Methods in the Representation Theory of Local Rings

该奖项支持一个项目，以推进交换诺瑟局部环的表示理论。这种环上的极大Cohen- Macaulay模的研究是从Artin代数的表示理论中发展出来的，该理论已经发展出了用于分类和表征非交换Artinian环的模理论的复杂理论技术。来自非交换理论的经典问题通常可以直接在可交换的高维框架中表述，并且该框架配备了自己独特的问题，专门的机制以及与代数几何的深刻而强大的联系。这两个领域之间的相互作用对双方来说都是非常有成效的。然而，阿提尼武器库中许多最强大的工具本质上是非交换的，因为它们甚至从交换输入产生非交换环，限制了它们在交换代数中的使用。非交换代数几何的最新理论进展已经开始提供方法，包括奇点的非交换解析的概念，类似于经典代数几何提供的方法。用Artinian理论的工具来描述这些非交换解析，将允许该领域的强大工具用于Cohen- Macaulay局部环上最大Cohen- Macaulay模的基本猜想。这个项目位于交换代数和非交换代数、组合学和代数几何领域的交叉点。交换代数的经典目标是描述多项式方程组的解集，方法是将代数小环与多项式方程组联系起来。另一方面，非交换代数发展了将环与有向图联系起来的理论，也称为颤波。这些主题之间的协同作用丰富了所有四个学科，并导致在机器人，统计学，密码学等不同领域的应用，特别是理论物理，其中非交换方法是量子力学，弦理论和基本粒子研究等学科的核心。

This award supports a program for advancement in the representation theory of commutative Noetherian local rings. The study of maximal Cohen--Macaulay modules over such rings has grown out of the theory of representations of Artin algebras, which has developed sophisticated theoretical techniques for classifying and characterizing module theories of non-commutative Artinian rings. Classical problems from the non-commutative theory can often be stated directly in the commutative higher-dimensional framework, and this framework comes equipped with its own unique problems, specialized machinery, and deep, powerful connections with algebraic geometry. The interplay between the two areas has been exceptionally productive for both. However, many of the most powerful tools of the Artinian arsenal are essentially noncommutative, in that they produce noncommutative rings even from commutative input, restricting their use in commutative algebra. Recent theoretical advances in non-commutative algebraic geometry have begun to provide methods, including the notion of a non-commutative resolution of singularities, similar to those provided by classical algebraic geometry. Describing these non-commutative resolutions in terms of tools from the Artinian theory will allow the powerful tools of that area to be brought to bear on fundamental conjectures about the maximal Cohen--Macaulay modules over Cohen--Macaulay local rings.This project lies at the intersection of the areas of commutative and non-commutative algebra, combinatorics, and algebraic geometry. The classical aim of commutative algebra is to describe the solution sets of systems of polynomial equations by associating to them algebraic gadgets known as rings. Non-commutative algebra, on the other hand, has developed theory for associating rings to directed graphs, also called quivers. The synergy among these topics has enriched all four subjects, and has led to applications in such varied fields as robotics, statistics, cryptography, and particularly theoretical physics, where non-commutative methods are central to such subjects as quantum mechanics, string theory and the study of fundamental particles.