Test Your Intuition 35: What is the Limiting Distance?

(Just heard it today from Sergiu Hart.)

At time t=0, point A is at the origin (0,0) and point B is distance 1 appart at (0,1). A moves to the right (on the x-axis) with velocity 1  and B moves with velocity 1 towards A.  What will be the distance between A and B when t goes to infinity?

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Beyond the g-conjecture – algebraic combinatorics of cellular spaces I

The g-conjecture for spheres is surely the one single conjecture I worked on more than on any other, and also here on the blog we had a sequence of posts about it by Eran Nevo (I,II,III,IV). Here is a great survey article  on the g-conjecture by Ed Swartz 35 Years and Counting and slides of a great lecture by Lou Billera “Even more intriguing, if rather less plausible…”.

June Huh talked about the conjecture on Numberphile attracting 183,893 views and counting. With hundreds of thousands new people interested in the g-conjecture and some highly capable teams of researchers who were interested even before the video, it is safer to say that the solution of the conjecture is imminent 🙂 and this is going to be exciting! So I want to say a few things  about the “G-program”, questions that go beyond the g-conjecture. (OK, I realize I need to split it into  parts and also that each of the items deserves a separate post in the future.)

I wrote a post to suplement Eran Nevo’s posts on how the g-conjecture came about, and here is a very brief summary:

Simplicial polytopes: Leading to the g-theorem we can start with Euler’s theorem, and its high dimensional extensions for general polytopes.  Next follow the Dehn-Sommerville relations for simplicial polytopes. Here are some main landmarks toward the g-conjecture and its solution for simplicial polytopes.

1. The upper bound theorem (UBT, conjectured by Motzkin, 1957, proved by Peter McMullen 1970),
2. The lower bound theorem (LBT, conjectured by Grunbaum in the 60s, proved by Barnette 1970),
3. In 1970, the generalized lower bound conjecture (GLBC) was conjectured by McMullen and Walkup, and the g-conjecture was proposed by McMullen.
4.  In 1979, the sufficiency part of the g-conjecture was proved by Billera and Lee.
5.  Also in 1979, the necessity part of the g-conjecture was proved by Stanley based on the hard Lefschetz theorem for toric varieties.
6.  In 1993 Peter McMullen found a direct geometric proof for the case of simplicial polytopes.
7. The GLBC (now, GLBT) consists of the linear inequalities  of the g-conjecture and an additional part about cases of equality. The equality part was proved in 2012 by Murai and Nevo (see this post).

Triangulated spheres: All of the above are conjectured to hold for simplicial spheres (and even to homological spheres in the most inclusive sense). Barnette’s proof for the LBT applies to triangulated spheres. The UBT was proved for spheres by Stanley (1975) pioneering the connection with commutative algebra. The g-conjecture for triangulated spheres is still open, and this is what we refer to these days as the “g-conjecture”.

Algebraic tools:  For the connection with commutative algebra and algebraic geometry pioneered by Stanley, see this post. The crucial linear algebraic property behind the g-conjecture is a linear algebraic statement called the Lefschetz property that is a  consequence of the Hard Lefschetz Theorem (when it applies).  McMullen’s 1993 proof validates the Lefschetz property in greater generality (for arbitrary simplicial polytopes rather than rational simplicial polytopes) via an inductive argument that relies on “Pachner moves”. Certain Hodge-Riemann inequalities also played an important  role in McMullen’s proof.  Hodge-Riemann’s inequalities play a crucial role also in the recent solution of Rota’s conjecture by Adiprasito, Huh and Katz. (See this post and this beautiful Quanta Magazine article by Kevin Hartnett.)

In this extra footage for his Numberphile video (viewes by 10,000) June mentions the hard Lefschetz theorem.

 

Eran Nevo (left)  and Karim Adiprasito 

Going Beyond the g-conjecture: The “G-Program”

Let be a class of cellular $(d-1)$-dimensional spheres.

We want to define -parameters forming the -vector of , that satisfies

(1) latex h_i(K)=h_{d-i}(K)$,

(2) . In other words if we let then . (The are the g-numbers giving the conjecture its name.)

(3) Some interesting non-linear inequalities in the spirit of those for the original conjecture.

If this program is not general enough we would like to consider arbitrary manifolds without boundary or even pseudomanifolds without boundary, to make some adaptation to allow boundary, and even  to allow the “cells” to be rather exotic. We also want to understand the equality cases for the inequalities and how the h-vectors reflect the combinatorics, topology, and algebra of the cellular space K.

