement of the embedded graph). Euler’s theorem asserts that

E – V + F = 2

## V – E + F = 2

and now note that every face must have at least three edges and every edge is contained in two faces and therefore $2E \ge 3F$, so 6=3V – 3E + 3F ≤ 3V – 3E +2E.

## 2. F ≤ 4E

Now let K be a two-dimensional simplicial complex and suppose that K can be embedded in $R^4$. Denote by E the number of edges of K and by F the number of 2-faces of K.

Here is a really great conjecture:

Conjecture:

## F ≤ 4E

A weaker version which is also widely open and very interesting is:

For some absolute constant C,

## F ≤ C E

Remarks: The conjecture extends to higher dimensions. If K is an r-dimensional simplicial complex that can be embedded into $R^{2r}$ then the conjecture is that

$f_r(K) \le C_rf_{r-1}(K),$

Where $C_r$ is a constant depending on r.  Here $f_i(K)$ is the number of i-dimensional faces of K. A stronger statement is that $C_r= r+2$. The conjecture also extends to polyhedral complexes and more general form of complexes. In the conjecture ‘embed’ refers to a topological embedding.

# Lionel Pournin found a combinatorial proof for Sleator-Tarjan-Thurston diameter result

I just saw in Claire Mathieu’s blog  “A CS professor blog” that a simple proof of the Sleator-Tarjan-Thurston’s diameter result for the graph of the associahedron was found by Lionel Pournin! Here are slides of his lecture “The diameters of associahedra” and link to the paper with the same title “The diameters of associahedra.” The original proof was based on hyperbolic volume computations and was quite difficult. (Here is an earlier post on the associahedron and an earlier mention of a connection found by Dehornoy with the Thompson group.)

# Karim Adiprasito: Flag simplicial complexes and the non-revisiting path conjecture

This post is authored by Karim Adiprasito

The past months have seen some exciting progress on diameter bounds for polytopes and polytopal complexes, both in the negative and in the positive direction.  Jesus de Loera and Steve Klee described simplicial polytopes which are not  weakly vertex decomposable and the existence of non weakly k-vertex decomposable polytopes for k up to about $\sqrt{d}$ was proved  by Hähnle, Klee, and Pilaud in the paper  Obstructions to weak decomposability for simplicial polytopes. In this post I want to outline a generalization of a beautiful result of Billera and Provan in support of the Hirsch conjecture.

I will consider the simplicial version of the Hirsch conjecture, dual to the classic formulation of Hirsch conjecture. Furthermore, I will consider the Hirsch conjecture, and the non-revisiting path conjecture, for general simplicial complexes, as opposed to the classical formulation for polytopes.

Theorem [Billera & Provan 79] The barycentric subdivision of a shellable simplicial complex satisfies the Hirsch Conjecture.

The barycentric subdivision of a shellable complex is vertex decomposable. The Hirsch diameter bound for vertex decomposable complexes, in turn, can be proven easily by induction.

This is particularly interesting since polytopes, the objects for which the Hirsch conjecture was originally formulated, are shellable. So while in general polytopes do not satisfy the Hirsch conjecture, their barycentric subdivisions always do! That was a great news!

Shellability is a strong combinatorial property that enables us to decompose a complex nicely, so it does not come as a surprise that it can be used to give some diameter bounds on complexes. Suprisingly, however, shellability is not needed at all!  And neither is the barycentric subdivision!

A simplicial complex Σ is called flag if it is the clique complex of its 1-skeleton. It is called normal if it is pure and for every face F of Σ of codimension two or more, Lk(F) is connected.

Theorem (Adiprasito and Benedetti):  Any flag and normal simplicial complex Σ satisfies the non-revisiting path conjecture and, in particular, it satisfies the Hirsch conjecture.

This generalizes the Billera–Provan result in three ways:

— The barycentric subdivision of a simplicial complex is flag, but not all flag complexes are obtained by barycentric subdivisions.

— Shellability imposes strong topological and combinatorial restrictions on a complex; A shellable complex is always homotopy equivalent to a wedge of spheres of the same dimension, and even if a pure complex is topologically nice (if, for example, it is a PL ball) it may not be shellable, as classic examples of Goodrick, Lickorish and Rudin show. Being normal still poses a restriction, but include a far wider class of complexes. For example, every triangulation of a (connected) manifold is normal, and so are all homology manifolds.

— Instead of proving the Hirsch conjecture, we can actually obtain the stronger conclusion that the complex satisfies the non-revisiting path conjecture, which for a given complex is stronger than the Hirsch conjecture.

