Dept. of Mathematics, Technion, Haifa, Israel.
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 numerous 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 the natural framework for formulating topological Helly type theorems. We will survey some old and new results on Leray complexes with combinatorial and geometrical applications. In particular, we’ll describe recent work on Leray numbers of projections and a topological Helly type theorem for unions. Joint work with Gil Kalai.