The typical algebraic shifting of a surface

Algebraic shifting, introduced by Kalai in 1984, is an operator that canonically associates a shifted complex to a given simplicial complex. This construction uses the Stanley-Reisner ring of the complex, or its exterior algebra analogue, and is useful especially in f-vector theory. We initiate a statistical study of exterior algebraic shifting, focusing on concentration phenomena for random triangulations of a fixed topological space. In particular, we prove concentration for (i) a uniform n-vertex refinement of any given graph, and for (ii) a random Delauney triangulation of any given Riemannian surface (compact, connected, without boundary), where the n vertices are sampled independently at random according to the volume measure. As a tool to prove concentration for surfaces, we prove a universality result on edge contractions: for every fixed surface triangulation K, every dense enough point set in the surface yields a Delaunay triangulation that edge contracts to K. Joint work with Denys Bulavka and Yuval Peled. All the relevant background will be reviewed in the talk.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/