Billerafest

I am unable to attend the conference taking place now at Cornell, but I send my warmest greetings to Lou from Jerusalem. The titles and abstracts of the lectures can be found here. Let me tell you about two theorems by Lou.

 

The first is the famous g-theorem: The g-theorem is a complete description of f-vectors (= vecors of face numbers) of simplicial d-polytopes. This characterization was proposed by Peter McMullen in 1970, and it was settled in two works. Billera and Carl Lee proved the sufficiency part of McMullen’s conjecture, namely for every sequence of numbers which satisfies McMullen’c conjecture they constructed a simplicial d-polytope P whose f-vector is the given sequence. Richard Stanley proved the necessity part based on the hard Lefschetz theorem in algebraic geometry. The assertion of the g-conjecture (the necessity part) for triangulations of spheres is open, and this is probably the one single problem I spent the most time on trying to solve. 

 

The second theorem is a beautiful theorem by Margaret Bayer and Billera. Consider general d-polytopes. for a set {0,1,2,…,d-1}, { } , , define the flag number as the number of chains of faces , where .  Bayer and Billera proved that the affine dimension of flag numbers of d-polytopes is where is the dth Fibonacci number. (, , , , etc.) The harder part of this theorem was to construct d-polytopes whose sequences of flag numbers are affinely independent.  The construction is simple: It is based on polytopes expressed by words of the form PBBPBPBBBPBP  where you start with a point, and P stands for “take a pyramid” and B stands for “take a bipyramid.” And the word starts with a P (to the left) and has no two consecutive B’s.

 

Let’s practice the notions of f-vectors and flag vectors on the 24-cell   . (The figure is a 3-dimensional projection into one of the facets of the polytope.)

 

 

 

This  4-polytope has 24 octahedral facets. It is self dual.

So . And . And , , etc. 

Brush Up Your Björner:

   

Just returning from a conference in Stockholm.

BRUSH UP YOUR BJÖRNER
(Cole Porter: Brush up your Shakespeare - new lyrics by Kimmo Eriksson)

The girls today in society
go for great mathematics, see.
So to win their hearts you must quote with ease
Pythagoras and Archimedes.
One must know Newton and Gauss and, hey,
what’s his name, eh? Poincaré!
Unless you know Grothendieck and DesCartes
no sweet lady will give you her heart.
But to really make them fall
and to make your love-life surrealist,
quote the greatest of them all:
a 60-year old comb’naturrealist.

Brush up your Björner,
start quoting him now!
Brush up your Björner,
and the ladies you will wow.

Many women will find it admirable
if you tell her she makes your CHIPS FIRABLE.
If your lady-friend still isn’t yielding
say you’ve got this big place: a TITS BUILDING.
If there still is some source of estrangement
make some cozier SUBSPACE ARRANGEMENT.
Brush up your Björner,
and they’ll all be charmed!

Jerusalem Combinatorics ‘93

Jerusalem Combinatorics ‘93 is the title of a conference I organized that took place fifteen years ago in May 9-17, 1993 at the Hebrew University of Jerusalem. It was a conference that was devoted to all areas of combinatorics. The other organizers were Noga Alon, Hélène Barcelo, Anders Björner, and Edna Wigderson. Altogether there were around thirty plenary talks, and about fifty additional invited talks in 10 sections representing various subareas of combinatorics. There were also four special talks for a large audience. Overall, it was a fruitful and a very nice event that I think people enjoyed.

A special aspect of the conference was the unusually large number of female speakers. 16 out of the 30 main plenary speakers were women, and also many of the additional speakers,  special session organizers, and other participants. The four large audience lecturers were Vera Sós, who talked about irregularities of distributions, Mireille Bousquet-Mélou who talked about polyominoes, Hillel Furstenberg who talked about Ergodic theory and Combinatorics, and Joan Birman who talked about the combinatorics of finite-type invariants for knots. 

A  collection of papers   by participants of the conference, edited by Barcelo and myself, appeared as Volume 178 of Contemporary  Mathematics.

The poster of the conference also has an interesting story behind it.

A Meeting at Marburg

  Just returning from a cozy two days discrete-math workshop in Marburg. A very nice mixture of participants and topics. The title of my talk was “Helly theorem, hypertrees and strange enumeration” and I plan to blog about it sometime soon. A few hours before taking off, Aner Shalev told me that a 1951 conjecture by Ore asserting that every element in a non abelian finite simple group is a commutator have just been proved by a group of four researchers - Aner himself and Liebeck, O’Brien and Tiep.  (Ore himself proved that for every element is a commutator.) The basis for a very complicated inductive proof required computer works and the final OK came four hours before Aner gave a lecture about it! 

The talks in Marburg were very interesting.

Day 1:Enumerative combinatorics techniques and results related to the asymptotic conjectured formula for the number of self avoiding random walks (a holy grail in statistical mechanics);