Department of Mathematics
Colloquium
by
Prof. Bernd Sturmfels

We discuss the recent interplay between the homological study of monomial ideals and extremal combinatorics. This represents joint work with Irena Peeva and Dave Bayer, as well as recent work by Peeva-Reiner-Welker and Morris-Hosten. A fundamental computational issue in commutative algebra is to determine the Hilbert series and the minimal free resolution of a monomialideal. If the ideal is generic, in a sense to be made precise, then the answer is given by a simplicial complex which we call the Scarf complex. The face numbers of the Scarf complex coincide with the Betti numbers of the monomial ideal, and our goal is to understand extremal behavior of these numbers. A first bound comes from enumerative combinatorics, namely, from the Upper Bound Theorem for Convex Polytopes. But this bound is not sharp if the given ideal is generated by 13 (or more) monomials in 4 (or more) variables. More refined answers are provided by extremal combinatorics, more specifically, by the dimension theory of partially ordered sets.

References:

1. S.Hosten and W.Morris: The order dimension of the complete graph, Preprint, January 1998.
2. I.Peeva, D.Bayer and B.Sturmfels: Monomial Resolutions, Math. Research Letters, to appear.
3. I.Peeva, V.Reiner and V.Welker: Homology of monomial ideals and W.Trotter: Combinatorics and Partially Ordered Sets. Dimension Theory, The Johns Hopkins University Press, Baltimore, 1992.order dimension of lattices, Preprint, November 1997.
4. W.Trotter: Combinatorics and Partially Ordered Sets. Dimension Theory, The Johns Hopkins University Press, Baltimore, 1992.
5. G.Ziegler: Lectures on Polytopes, Springer Verlag, New York, 1994.