Methods in the Representation Theory of Local Rings

该奖项支持了交换诺特局部环的代表性理论的进步计划。 对这类环上极大Cohen-Macaulay模的研究是从Artin代数的表示理论发展而来的，它发展了分类和刻画非交换Artin环上模理论的成熟理论方法。 来自非交换理论的经典问题通常可以直接在交换的高维框架中陈述，这个框架配备了自己独特的问题，专门的机器，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两者都非常有成效。然而，许多最强大的阿廷工具本质上是非交换的，因为它们从交换输入产生非交换环，限制了它们在交换代数中的应用。 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的解决方案，将使该领域的有力工具能够应用于关于局部环上极大的Cohen-Macaulay模的基本理论，这个项目位于交换和非交换代数、组合数学和代数几何领域的交叉点。 交换代数的经典目标是通过将代数小工具（称为环）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（也称为箭图）联系起来的理论。 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，特别是理论物理学等不同领域的应用。

This award supports a program for advancement in the representationtheory of commutative Noetherian local rings. The study of maximalCohen--Macaulay modules over such rings has grown out of the theory ofrepresentations of Artin algebras, which has developed sophisticatedtheoretical techniques for classifying and characterizing moduletheories of non-commutative Artinian rings. Classical problems fromthe non-commutative theory can often be stated directly in thecommutative higher-dimensional framework, and this framework comesequipped with its own unique problems, specialized machinery, anddeep, powerful connections with algebraic geometry. The interplaybetween the two areas has been exceptionally productive for both.However, many of the most powerful tools of the Artinian arsenal areessentially noncommutative, in that they produce noncommutative ringseven from commutative input, restricting their use in commutativealgebra. Recent theoretical advances in non-commutative algebraicgeometry have begun to provide methods, including the notion of anon-commutative resolution of singularities, similar to those providedby classical algebraic geometry. Describing these non-commutativeresolutions in terms of tools from the Artinian theory will allow thepowerful tools of that area to be brought to bear on fundamentalconjectures about the maximal Cohen--Macaulay modules overCohen--Macaulay local rings.This project lies at the intersection of the areas of commutative andnon-commutative algebra, combinatorics, and algebraic geometry. Theclassical aim of commutative algebra is to describe the solution setsof systems of polynomial equations by associating to them algebraicgadgets known as rings. Non-commutative algebra, on the other hand,has developed theory for associating rings to directed graphs, alsocalled quivers. The synergy among these topics has enriched all foursubjects, and has led to applications in such varied fields asrobotics, statistics, cryptography, and particularly theoreticalphysics.