Helly type theorems for Leray complexes

Helly's theorem asserts that if a finite family of convex sets in d-space has the property that every subfamily of cardinality at most d+1 has a common point, then the whole family has a common point.

Leray complexes are, roughly speaking, simplicial complexes of bounded hereditary homological dimension, and they arise naturally in the study of topological extensions of Helly's theorem and its relatives. We will describe some old and new results on Leray complexes with geometrical and combinatorial applications. In particular we'll discuss a topological extension of the Lovasz colorful Helly theorem (conjectured by Barany), and a proof of a conjecture of Terai on monomial ideals.

Joint work with Gil Kalai.