Some classes of cellular spheres: The class of simplicial spheres (and polytopes) contains the classes of flag spheres (and polytopes) and completely balanced spheres (and polytopes). The class of regular CW spheres with the intersection property, (including general polytopes) contains the classes of cubical spheres (and polytopes) and centrally symmetric spheres (and polytopes). Regular CW spheres without the intersection property include those coming from Bruhat intervals of Coxeter groups.  (The class of zonotopes is also very important but its connections to the present story are more subtle.)

From the algebraic geometry side, simplices correspond to complex projectice spaces, the h-vectors for simplicial polytopes correspond to Betti numbers that measure how more complicated a general smooth (toric) variety is. For general (rational) polytopes, you get more complexity in terms of complicated singularities, and for objects like intervals in Bruhat orders you have even more complicated singularities. The varieties exist only for small fragments of the pictures and beyond that you need to do some sort of “algebraic geometry without varieties”.

A) The original g-conjecture: simplicial polytopes and simplicial spheres

Again, let me refer the reader to the four posts by Eran Nevo (I,II,III,IV). I will repeat the definition of h-vectors and g-vectors at the end of the post.

Problem 1: Prove the g-conjecture for triangulated sphere.

Again, the most promising avenue towards a proof is by proving the Lefschetz property for face rings associated with triangulated spheres.

Problem 2: Understand the cases of equality for the Macualay (non-linear) inequalities?

Problem 3: Show that the gap in the inequalities tends to infinity for every sequence of simplicial polytopes tending to a smooth body. Formulate and prove an extension to spheres.

In the polytope case this problem was settled. This has been proved (for polytopes, see this paper and this post) by Karim Adiprasito, Eran Nevo,  and José Alejandro Samper for the linear inequalities (see picture below). The result, referred to as the “geometric LBT” asserts that for a sequence of simplicial polytopes tending to a smooth body,   More recently, Karim Adiprasito managed to prove it for the nonlinear inequalities. For triangulated spheres, finding the relevant notion of limits for triangulations would be a place to start.

The Lefschetz property (when it holds) allows to associate to every simplicial sphere S, a (shifted) order of monomials  M(S) with monomials of degree , of degree . There is also a construction (Kalai, 1988) that associates a triangulated sphere S(M) (called “squeezed sphere”) with every such M. Satoshi Murai proved that M(S(M))=M.

Problem 4: Study M(S)

B) General polytopes: The toric h-vector and g-vector

Intersection homology allows the definition of h- and g- polynomials for general polytopes. Those are linear combinations of flag numbers.  The post Euler’s formula, Fibonacci, the Bayer-Billera theorem, and Fine’s cd-index is very relevant.

Again, I will repeat a definition of h-vectors and g-vectors at the end of the post. HLT for intersection homology shows the nonnegativity of the g-polynomials for rational polytopes. Kalle Karu (relying on works by Barthel, Brasselet, Fieseler and Kaup and by Bressler and Luntsand and on McMullen’s 1998 proof) proved it for general polytopes!

Problem 5: Show that the toric g-numbers are nonnegative for every strongly regular CW-sphere.  (In particular for all polyhedral spheres.)

Recall that  a regular CW complex is a CW-complex where the closure of an open cell is homeomorphic to a close ball.  Regular CW spheres such that the intersection of two faces is a face are also called strongly regular CW complexes.

A big open problem which is open even for polytopes is

Problem 6: Show that the toric g-numbers form an M-vector.

Problem 6 is of great interest also for rational polytopes. The difficulty is that intersection homology does not admit a ring structure. One approach is to introduce some additional ring structure while another approach is to try to derive the M-inequalities from weaker algebraic or combinatorial conditions.

Problem 7: Understand the cases of equality for the linear inequalities and the non linear inequalities.

Recent advances by Adiprasito and Nevo in their paper QGLBT for polytopes toward the equality case for the toric GLBC. They also extended the geometric LBT to general polytopes.

Problem 8 (Jonathan Fine): Extend the toric g-numbers to a Fibonacci number of sharp nonnegative parameters, based, perhaps, on a Fibonnaci number of homology groups.

Interesting and rather mysterious duality relations were discovered for g-numbers. They were found to be  related to Koszul duality and to Mirror symmetry. See this recent post. of that post is .

Problem 9 : Understand further the connections between toric g-numbers and related invariants of polytopes and their dual.

Closely related issues are: flag vectors and the Bilerra-Bayer theorem (see this post), nonnegative flag inequalities, the cd-index, FLAGTOOL, and Braden-MacPherson theorem. An excellent 2006 paper on toric h- and g- numbers and related combinatorics and algebraic-geometry is by Tom Braden:  Remarks on the combinatorial intersection cohomology of fans.

