Methods in the Representation Theory of Local Rings

点击翻译按钮获取中文摘要

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.