Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Families of level algebras by Anthony IARROBINO, Northeastern University Abstract: Let R be a polynomial ring in r variables over a field K. Let H = (1,r, ...,hj), hj > 0, be a level O-sequence, a sequence of integers that occurs as the Hilbert function of some standard Artinian graded level quotient A = R/I, and denote by LevAlg(H ) the family of all such level quotients of R. We first report on work joint with M. Boij concerning the existence of many reducible families LevAlg(H ) of height three (r =3) type two (hj =2) level algebras. We then describe briefly some recent advances, and propose several problems in the study of level algebras.