Mathematical Sciences: Classification and Moduli Problems inAlgebraic Geometry

Morrison教授将研究四维孤立终端商奇点的分类问题，以及将这种分类扩展到高维的可能性。大量的计算机辅助计算已经并将继续用于寻找这些奇点。与David Morrison合作，他将研究K3曲面的Hilbert点的稳定性，并将利用这项研究的结果来描述这种曲面的模空间的紧化。他将与Dave Bayer一起实现一种普遍标准的构造射影品种状态多面体的基算法，并用它来生成和研究这种多面体。已经找到了一种计算这些多面体的优越算法，并将在这些研究中得到应用。在这些例子的基础上，他将试图找到适合于研究射影变的希尔伯特点稳定性的希尔伯特-芒福德判据的几何细化。在与拜耳的进一步合作中，他将使用这些例子来研究品种的状态多面体与品种理想的标准或Grobner基计算的复杂性之间的关系。

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.