Problem 10: Understand (all) linear inequalities among flag numbers of -polytopes and polyhedral -spheres.

Warning: I used Whitehead notion “strongly regular CW complexes”, and “regular CW complexes with the lattice property” and “regular CW-complexes with the intersection property” for the same mathematical object. Sorry.

C) Regular CW-spheres and Kazhdan-Lusztig polynomials

What happens when you give up also the lattice property?  For Bruhat intervals of affine Coxeter groups the Kazhdan Luztig polynomial can be seen as subtle extension of the toric g-vectors adding additional layers of complexity. Of coursem historically Kazhdan-Lustig polynomials came before the toric g-vectors. (This time I will not repeat the definition and refer the readers to the original paper by Kazhdan and Lustig, this paper by Dyer and   this paper by Brenti. Caselli, and  Marietti.) It is known that for Bruhat intervals with the lattice property the KL-polynomial coincide with the toric g-vector. Can one define h-vectors for more general regular CW spheres?

Problem (fantasy) 12: Extend the Kazhdan-Luztig polynomials (and show positivity of the coefficients) to all or to a large class of regular CW spheres.

This is a good fantasy with a caveat: It is not even known that KL-polynomials depend just on the regular CW sphere described by the Bruhat interval. This is a well known conjecture on its own.

Problem 11: Prove that Kazhdan-Lustig polynomials are invariants of the regular CW-sphere described by the Bruhat interval.

A more famous conjecture was to prove that the coefficients of KL-polynomials are non negative for all Bruhat intervals and not only in cases where one can apply intersection homology of Schubert varieties associated with Weil groups. (This is  analogous to moving from rational polytopes to general polytopes.) In a major 2012 breakthrough, this has  been proved by Ben Elias and Geordie Williamson following a program initiated by Wolfgang Soergel.

There is vast literature on KL-polynomials further extensions and combinatorial aspects. The combinatorics of regular, but not strongly regular CW complexes is again related to the cd-index and let me also mention that there is interesting combinatorics (and a good opportunity for the G-program and fantasy 12) for simplicial posets, namely when you insist on lower intervals to be Boolean but give up the lattice property.

D) Cubical polytopes and spheres: Adin’s h-vectors

A cubical polytope (sphere) is a polytope all whose faces are combinatorial cubes. Cubical complexes are important and mysterious objects. (They play a crucial role in some recent developments in 3D topology, see here.) Ron Adin defined h-numbers and formulated a “GLBC ineqequalities  for cubical polytopes (and spheres). Again, I will repeat Adin’s definition of h-vectors and g-vectors at the end of the post.

Problem 13: Prove Adin’s conjecture for cubical polytopes and spheres.

A recent paper by Adin, Kalmanovich, and Nevo On the cone of f -vectors of cubical polytopes shows that if Adin’s conjecture is valid, it describes the full cone spanned by linear inequalities among  face number of cubical d-polytopes.

Problem 14: Explore an UBT  for cubical polytopes and non-linear inequality for Adin’s numbers.

Problem 15: Extend (in the toric spirit) Adin’s invariant to polytopes and spheres with the property that every two-dimensional face has at least four edges (or just an even number of edges).

 

Coming next on the G-program:

• E) Centrally-symmetric polytopes and spheres;
• F) Flag polytopes and spheres – the Charney-Davis conjecture and the -conjecture;
• G) Completely balanced polytopes and spheres;
• H) Polytope pairs and polytopes with one non-simplex facet;
• I) The Batchi-Datta conjecture;
• J) Sharper versions of the generalized lower bound inequalities and further applications of the Lefschetz property.
• K) Stanley’s local theory
• L) Simplicial posets and ASL.
• M) Minkowski sums of polytopes;
• N) Section of a given polytope;
• O) Integer points in polytopes and associated polynomials
• P) Grunbaum’s conjecture and the GUBT (a related old post); (Let me mention that Karim Adiprasito reported recently on progress on the Grunbaum’s conjecture which is related to the g-conjecture.)
• Q) The Welzl-Wagner framework and early continuous analogs;
• R) triangulations of manifolds and pseudomanifolds.

I am sure that I missed, overlooked, forgot, or wasn’t aware of several things that I should mention. Please comment here or alert me about them. Also most of the problems I described are on the basic combinatorics level of the theory and Karim promised to contribute some problems on the algebraic level for a later post in this series.

