Roy
Meshulam, Technion - Israel Institute of Technology
Topological Helly type theorems |
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'll describe some old and new results on Leray complexes
with geometrical and combinatorial applications. In particular we'll
discuss
1) A topological extension of the Lovász colorful Helly
theorem (conjectured by Bárány).
2) Relations with commutative
algebra and a proof of a conjecture of Terai on monomial ideals.
3) Covering vs. fractional covering in hypergraphs with bounded Leray
numbers.
(Joint work with Gil Kalai.)