Mathematical Sciences: Resolution of Singularities of an Algebraic Variety over a Field of Characteristic p.

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The research in this proposal concerns the removing (or resolution of) singularities of an algebraic variety by birational transformations. This problem was solved 23 years ago by Hironaka in the case of characteristic zero. The case of characteristic p 0 has resisted all attempts since then. This problem is the focus of this research. In algebraic geometry many of the geometric objects one studies (varieties) have singularities, i.e. corners or cusps. It is a very important problem to remove these in certain prescribed ways. The principal investigator has the first really new idea on this problem in many years. If he solves this problem or even makes significant headway on it, it would be a significant advance. One can only wait and see on a problem of this depth.