Mathematical Sciences: Classification and Moduli Problems inAlgebraic Geometry

莫里森教授将进行研究， 孤立终商分类中的问题 四维空间中的奇异点和可能的扩展 分类到更高的维度。 广泛的计算机辅助 计算已经并将继续用于搜索 对于这些奇点。 与大卫莫里森合作， 他将研究K3曲面的希尔伯特点的稳定性， 将使用本研究的结果来描述紧致化 这类曲面的模空间。 与戴夫拜耳，他将 实现一个通用的标准基础算法， 射影簇的状态多面体的构造及应用 它生产和研究这样的多面体。 一个上级算法， 计算这些多面体已被发现，并将被利用， 这些研究。 在这些例子的基础上，他将 试图找到希尔伯特-芒福德的几何改进 的Hilbert点的稳定性的研究 投射簇 在与拜耳的进一步合作中，他将 用这些例子来研究 国家多面体的品种和复杂性的标准或 Grobner基计算的理想的品种。

Professor Morrison will carry out research into several questions in the classification of isolated terminal quotient singularities in dimension four and possible extensions of this classification to higher dimensions. Extensive computer assisted calculations have been and will continue to be used in the search for these singularities. In collaboration with David Morrison, he will study the stability of Hilbert points of K3 surfaces and will use the results of this study to describe compactifications of moduli spaces of such surfaces. With Dave Bayer, he will implement a universally standard basis algorithm for the construction of state polytopes of projective varieties and use it to produce and study such polytopes. A superior algorithm for computing these polytopes has been found and will be exploited in these researches. On the bases of these examples, he will attempt to find a geometric refinement of the Hilbert-Mumford criterion suited to the study of stability of Hilbert points of projective varieties. In further joint work with Bayer, he will use these examples to investigate the relationship between the state polytopes of a variety and the complexity of standard or Grobner basis calculations for the ideal of the variety.