Methods in the Representation Theory of Local Rings

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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.