STANLEYLAND-enumerative algebraic combinatorics

Some definitions

General polytopes: Toric h and g

Consider general d-polytopes. For a set {0,1,2,…,d-1}, { } , , define the flag number as the number of chains of faces , where . If is a dimensional polytope and is an dimensional polytope, their direct sum is obtained by embedding P and Q in two orthogonal subspaces such that the origin is an interior point of both P and Q, and taking the convex hull. The free join is the convex hull of P and Q when they are embedded in two skew affine spaces of dimensions d and e in a (d+e+1)-dimenional space.

For the original recursive definition of toric h-vectors see Stanley’s 1987 paper Generalized h-vectors, intersection cohomology of toric varieties, and related results. A simple axiomatic definition of the toric -vectors (taken from my 1988 paper A new basis of polytopes) is the following: We consider two polynomials and $g[P](x)$, with the properties

(A1) and , where .

(A2) if is a point.

(A3) The coefficients $h_i$ of are linear combinations of flag numbers.

(A4)

(A5)

(A1)-(A5) determine uniquely the polynomials h and g. In the simplicial case  all flag numbers are determined by ordinary face numbers and this gives the definition described below.

Cubical polytopes: Adin’s short h-vector and (long) h-vector

Adin’s short h-vector is defined as follows:

Adin’s (long) h-vector is defined by , and , for .

The short and long -vectors are defined by taking differences as in the simplicial case.
,
for ;
. for
.

 

The simplicial case: h-vectors  g-vectors, and the g-conjecture

The –vector (face vector) of  a cellular complex is where is the number of -dimensional faces of . For example, if is the -simplex then . Let be a -dimensional simplicial complex. the –vector of is  defined by

 

Let , for . is called the -vector of . The Dehn-Sommerville relations state that when is a sphere for every . (This result can be proved combinatorially, for the larger family of Eulerian posets, and, for rational simplicial polytopes  it reflects  Poincare duality for the associated toric variety.)

We say that a vector is an -vector ( for Macaulay) if it is the -vector of a multicomples, i.e. of a collection of multisets (elements can repeat!) closed under inclusion. For example, is an -vector, as is demonstrated by the multicomplex , written in monomial notation – the exponent tells how many copies of to take. Macaulay gave a numerical characterization of such vectors. (The proof uses compression – see this post for a general description of the method.)

The -conjecture:

The following are equivalent:

(i) The vector is the -vector of a simplicial -polytope

(ii) The vector is the -vector of a triangulated -sphere.

(iii)

(a) for every

(b)   is an -vector.

Last observation: explained to the general public, the most confusing aspect of the story is when you draw a triangle and refer to it as a “sphere”.

A description of the Adiprasito, Nevo,  and Samper paper.

Our chair Elon Lindenstrauss promised to ease the financial burdon caused by my habit of paying for Karim Adiprasito coffee’s  once after every breakthrough, so I started to collect the receipts.

An Interview with Yisrael (Robert) Aumann

I was privileged to join Menachem Yaari and Sergiu Hart in interviewing Yisrael Aumann.  The interview is in Hebrew. It is an initiative of the Israel Academy of Sciences and the Humanities.

For our non Hebrew speakers here is in English an interview with Aumann by Sergiu Hart from 2005, and here is a paper  that I wrote in 2006  Science, beliefs and knowledge: a personal reflection on Robert J. Aumann’s approach.

 

Aumann explained to us some details of his Ph D thesis on knot theory,

Can game theory help resolving human conflicts and the suffering and waste caused by them?

Entomologist observe ants but dont tell them what to do.

Igor Pak is Giving the 2018 Erdős Lectures in Discrete Mathematics and Theoretical Computer Science

Update: The lectures this week are cancelled.  They will be given at a later date.

Next week Igor Pak will give the 2018 Erdős Lectures

 

• Monday Jun 18 2018

  Combinatorics — Erdos lecture: Igor Pak (UCLA) “Counting linear extensions”

  11:00am to 12:30pm

  Location:

  IIAS, Eilat hall, Feldman bldg (top floor), Givat Ram

  I will survey various known and recent results on counting the number of linear extensions of finite posets. I will emphasize the asymptotic and complexity aspects for special families, where the problem is especially elegant yet remains #P-complete.

• Wednesday Jun20, 2018

  CS Theory — Erdős Lecture: Igor Pak (UCLA) “Counting Young tableaux”

  Igor Pak (UCLA)

  10:30am to 12:00pm

  Location:

  Rothberg (CS building) B-220

  The number of standard Young tableaux of skew shape is a mesmerizing special case of the number of linear extensions of posets, that is important for applications in representation theory and algebraic geometry.  In this case there is a determinant formula, but finding their asymptotics is a difficult challenge.  I will survey some of the many beautiful result on the subject, explain some surprising product formulas, connections to Selberg integral, lozenge tilings and certain particle systems.

