Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Toward a numerical theory for the Betti numbers of ideals of fat points by Brian HARBOURNE, University of Nebraska Abstract: Given multiplicities m1, ..., mn of general points p1, ..., pn of P2, there is an increasingly well-supported conjecture which, in terms of the mi only, gives the Hilbert function of the ideal I defining the fat point subscheme m1p1 + ... + mnpn. No such conjecture for the Betti numbers of the minimal free resolution of I has ever been given. Using a connection between the Betti numbers and splittings of certain rank 2 bundles on P1, we give a conjecture for the Betti numbers in many cases. We also formulate a stable version of the problem; our conjecture, it true, would give a complete solution of the stable problem. (This is joint work with Alessandro Gimigliano and Monica Idà.)