Topics in Commutative Algebra, Singularities, and Valuations

本专题将探讨代数几何及相关数学领域的基础问题。代数几何是研究多项式方程的解的学科;它在数学、科学和工程的其他领域有许多应用。该项目将侧重于任意域上的代数变换的结构以及真实的和复杂的解析变换。本计画的研究领域包括交换代数、代数几何、赋值理论、奇异性理论等，其研究成果具有实际应用的潜力。代数映射和解析映射的局部结构将被研究，特别是关于映射的分支和好的稳定形式，在沿着一个赋值或一个顶点足够的爆破之后。激发问题是局部一致化，解决奇异性和正特征的equisingularity，以及代数和解析映射的整体结构。

This project will investigate foundational problems in algebraic geometry and related areas of mathematics. Algebraic geometry is the study of solutions to polynomial equations; it has many applications in other areas of mathematics, the sciences, and engineering. The project will focus on the structure of algebraic transformations over arbitrary fields and real and complex analytic transformations. The results of this research have potential for practical application to computer science.The research in this project is in the areas of commutative algebra, algebraic geometry, valuation theory, and singularity theory. The local structure of algebraic and analytic mappings will be investigated, especially with regard to ramification and good stable forms of mappings after enough blowups along a valuation or an étoile. Motivating problems are local uniformization, resolution of singularities and equisingularity in positive characteristic, and the global structure of algebraic and analytic maps.