Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Syzygies: Geometry, Combinatorics and Low Complexity by Sorin POPESCU, State University of New York at Stony Brook Abstract: I will discuss several classical and recent (some conjectural) geometric bounds for the complexity of the equations and syzygies of a projective variety, as expressed by the Castelnuovo-Mumford regularity of their defining ideal. I will also discuss recent results and work in progress relating the geometry and the combinatorics of homogeneous ideals and projective schemes of small regularity, with a focus on the open subset of the Hilbert scheme parametrizing 2-regular schemes.