Topics in Commutative Algebra, Singularities, and Valuations

这个项目将研究代数几何和相关数学领域的基本问题。代数几何是研究多项式方程的解；它在数学、科学和工程的其他领域有许多应用。该项目将专注于任意域上的代数变换以及实数和复数解析变换的结构。本课题的研究内容涉及交换代数、代数几何、赋值理论、奇点理论等多个领域。我们将研究代数映射和解析映射的局部结构，特别是关于沿赋值或Aétoile有足够爆破的映射的分支和良好的稳定形式。激励问题是局部一致性、正特征中奇异性和等价性的分解以及代数映射和解析映射的全局结构。

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.