Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Empty simplices of polytopes and graded Betti numbers by Uwe NAGEL, University of Kentucky Abstract: The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope can be interpreted as a problem about the first graded Betti numbers of the polytope. The proof of the conjecture allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we discuss optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.