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.
