Topics in Singularity Theory and Commutative Algebra

Cutkosky将研究四个项目：正特征曲面的环面化、局部单项化和分支、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，通过爆破整环和值域上的非奇异子簇，可以使簇的任何显性态射成为环面映射或局部单项映射。Cutkosky将把他最近证明猜想的工作推广到3倍，从而得到猜想的一般证明。关于正特征中的分支问题，作者用Oliver Pilant找到了特征p曲面芽之间映射的稳定形式.Cutkosky建议改进这些形式的表述，并将它们应用于研究特征p中曲面映射的分支.Cutkosky将继续他的工作，计算理想幂函子上Hilbert型函数的渐近性质.Cutkosky还试图理解半群，它可以作为支配Notherian局部域的(可能Noether)赋值的值半群出现。代数在现代科学和技术中越来越重要。卡特科斯基将在交换代数和代数几何的一般领域中研究几个具有理论意义的问题。主要目的是了解多项式映射的结构和多项式方程组。卡特科斯基将通过这个项目培养本科生和研究生，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.