• Thursday, Jun21, 2018

  Colloquium: Erdos lecture – Igor Pak (UCLA) “Counting integer points in polytopes”

  2:30pm to 3:30pm

  Location:

  Manchester Building (Hall 2), Hebrew University Jerusalem

  Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From a computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc.?

Vera T. Sòs, Doctor Philisophiae Honoris Causa, Hebrew University of Jerusalem

Videotaped by Ehud Friedgut.

Sailing into High Dimensions

On June 20 at 13:30 I talk here at HUJI about Sailing into high dimensions. (Thanks to Smadar Bergman for the poster.)

Conference in Singapore, Vietnam, Appeasement, Restorative Justice, Laws of History, and Neutrinos

Eliezer Rabinovici

Some weeks ago I returned from a beautiful trip to Singapore and Vietnam. For both me and my wife this was the first trip to these very interesting countries. In Singapore I took part in a very unusual scientific meeting Laws: Rigidity and Dynamics, and my old-time friend Eliezer Rabinovici was one of the organizers. Eliezer, a prominent theoretical physicist,  was the force behind SESAME, a unique scientific endeavor cospnsored  by Jordan-Iran-Israel-Pakistan-Egypt-Cyprus-The Palestinian authority-Turkey which we previously mentioned in this post.  There is something amazing about a scientific project in Jordan sponsored jointly by Israel and Iran with the involvment of Turkey, Egypt, and Pakistan.

(Some background for readers unfamiliar with the situation in the Middle East: these days, as in the last decades, and even more so than before, there is some hostility and tension between Israel and Iran.)

Eliezer was also the main force behind Isreal joining CERN where he served as a vice-president. As having interest in both  theoretical aspect of soccer, and in actually playing, I was especially amazed by the festive soccer game organized by Eliezer between the two top Israeli teams, which included, in the break, a historic game between the HUJI soccer team and the team of the Knesset, the Israeli parliament.  (Our team won big-time.)

Not everyone shares my excitement about SESAME. What do you think?

Singapore’s  investments in education, science, art, and public housing are impressive

(Compare the sculpture with this picture. )

Darkest hours and appeasement

On the flight to Singapore I saw the excellent film “darkest hours” about the early weeks of World War II and Winston Churchill’s  early days as prime minister. Also in the guided tours in Singapore a lot of attention was given to the surrender of Singapore and the war memories, and needless to say that the memory of war was very much present while we were traveling in Vietnam. The Churchill movie was truly excellent, bringing us back to the appeasement policy of the two prime ministers before him, and the tragic and disastrous pursuit  of peace. I was wondering if the great military weakness of England (and France) in 1939 was a logical consequence of the appeasement policy/philosophy. Can you have a combined policy of strength and appeasement? Is appeasement ever justified?

Laws and predictions in History

The conference I took part in in Singapore was the  3rd UBIAS Intercontinental Academia (ICA) – Laws: Rigidity and Dynamics, In short, it was a conference about “laws”. (The first conference of this kind was about “time” and the second about “human dignity”.) There were altogether 16 exciting mentors and 18 brilliant fellows from all areas of academia.

Nobel laureate  physicist David Gross was one of the mentors. (Ada Yonath was another mentor, but she arrived shortly before I left.) Both David Gross and Eliezer Rabinovici were pushing for some sort of physics-style laws or at least physics-style scientific exploration in History. I am quite skeptical about this idea and it even took me some time to realize that David and Eliezer were serious about it. (I suppose that the chaotic nature of historical developments is one good reason for skepticism.) Davis Gross (whom I found to be rather nice on a personal level) self-described  his view on academia and science as “arrogant” and his bold and provocative statements provided much food for thought and discussion throughout the conference. Anyway, when it comes to history, IAS historian  Patrick Geary gave quite an appealing argument for what we can expect and what we cannot expect from historical laws, with some very interesting examples.

Penelope (Penny) Andrews

The afternoon session of March 20

Constitution and Love – The South African constitution and restorative justice

Penelope (Penny) Andrews who had just arrived after a 24 hours flight from Cape Town gave a beautiful (and moving) talk entitled “The ‘casserole’ constitution – South African constitution and international law”. The talk was about the creation of South African human-rights based constitution,  and the struggle of South African democracy afterwards.  I strongly recommend watching the lecture. I was scheduled to be one of the responders.

