Speaker:  Eran Nevo, Cornell
Title:   Upper bounds on face numbers of flag spheres
Time: 2:30 PM, Monday, November 2, 2009
Place:  Malott 206

Abstract:  A flag sphere is a simplicial complex whose faces are the cliques of a graph and whose geometric realization is homeomorphic to a sphere.  Flag spheres arise naturally in combinatorics.  A conjecture of Gal, that the so called gamma-vector of a flag sphere is nonnegative, will imply the Charney-Davis conjecture, and in turn a conjecture of Hopf in Riemanian geometry, for the cubical case. Gal's conjecture is about a lower bound on the number of  faces of a given dimension in terms of the numbers of faces of lower dimensions. We conjecture upper bounds too. The most optimistic conjecture is:
Conjecture: The gamma-vector of a flag (homology) sphere is the (face) f-vector of a flag complex. 
We prove this conjecture for several infinite families, including Coxeter complexes and flag spheres with few vertices compared to their dimension.

This is joint work with Kyle Petersen.

October 27 2009