Uniformity in Commutative Algebra

交换代数和代数几何是研究非线性方程组的代数和几何性质。相比之下，线性代数，科学的基本工具，是研究线性方程组。通常线性方程不足以描述复杂的系统，必须使用更高次的多项式。交换代数提供了多项式方程组的模型，称为环，其中可以进行加法和乘法。在这个建议中研究的环通常来自多项式方程组。 通过研究这个模型的性质，人们可以更好地理解原始方程组。主要有两种方法。一个是理解这种环上的模理论。模是方程成立的空间的一种特殊表示。研究这些模型是研究方程的一种非常有效的方法。另一个主要的技术方法是研究环中相同的基本方程，这些方程以素数为模约化。在这样的系统中，算术变得更容易。例如，模2意味着每个偶数都被认为是0，所有奇数都被认为是1。这有许多深刻的优势，在整个项目中使用。这个项目有一个重要的教育部分。PI已经担任了许多研究人员的博士生导师和博士后导师，并将继续这一水平的活动。PI将研究交换代数的几个方面，这些方面都属于一致性的范畴。 提出了关于理想的符号幂及其与正则幂的关系的三个基本问题。已经取得了重要的部分成果。PI还将研究Stillman关于方程复杂性的一个基本问题。 各种其他问题将研究，包括分类的Golod环，存在的刚性模块，结构的迭代socles，和发展的Frobenius贝蒂数及其性质。其中几个项目旨在与研究生，博士后和年轻的研究人员合作。

Commutative algebra and algebraic geometry are the study of the algebraic and geometric properties of systems of non-linear equations. By contrast, linear algebra, a fundamental tool of the sciences, is the study of linear systems of equations. Often linear equations do not suffice to describe complex systems, and higher degree polynomials must be used. Commutative algebras provide models, called rings, of systems of polynomial equations where one can add and multiply. The rings studied in this proposal usually come from a system of polynomial equations. By studying the properties of this model, one can then better understand the original system of equations. There are two main methods. One is to understand the theory of modules over such rings. Modules are a type of special representation of spaces where the equations hold. Studying these models has been an extremely effective way to study equations. The other main technical method is to study the same basic equations in rings which are reduced modulo a prime number. In such a system, arithmetic becomes easier. For instance modulo 2 means that every even number is thought of as 0, and all odd numbers as 1. This has a number of profound advantages which are used throughout this project. This project has a substantial educational component. The PI has served as PhD advisor and postdoc mentor to many researchers, and will continue this level of activity.The PI will investigate several aspects of commutative algebras which all fall under the rubric of uniformity. Three fundamental questions on symbolic powers of ideals and their relationship to regular powers are proposed. Significant partial results have been obtained. The PI will also investigate a fundamental question of Stillman concerning the complexity of equations. A variety of other problems will be studied, including classification of Golod rings, existence of rigid modules, the structure of iterated socles, and the development of Frobenius Betti numbers and their properties. Several of the projects are intended for collaboration with graduate students, postdocs, and young researchers.