Polytopes and spheres: the enumeration and reconstruction problems

Seminar in Mathematics

Consider a simplicial d-polytope P or a simplicial (d-1)-sphere P with n vertices. What are the possible numbers of faces in each dimension? What partial information about P is enough to reconstruct P up to certain equivalences?
In this talk, I will introduce the theory of stress spaces developed by Lee. I will report on recent progress on conjectures of Kalai asserting that under certain conditions one can determine P from the space of affine stresses of P ---- a higher-dimensional analog of the set of affine dependencies of vertices of P. This in turn leads to new results in the face enumeration of polytopes and spheres; for example, a strengthening of (the numerical part of) the g-theorem.
Joint work with Satoshi Murai and Isabella Novik.