Studies in Commutative Algebra

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Huneke will continue investigating several open questions concerning the theory of Noetherian rings, especially local Noetherian rings or polynomial rings. There are several main thrusts to this work, including investigating singularities via reduction to prime characteristic. Other problems deal with the uniform behavior of symbolic powers on multiple levels, from points in projective space, to primes in local rings, to square-free monomial ideals. A major effort is proposed to understand non-commutative crepant resolutions in broader classes of rings, and to answer several questions concerning the concept of height.The proposed research concerns the theory of commutative rings, which are higher abstract systems where one can add and multiply. The rings arising in this proposal usually comes from a system of polynomial equations. The ring is a type of abstract model where solutions to the equations exist. By studying the properties of this model, one can then better understand the original system of equations. There are two main methods.One is to understand the theory of modules over such rings. Modules are a type of special representation of spaces where the equations hold. Studying these models has been an extremely effective way to study equations. The other main technical method is to study the same basic equations in rings which are reduced modulo a prime number. In such a system, arithmetic becomes easier. For instance modulo 2 means that every even number is thought of as 0, and all odd numbers as 1. This has a number of profound advantages which are used in this proposal.