A geometric proof of our theorem appeared in a recent paper “Metric geometry and collapsibility”  with Bruno Benedetti. . I will give here a short combinatorial proof.

# Tokyo, Kyoto, and Nagoya

Near Nagoya: Firework festival; Kyoto: with Gunter Ziegler; with Takayuki Hibi, Hibi, Marge Bayer, Curtis Green and Richard Stanly; Tokyo: Peter Frankl; crowded crossing. Added later: Mazi and I at the same restaurant taken by Stanley.

I just returned from a trip to Japan to the FPSAC 2012 at Nagoya and a workshop on convex polytopes in Kyoto. As in my first visit to Osaka in 1999 I found Japan stunning, and this time I was able to share the experience with my wife.

Kyoto: The week before FPSAC there was a workshop at RIMS devoted to convex polytopes. Some highlights: A classical result by Venkov and McMullen characterizes polytopes whose translates tile the Euclidean d-space.  Sinai Robins talked about his work with Nick Gravin and Dima Shiryaev about k-tilings (every point is covered k times).  Not a full characterization yet but already some impressive results. Gunter Ziegler talked about his work with Karim Adiprasito disproving an old conjecture by Shephard which asserts that there are only finitely many projectively unique d-polytopes for every dimension d. This is false above dimension 81.  Eran Nevo talked about his solution with Satoshi Murai of the generalized lower bound conjecture and some subsequent works on triangulated manifolds. There were several talks relating Ehrhard polynomials of polytopes with enumerative combinatorics,  and several talks on algebraic geometry, toric varieties, Fano varieties, and mirror symmetry. Here are the slides of my lecture entitled open problems on convex polytopes I’d love to see solved (but some further explanations for some parts are needed). I hope to return to some of the topics of the workshop in a later post.

Nagoya: FPSAC (Formal power series and algebraic combinatorics) is a central annual  combinatorial meeting with special emphasis on enumerative and algebraic combinatorics. This year it was the 24th such event although I could have swear that I took part in an FPSAC in Montreal already in 1985 but apparently this was a conference with a similar flavor and different name. Much is going on in enumerative and algebraic combinatorics. Cluster algebras, miraculously discovered a decade ago by Fomin and Zelevinsky are going strong in combinatorics as well as in other areas. Alternating sign matrices continue to offer amazing problems and answers. There were quite a few lectures on representation theory, symmetric functions, random matrices, and also on relations of enumerative combinatorics with hyperplane arrangements with polytopes and with physics. There were lectures involving massive computations and new computerized method and packages for symbolic computations  were described. Here are the slides of my lecture entitled Discrete isoperimetry: problems, results, applications and methods. The program page now contains slides for most lectures.

### What are alternating sign matrices?

I suppose that you all know what a permutation matrix is (an n by n matrix with 0,1 entries and one non zero entry in each row and each column) and alternating sign matrices are amazing generalization of permutation matrices discovered by William Mills, David Robbins, and Howard Rumsey. (See also this article by Bressoud and Propp) They are integral n by n matrices with 0,1 and -1 entries. In every row and every columns  the non zero aliments (and there must be at least one such element)  should alternate between 1 and -1 and start and end with a ‘1’. Alternating sign matrices are in one to one correspondence with monotone triangle. Thos are triangles of integers starting with a row: 1,2,…,n. With the following rules: (a) all rows are strictly increasing; (b) every element in row i is weakly between the two elements above it.

The amazing thing is that the number of alternating sign matrices of order n is precisely

## 1!4!7!…(3n-2)!/n!(n+1)!…(2n-1)!

This was a conjecture by Mills, Robbins and Rumsey and it took over a decade until it was proved by Doron Zeilberger. Later Greg Kuperberg found a short proof and by now several proofs are known. If you have some information or ideas on alternating sign matrices that you would like to share or some questions about them, or about anything else, please contribute a comment.

# Satoshi Murai and Eran Nevo proved the Generalized Lower Bound Conjecture.

Satoshi Murai and Eran Nevo have just proved the 1971 generalized lower bound conjecture of McMullen and Walkup, in their  paper On the generalized lower bound conjecture for polytopes and spheres . Let me tell you a little about it. For more background see the post: How the g-conjecture came about.

### Face numbers and h-numbers

Let P be a (d-1)-dimensional simplicial polytope and let $f_i(P)$ be the number of $i$-dimensional faces of P. The $f$-vector (face vector) of P is the vector $f(P)=(f_{-1}(P),f_0(P),f_1(P),...)$.

