Roy Meshulam: Leray numbers and monomial ideals

Abstract: The Leray number is a natural topologically
based invariant of a simplicial complex. Originally defined
by Wegner (1975) in connection with Helly's theorem,
it later proved useful in other areas mathematics.
We'll describe two recent applications of Leray numbers:
1) A proof of Terai's conjecture on the Castlenuovo-Mumford
regularity of monomial ideals.
2) A topological version of Helly's theorem for unions.