Wednesday, October 6th, 2010
3:00pm - 4:00pm, in Science 2-066

Gregory Maloney

UMass Boston

A complete classification of finite rank dimension groups by their representations in ordered real vector spaces

Abstract: We study finite rank dimension groups, as well as finite dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite dimensional ordered real vector space with Riesz interpolation via a canonical embedding. We then characterize which of the subgroups of a finite dimensional ordered real vector space have interpolation (and are therefore dimension groups). This is joint work with Aaron Tikuisis.