Abstract: Helly's theorem asserts that if a finite family of convex sets in d-space has an empty intersection, then there exists a subfamily of cardinality at most d+1 with an empty intersection. Helly's theorem and its extensions play a central role in discrete and computational geometry. It is of considerable interest to understand the role of convexity in these results, and to find suitable topological extensions. The class of d-Leray complexes (introduced by Wegner in 1975) is a natural framework for formulating topological Helly type theorems. We will survey some old and new results on Leray complexes with geometrical and algebraic applications. In particular, we'll describe a result on projections of d-Leray complexes with two applications:
1) A topological extension of Helly's theorem for unions.
2) A proof of Terai's conjecture on the Castelnuovo-Mumford regularity of monomial ideals.

Joint work with Gil Kalai.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006