Singularity Theory and Commutative Algebra

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Cutkosky proposes to investigate several problems involving the interaction of algebraic geometry and commutative algebra. Asymptotic properties of high powers of ideals will be investigated. Good stable properties tend to hold for high powers. Toroidalization of mappings of algebraic varieties will be studied, continuing previous work of the PI. Ramification of varieties in positive characteristic will be investigated, with an eye towards local uniformization. The author will investigate the related problem of classification of semigroups of valuations on a noetherian ring, and the construction of generating sequences of valuations.Algebra is of increasing importance in modern science and technology. Cutkosky will investigate several problems of theoretical importance in algebra. Undergraduate and graduate students will be trained and young researchers will be mentored through this project.