What do I have to contribute to a discussion on the South African constitution? I had one personal story to tell: My father’s parents left from Lithuania for Israel (or Palestine as it was called under the British rule) in 1923 and my grandmother’s brother Hirsch stayed there.  During WWII  Hirsch’s daughter (my father’s cousin) Dora Love (Rabinowitz) was moved from one concentration camp to another, and lost her sister and her mother in the Nazi camps. After the war she met and married an English soldier Frank Love and they both moved to South Africa. Dora was a teacher at a Hebrew school in Johannesburg (among her students were Peter and Neil Sarnak), and also spent much of her life lecturing about the Holocaust. Her younger daughter Janet Love joined Mandela’s ANC at a young age, was considered a “terrorist” for more than a decade and had to flee South Africa. She eventually returned to South Africa and actually took part in the negotiations for the new constitution. After that she served in the South African parliament for five years and later on several governmental posts, now serving as the South Africa Human Rights Commissioner. (This story took Penny Andrews by surprise and it turned out she knows Janet well.)

Besides that I prepared one comment on the issue.  My comment/question/suggestion was about applying South Africas’s famous Truth and Reconciliation policy  in the criminal law. Can we apply similar principles there? I was especially thinking about the situation regarding sexual harassments and other sexual crimes which seem to be very wide spread. Here, as in other aspects of criminal law, it is not clear at all whether the instinctive demand for harsher punishments is correct. Restorative justice seems of relevance. (I suppose this is also an appeasing approach of some sort.)  Penny Andrews gave a very thoughtful answer including on cases in SA and elsewhere where a restorative system of justice is implemented.

  

Janet Love (right). The Love family visited Israel in the late 60s (and brought me my first jacket). On the left is Seth, Janet’s older brother.  (I added more pictures at the end of the post.)

My talk

My own talk was about quantum computing. Given the wide interests of the audience I promised to be non technical and to have less mathematics in my talk compared to the talk about history by Patrick Geary. It is a good time to revisit this issue here on the blog but let me delay it to another post. David Gross asked me if I really thought that my hand-waving computer-science argument against quantum computers applies to show infeasibility of structures that serious physicists had given much thought to.  (He mainly referred to topological quantum computing.) My answer was “yes”.

Atul Parikh did a brilliant job in responding to me. Here is a link to a video of this session. For another session with two great talks by Atul Parikh and Ada Yonath see this video. A session with Michal Feldman’s conference-opening talk and Patrick Geary’s talk is here. A session with talks by Partha Dasgupta and David Gross is here. Combinatorialists may  enjoy the introduction by Robin Mason to Frank Ramsey at the beginning. (Have a look also at David and others’ response t0 Partha’s talk and Partha’s rejoiner.)  The entire collection of videotaped sessions is here. There are many interesting talks and discussions.

 

 

Sights from Vietnam

Mathematicians’  lunch at Hanoi. From right Phan Ha Duong, me, Ngo Viet Trung, Nguyen Viet Dung, and Le Tuan Hoa. Below: Halong Bay

 

 

Added pictures

Janet Love, Frank Love, Seth Love, Tami Kalai (my sister), Dora Love, Hanoch Kalai (my father), near our home in Jerusalem. Below, a few years later with the jacket.

 

 

 

 

A Mysterious Duality Relation for 4-dimensional Polytopes.

Two dimensions

Before we talk about 4 dimensions let us recall some basic facts about 2 dimensions:

A planar polygon has the same number of vertices and edges.

This fact, which just asserts that the Euler characteristic of is zero, can be reformulated as: Polygons and their duals have the same number of vertices.

A linear algebra statement

The  spaces of affine dependencies of the vertices  for a polygon P and its dual P* have the same dimension.

I am not aware of a pairing or an isomorphism for demonstrating this equality. (See, however Tom Braden’s comment.)

Four dimensions

Given a 4-dimensional polytope P define

Here, is the number of -faces of . is the number of chains of the form 0-face 2-face. In other words it is the sum over all 2-faces of P, of the number of their vertices. In 1987 I discovered the following:

Theorem [Mysterious Four-dimensional Duality]: Let P and P* be dual four dimensional polytopes then

(1)   γ(P)=γ(P*)

Here is an example: Let be the four dimensional cube. In this case , , , , and since every 2-face has 4 vertices . . is the 4-dimensional cross-polytope with  , , , , and since every 2-face has 3 vertices . . Viola!

The proof of (1) is quite a simple consequence of Euler’s theorem.

For the 120-cell and its dual the 600-cell .

Frameworks, rigidity and Alexandrov-Whiteley’s theorem.

