Date:

Mon, 21/05/201811:00-12:30

Location:

IIAS, Eilat hall, Feldman Building, Givat Ram

First speaker: Daniel kalmanovich, HU

Title: On the face numbers of cubical polytopes

Abstract:

Understanding the possible face numbers of polytopes, and of subfamilies of interest, is a fundamental question.

The celebrated g-theorem, conjectured by McMullen in 1971 and proved by Stanley (necessity) and by Billera and Lee (sufficiency) in 1980-81, characterizes the f-vectors of simplicial polytopes.

We consider f-vectors of cubical polytopes (these are polytopes in which each proper face is combinatorially a cube), where no such characterization is known, and investigate the following question:

What is the minimal closed cone containing all f-vectors of cubical d-polytopes?

We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan from 1997. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.

Based on joint work with Ron Adin and Eran Nevo.----------

Second speaker: Or Raz, HU

Title: Symplectic geometric lattices are shellable

Abstract:

Geometric lattices were extensively studied, in part because they

correspond to finite matroids. I define a symplectic geometric lattice

in a set theoretic way similar to their classical counterparts,

proving a few properties, for which the main one is a characterization

of these lattices by atom orderings of which being shellable is a

corollary of.

In the same way that all geometric lattices are attained as

sub-lattices of the the face lattice of the simplex, we have our

lattices be sub-lattices of the face lattice of the Cross polytop.

This makes them a natural candidate for the lattice of flats for

symplectic matroids.

----------

Title: On the face numbers of cubical polytopes

Abstract:

Understanding the possible face numbers of polytopes, and of subfamilies of interest, is a fundamental question.

The celebrated g-theorem, conjectured by McMullen in 1971 and proved by Stanley (necessity) and by Billera and Lee (sufficiency) in 1980-81, characterizes the f-vectors of simplicial polytopes.

We consider f-vectors of cubical polytopes (these are polytopes in which each proper face is combinatorially a cube), where no such characterization is known, and investigate the following question:

What is the minimal closed cone containing all f-vectors of cubical d-polytopes?

We construct cubical polytopes showing that this cone, expressed in the cubical g-vector coordinates, contains the nonnegative g-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan from 1997. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.

Based on joint work with Ron Adin and Eran Nevo.----------

Second speaker: Or Raz, HU

Title: Symplectic geometric lattices are shellable

Abstract:

Geometric lattices were extensively studied, in part because they

correspond to finite matroids. I define a symplectic geometric lattice

in a set theoretic way similar to their classical counterparts,

proving a few properties, for which the main one is a characterization

of these lattices by atom orderings of which being shellable is a

corollary of.

In the same way that all geometric lattices are attained as

sub-lattices of the the face lattice of the simplex, we have our

lattices be sub-lattices of the face lattice of the Cross polytop.

This makes them a natural candidate for the lattice of flats for

symplectic matroids.

----------