Topics in Singularity Theory and Commutative Algebra

Cutkosky将研究四个项目：环面化、正特征曲面的局部一元化和分枝、理想幂的渐近性质和局部环上赋值的半群。在环面化中，主要的猜想是通过在定域和值域上吹出非奇异的子变种，来证明任何显性模态变体都可以变成环面映射或局部单项式映射。Cutkosky将扩展他最近证明3倍猜想的工作，朝着猜想的一般证明方向发展。在正特征的分枝问题上，作者利用Oliver Piltant找到了特征p曲面胚芽间映射的稳定形式。Cutkosky建议完善这些形式的表述，并将其应用于特征p的曲面映射的分支研究。Cutkosky将继续他的工作，计算希尔伯特型函数在理想幂函数上的渐近性质。Cutkosky还试图理解可以作为支配noether局部域的（可能的noether）估值的值半群出现的半群。代数在现代科学技术中占有越来越重要的地位。卡特考斯基将研究交换代数和代数几何中几个重要的理论问题。主要目的是了解多项式映射的结构，以及多项式方程组。Cutkosky将通过这个项目培训本科生和研究生，并指导年轻的研究人员。

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.