The toric Hilbert scheme
Abstract: Toric ideals are the defining ideals of semigroup algebras and toric varieties. These ideals lie at the intersection of combinatorics and algebraic geometry, and have applications to fields as diverse as integer programming and mirror symmetry.
A natural object which arises in the study of toric ideals is the toric Hilbert scheme, which parameterizes all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, the toric Hilbert scheme is not know to be connected. I will describe an approach to this question, which highlights the connections to geometric combinatorics.