Methods in the Representation Theory of Local Rings

这一奖项支持在交换Notherian局部环的表示理论方面取得进展的计划。这类环上的极大Cohen-Macaulay模的研究源于Artin代数的表示理论，它发展了用于分类和刻画非交换Artin环的模理论的成熟的理论方法。来自非对易理论的经典问题通常可以在交换的高维框架中直接描述，并且这个框架配备了自己独特的问题、专门的机械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格外富有成效。然而，Artin武器库中许多最强大的工具本质上是非对易的，因为它们即使从交换输入产生非对易环，限制了它们在交换代数中的使用。最近非对易代数几何的理论进展已经开始提供方法，包括非对易奇点分解的概念，类似于经典代数几何所提供的方法。用Artin理论中的工具来描述这些非对易分解将允许该领域的强大工具应用于关于Cohen-Macaulay局部环上的极大Cohen-Macaulay模的基本猜想。这个项目位于对易和非对易代数、组合学和代数几何领域的交集。交换代数的经典目的是通过将称为环的代数小工具与多项式方程组相关联来描述多项式方程组的解集。另一方面，非交换代数发展了将环与有向图相关联的理论，也称为箭图。这些主题之间的协同作用丰富了所有四门学科，并导致在机器人、统计学、密码学，特别是理论物理等不同领域的应用，其中非对易方法是量子力学、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.