Colloquium: Hailun Zheng

January 29, 2024 @ 3:30 pm – 4:30 pm

Colloquium in Keller 303

The speaker is Dr. Hailun Zheng, from the University of Houston-Downtown.

Title: Polytope and spheres: the enumeration and reconstruction problems

Abstract: 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 reconstruct 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; in particular, a strengthening of (the numerical part of) the g-theorem.

Joint work with Satoshi Murai and Isabella Novik.