Face numbers of simplicial d-polytopes  are nicely expressed via certain linear combinations called the h-numbers. Those are defined by the relation:

$\sum_{0\leq i\leq d}h_i(P)x^{d-i}= \sum_{0\leq i\leq d}f_{i-1}(P)(x-1)^{d-i}.$

What’s called “Stanley’s trick” is a convenient way to practically compute one from the other, as illustrated in the difference table below, taken from Ziegler’s book Lectures on Polytopes’, p.251:

1

1           6

1          5            12

1          4           7            8

h= (1        3          3            1)

Here, we start with the $f$-vector of the Octahedron (1,6,12,8) (bold face entries) and take differences as shown in this picture to end with the $h$-vector (1,3,3,1).

The Euler-Poincare relation asserts that $h_d(P)=(-1)^{d-1}\tilde{\chi}(P)=1=h_0(P)$. More is true. The Dehn-Sommerville relations state that $h(P)$ is symmetric, i.e. $h_i(P)=h_{d-i}(P)$ for every $0\leq i\leq d$.

### The generalized lower bound conjecture

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture (GLBC):

Let P be a simplicial d-polytope. Then

(A) the h-vector of P, $(h_0,h_1,...,h_d)$ satisfies $h_0 \leq h_1 \leq ... \leq h_{\lfloor d/2 \rfloor}$.

(B) If $h_{r-1}=h_r$ for some $r \leq d/2$ then $P$ can be triangulated without introducing simplices of dimension $\leq d-r$.

The first part of the conjecture was solved by Stanley in 1980 using the Hard Lefschetz theorem for toric varieties. This was part of the g-theorem that we discussed extensively in a series of posts (II’, II, IIIB). In their paper, Murai and Nevo give a proof of part (B). This is remarkable!

Earlier posts on the g-conjecture:

I: (Eran Nevo) The g-conjecture I

I’ How the g-conjecture came about

II (Eran Nevo) The g-conjecture II: The commutative-algebra connection

III (Eran Nevo) The g-conjecture III: Algebraic shifting

B: Billerafest

# Updates, Boolean Functions Conference, and a Surprising Application to Polytope Theory

## The Debate continues

The debate between Aram Harrow and me on Godel Lost letter and P=NP (GLL) regarding quantum fault tolerance continues. The first post entitled Perpetual  motions of the 21th century featured mainly my work, with a short response by Aram. Aram posted two of his three rebuttal posts which included also short rejoiners by me. Aram’s first post entitled Flying machines of the 21th century dealt with the question “How can it be that quantum error correction is impossible while classical error correction is possible.” Aram’s  second post entitled Nature does not conspire deals with the issue of malicious correlated errors.  A third post by Aram is coming and  the discussion is quite interesting. Stay tuned. In between our posts GLL had several other related posts like Is this a quantum computer? on how to tell that you really have a genuine quantum computer , and Quantum ground day that summarized the comments to the first post.

## Virgin Island Boolean Functions

In the beginning of February I spend a week in a great symposium on Analysis of Boolean Functions, one among several conferences supported  by the Simons foundation, that took place at St. John of the Virgin Islands.

Irit Dinur and me

Ryan O’Donnell who along with Elchanan Mossel and Krzysztof Oleszkiewicz (the team of “majority is stablest” theorem) organized the meeting, live blogged about it on his blog. There are also planned scribes of the lectures and videos. I posed the following problem (which arose naturally from works presented in the meeting): What can be said about circuits with n inputs representing n Gaussian random variables, and gates of the form: linear functions, max and min.

## A surprising application of a theorem on convex polytopes.

(Told to me by Moritz Schmitt and Gunter Ziegler)

A theorem I proved with Peter Kleinschmidt and Gunter Meisinger asserts that every rational polytope of dimension d>8 contains a 3-face with at most 78 vertices or 78 facets. (A later theorem of Karu shows that our proof applies to all polytopes.) You would not expect to find a real life application for such a theorem. But a surprising application has just been given.

Before talking about the application let me say a few more words about the theorem. The point is that there is a finite list of 3-polytopes so that every polytope of a large enough dimension (as it turns out, eight or more) has a 3-face in the list. It is conjectured that a similar theorem holds for k-faces, and  it is also conjectured that if the dimension is sufficiently high you can reduce the list to two polytopes: the simplex and the cube. These conjectures are still open. (See this post  for related open problems about polytopes.) For k=2, it follows from Euler’s theorem that every three-dimensional polytope contains a face which is a triangle, quadrangle, or pentagon, and in dimension five and up, every polytope has a 2-face which is a triangle or a rectangle.