A framework based on a -polytope is obtained from by adding diagonals which triangulate every 2-face of . Adding the diagonals leads to an infinitesimally rigid framework. This was proved by Alexandrov for 3 dimension and by Walter Whiteley in higher dimensions.

Walter Whiteley

Theorem (Alexandrov-Whiteley): For , every famework based on is infinitesimmaly rigid.

It follows from the Alexandrov-Whiteley theorem that for a four-dimensional polytope , is the dimension of the space of stresses of every  framework based on . (An affine stress is an assignment of weights to edges so that every vertex lies in “equilibrium”.) Whiteley’s theorem implies that for every 4-dimensional polytope .

4-dimensional stress-duality: Let P and P* be dual four dimensional polytopes. Then the space of affine stresses for a frameworks based on P and on P* have the same dimension.

Again, I am not aware of a pairing or an isomorphism that demonstrates this equality between dimensions. (See, however Tom Braden’s comment.)

Remark: Given a polytope ,  let be the number of edges in a framework based on . We can define for every -polytope, . Whiteley’s theorem implies that for every . (For by Euler’s theorem .)

Just as a polytope in dimension greater than 2 need not have the same number of vertices as its dual, it is also no longer true in dimensions greater than 4 that equals . However, it is true in every dimension that if and only if .

Toric varieties

Let be a toric variety based on a rational 4-dimensional polytope . The dimension of the primitive part of the 4th intersection homology group of is equal to .  Our duality theorem thus asserts that the primitive part of the 4th intersection homology group of has the same dimension as the primitive part of the 4th intersection homology group of .

Some references: Whiteley’s theorem (Whiteley, 1984); The 4-d duality relation (Kalai, 1987);  A more general relation (Bayer and Klapper, 1991); An even more general relation (Stanley; 1992) Related theorems on combinatorics of polytopes (Braden, 2006);  Connection to Mirror symmetry (Batyrev and Borisov, 1995); Connection to Koszul duality (Braden, 2007); Rigidity and polarity in 3-d (Whiteley 1987)

 

Duncan Dauvergne and Bálint Virág Settled the Random Sorting Networks Conjectures

A short summary: The beautiful decade-old conjectures on random sorting networks by  Omer Angel, Alexander Holroyd, Dan Romik, and Bálint Virág, have now been settled by Duncan Dauvergne and Bálint Virág in two papers: Circular support in random sorting networks, by Dauvergne and Virág, and The Archimedean limit of random sorting networks by Duncan Dauvergne.

 

Dan Romik, see also Romik’s mathematical gallery and Romik’s moving sofa page

Sorting networks

Sorting is one of the most important algorithmic tasks and it is closely related to much beautiful mathematics. In this post, a sorting network is a shortest path from 12…n to n…21 in the Cayley graph of S_n generated by nearest-neighbour swaps. In other words, a sorting networks is a sequence of transpositions of the form whose product is the permutation .

Sorting networks, as we just defined,  appear in other contexts and under other names. Also  the term “sorting networks” is sometimes used in a different way, and in particular, allows swaps that are not nearest neighbors. (See the slides for Uri Zwick’s lecture on sorting networks.)

Richard Stanley proved a remarkable formula

for the number of sorting networks or reduced decomposition of the permutation n…21.  (We mentioned it in this post. )

We can also think about sorting networks as  maximal chains in the weak Bruhat order of the symmetric group.  Another related notion in combinatorial geometry is that of  “(simple) allowable sequence of permutations”. an allowable sequence of permutation is a sequence of permutations starting with  12…n and ending with n…21 such that each permotation in the sequence is obtained by the previous one by reverseing one set or several disjoint sets of consecutive elements. See, e.g., this paper of Hagit last on two proofs of the Sylvester-Gallai theorem via allowable sequence of permutations.

Random Sorting networks

Alexander Holroyd’s videotaped lecture on random sorting networks. Holroyd’s random sorting gallery (pictures below are taken from there), and his six other amazing mathematical gallerys. 

Random sorting networks were considered a decade ago in a beautiful paper by Angel, Holroyd, Romik, and Virág . In this paper some brave conjectures were made

Conjecture 1: the trajectories of individual particles converge to random sine curves,

Conjecture 2:  the permutation matrix at half-time converges to the projected surface measure of the 2-sphere.

There are more conjectures! I will just show you one additional picture

The rhombic tiling corresponding to a uniform random sorting network:

The works by Dauvergne and Virág

In two recent breakthrough papers all those conjectures were proved. The papers are

1. Circular support in random sorting networks, by Dauvergne and Virag

…in which they show the tight support bound for the circle seen in the halfway permutation, as well as the tight Lipschitz bound for trajectories.

