Topics in Singularity Theory and Commutative Algebra

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Cutkosky will investigate four projects: toroidalization, local monomialization and ramification of surfaces in positive characteristic, asymptotic properties of powers of ideals and semigroups of valuations on local rings. In toroidalization, the main conjecture is to show that any dominant morphismof varieties can be made into a toroidal mapping, or a local monomial mapping, by blowing up nonsingular subvarieties in the domain and range. Cutkosky will extend his recent work proving the conjecture for 3-folds,towards a general proof of the conjecture. On the problem of ramification in positive characteristic,the proposer has found with Oliver Piltant stable forms of mappings between germs of characteristic p surfaces.Cutkosky proposes to refine the formulation of these forms, and apply them to a study of ramification of maps of surfaces in characteristic p. Cutkosky will continue his work on computing asymptotic properties of Hilbert type functions on functors of powers of ideals. Cutkosky also seeks to understand the semigroups which canoccur as the value semigroup of a (possible noetherian) valuation dominating a noetherian local domain.Algebra is of increasing importance in modern science and technology. Cutkosky will investigateseveral problems of theoretical importance in the general area of Commutative Algebra and Algebraic Geometry.The main objective is to understand the structure of polynomial mappings, and systems of polynomial equations. Cutkosky will train undergraduate and graduate students, and mentor young researchers through this project.