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STATEMENT OF RESEARCH INTERESTS

My research is in the area of algebraic combinatorics, with an emphasis on problems from commutative algebra and algebraic geometry. The connection between algebra and combinatorics has had many implications in both fields. In combinatorics the highlights include Stanley’s proofs of the upper bound conjecture [21] and the g-theorem [22] using the theory of Cohen-Macaulay rings and toric varieties. In algebraic geometry the theory of toric varieties provides an important class of examples, with connections to still more areas, such as integer programming [4], [25], and mirror symmetry in mathematical physics [3], [6]. A less obvious connection between the areas is through the theory of Hilbert schemes. This connection is most spectacularly illustrated in Mark Haiman’s recent (2000) proof of the n!-conjecture of algebraic combinatorics [10] using the Hilbert scheme of n points in the affine plane. Both in my thesis, and in more recent work, I have been studying similar interactions with Hilbert schemes. I use computers extensively in my research, both to develop conjectures and to produce examples. This involves both using computational algebra packages such as Macaulay 2 [9] and Maple, and writing code in C. The theory of Gröbner bases also plays an important in my work. A unifying theme in my research is the study of toric ideals and their initial ideals. The Baues problem in geometric combinatorics also plays an important role. One version of the Baues problem concerns polyhedral subdivisions of the convex hull of a point configuration. Of interest is the homotopy type of the poset of all subdivisions, and whether the set of all triangulations (the finest possible subdivisions) is connected by operations known as bistellar flips. Part of the Baues graph of a configuration of six point in the plane is shown in Figure 1. Variants of such flips are of interest in computational geometry, and in low dimensional topology, where they are known as Pachner moves. An excellent survey is [18]. My work concerns algebraic analogues of this question.