Mathematical Sciences: Commutative Algebra

本课题主要研究局部环上的交换代数及其奇点。主题包括Cohen-Macaulay局部环中m-素数理想的Hilbert函数和Hilbert多项式，以及局部环的双态扩张和完备化。这项工作在代数几何中有具体的应用，特别是在奇点分解问题上。交换环是具有交换加法和交换乘法的代数结构。这些结构在整个数学中都存在。常见的例子包括多项式环和有理数扩域中的代数整数环。这个项目涉及到关于交换环结构的各种问题。

This project is concerned with research in commutative algebra on local rings and their singularities. Topics include the Hilbert function and Hilbert polynomial of an m-primary ideal in a Cohen-Macaulay local ring, and birational extensions and completions of local domains. The work has specific applications to algebraic geometry, particularly to the problem of resolution of singularities. Commutative rings are algebraic structures possessing a commutative addition and a commutative multiplication. These structures occur throughout mathematics. Common examples include polynomial rings, and rings of algebraic integers in extension fields of the rationals. This project is concerned with a variety of questions on the structure of commutative rings.