Studies in Commutative Algebra

Huneke将继续研究有关Notherian环理论的几个公开问题，特别是局部Notherian环或多项式环。这项工作有几个主要的推动力，包括通过约化为素数特征来研究奇点。其他问题涉及符号幂在多个层次上的一致行为，从射影空间中的点，到局部环中的素数，再到无平方的单项理想。主要的工作是在更广泛的环类中理解非交换的可逆分解，并回答与高度概念有关的几个问题。所提出的研究涉及交换环的理论，这是一个可以进行加法和乘法的更高抽象系统。在这个提议中出现的环通常来自多项式方程组。环是一种抽象模型，其中存在方程的解。通过研究这个模型的性质，人们可以更好地理解原始的方程组。有两种主要的方法，一种是理解这种环上的模理论。模是方程所在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。另一种主要的技术方法是研究模为素数的环中相同的基本方程。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，而所有奇数都被认为是1。这有许多在这个提议中使用的深刻的优点。

Huneke will continue investigating several open questions concerning the theory of Noetherian rings, especially local Noetherian rings or polynomial rings. There are several main thrusts to this work, including investigating singularities via reduction to prime characteristic. Other problems deal with the uniform behavior of symbolic powers on multiple levels, from points in projective space, to primes in local rings, to square-free monomial ideals. A major effort is proposed to understand non-commutative crepant resolutions in broader classes of rings, and to answer several questions concerning the concept of height.The proposed research concerns the theory of commutative rings, which are higher abstract systems where one can add and multiply. The rings arising in this proposal usually comes from a system of polynomial equations. The ring is a type of abstract model where solutions to the equations exist. 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 in this proposal.