Combinatorics Seminar

An important invariant in the study of face numbers of simplicial d-polytopes is the g-vector. The generalized lower bound theorem states that g_i is nonnegative for any simplicial polytope and characterizes the case of equality. Much less is known for centrally symmetric polytopes. A seminal work is established by Stanley thirty years ago, where he proved that for any centrally symmetric simplicial d-polytope P with d at least 3 and i between 1 and d/2, we have g_i(P) \geq \binom{d}{i}-\binom{d}{i-1}.

In this talk, I will introduce the rigidity theory of frameworks, and show how to apply this machinery to give a characterization of centrally symmetric d-polytopes with which satisfy g_2=\binom{d}{2}-d. This is joint work with Steve Klee, Eran Nevo and Isabella Novik. Speaker(s): Hailun Zheng (University of Michigan)