2. The Archimedean limit of random sorting networks, by Dauvergne.

…in which all the 2007 conjectures are proved. Here is the abstract:

A sorting network (also known as a reduced decomposition of the reverse permutation), is a shortest path from 12⋯n to n⋯21 in the Cayley graph of the symmetric group Sn generated by adjacent transpositions. We prove that in a uniform random n-element sorting network , that all particle trajectories are close to sine curves with high probability. We also find the weak limit of the time-t permutation matrix measures of . As a corollary of these results, we show that if is embedded into via the map τ↦(τ(1),τ(2),…τ(n)), then with high probability, the path σn is close to a great circle on a particular (n−2)-dimensional sphere in . These results prove conjectures of Angel, Holroyd, Romik, and Virag.

These papers build on previous results, including some in the 2007 paper, recent results by Gorin and Rahman from Random sorting networks: local statistics via random matrix laws, as well as those of Angel, Dauvergne, Holroyd and Virág from their paper on the local limit of random sorting networks.

The halving (pseudo)lines problem

Let me mention an important conjecture on sorting networks,

Conjecture: For every k, and the number of appearances of the transposition (k,k+1) in every sorting network is .

This is closely related to the halving line problem. The best lower bound (Klawe, Paterson, and Pippenger) behaves like . A geometric construction giving this bound  is a famous theorem by Geza Toth.   The best known upper bound by Tamal Dey is .

I am amazed that I did not blog about the halving lines problem and about allowable sequences of permutations in the past. (I only briefly mentioned the problem and allowable sequences of permutations in one earlier post on a certain beautiful geometric discrepancy problem.)

The permutahedron makes an appearance

 

The permutahedron is the Cayley graph of the symmetric group S_n generated by the nearest-neighbour swaps (12), (23), (34) and (n-1n). (Here .) Are there analogous phenomena for the associahedron? one can ask.

Zur Luria on the n-Queens Problem

(From Wikipedia )

The eight queens puzzle is the famous problem of placing eight chess queens on a chessboard so that no two queens threaten each other. The questions if this can be done and  in how many different ways, as well as the extension to n queens on a n × n chessboard  was raised already in the mid nineteen century.  Apparently Gauss was interested in the problem and figured out that the number of solutions for the 8 × 8  board is 92, and  Zur Luria gave a beautiful lecture about  his new results on the number of solutions in our seminar earlier this week. The lecture follows Luria’s paper New bounds on the n-queens’s problem.

Before getting to Zur’s result a little more on the history taken from Wikipedia: “Chess composer Max Bezzel published the eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850. Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n × n squares. Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n-queens version. In 1874, S. Gunther proposed a method using determinants to find solutions… In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. He published a highly detailed description of a depth-first backtracking algorithm.” For more on the history and the problem itself see the 2009 survey by Bell and Stevens.  Let me also mention the American Mathematical Monthly paper The n-queens problem by Igor Rivin, Ilan Vardi and Paul Zimermmann.  (In preparing this post I realized that there are many papers written on the problem.)

Let be the number of ways to place n non attacking queens on an n by n board, and let be the number of ways to place n non-attacking queens on an n by n toroidal board.   is sequence A000170 in the On-Line Encyclopedia of Integer Sequences.  The toroidal case, also referred to as the modular n-queens problem was asked by Polya in 1918. Polya proved that iff is 1 or 5 modulo 6. (Clearly, .)

Here are Zur Luria’s new results:

Theorem 1: 

Theorem 2: , for some .

As far as I know, these are the first non-trivial upper bounds.

Theorem 3: For some constant , if is of the form then .

Update: In the lecture Zur noted that the proof actually works for all odd n such that -1 is a quadratic residue mod n.

Theorem 3 proves (for infinitly many integers) a conjecture by Rivin, Vardi, and Zimmermann, who proved exponential lower bounds. Luria’s beautiful proof can be seen (and this is how Zur Luria views it) as a simple example of the method of algebraic absorbers, and Zur mentions  an earler application of a similar methods is in a paper of Potapov on counting Latin hypercubes and related objects.  Rivin, Vardi and Zimmermann conjectured that the exponential generating function of has a closed form and understanding this generating function is a very interesting problem. Luria conjectures that his upper bound for is sharp.

Luria’s conjecture: .

In view of recent progress in constructing and counting combinatorial designs this conjecture may be within reach. Zur also conjectures that for , 3 should be replaced by some .

More on chess on this blog: Chess can be a Game of Luck,  Amir Ban on Deep Junior, and quite a few other posts.