Abstracts

• (with Rekha R. Thomas) Combinatorics of the Toric Hilbert Scheme . Submitted. The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the usual Hilbert scheme it is unknown whether toric Hilbert schemes are connected. In this paper we construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. We use these graphs to provide a family of corank three matrices with toric Hilbert schemes of arbitrarily high dimension. The flip graph is then related to the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with observations that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme.
• Antichains of Monomial Ideals are Finite . To appear in Proceedings of the American Mathematical Society. The main result of this paper is that all antichains are finite in the poset of monomial ideals in a polynomial ring, ordered by inclusion. We present several corollaries of this result, both simpler proofs of results already in the literature and new results. One natural generalization to more abstract posets is shown to be false.
• Boolean Term Orders and the Root System B_n. Order 15 (3):279-295, 1998 . A Boolean term order is a total order on subsets of [n]={1,...,n}$ such that the empty set < a for all nonempty a contained in [n], and a < b if and only if a union c < b union c for all c not intersecting a or b. Boolean term orders arise in several different areas of mathematics, including Groebner basis theory for the exterior algebra, and comparative probability. The main result of this paper is that Boolean term orders correspond to one element extensions of the oriented matroid M(B_n), where B_n is the root system {e_i:1 <= i <= n } union {e_i +/- e_j :1 <= i < j <= n }. This establishes Boolean term orders in the frame work of the Baues problem. We also define a notion of coherence for a Boolean term order, a nd a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of Boolean term orders for small values of n.
• (with Timothy Sturge and William Baritompa) Equivalent Methods for Global Optimization. In State of the Art in Global Optimization (Princeton, NJ, 1995), 201--211, Nonconvex Optim. Appl., 7, Kluwer Acad. Publ., Dordrecht, 1996. In this paper we show that many global optimization algorithms which use information from more than one function evaluation to form an upper envelope for the function are equivalent to ones which use information from only the most recently evaluated point. As a special case of this we show that a two point smoothed improvement of the algorithm of Breiman and Cutler is equivalent to the one point algorithm of Baritompa and Cutler. This comprised the (undergraduate) Honours III thesis of Sturge and myself under the supervision of Baritompa.