Uniformity in Commutative Algebra

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

Commutative algebra and algebraic geometry are the study of the algebraic and geometric properties of systems of non-linear equations. By contrast, linear algebra, a fundamental tool of the sciences, is the study of linear systems of equations. Often linear equations do not suffice to describe complex systems, and higher degree polynomials must be used. Commutative algebras provide models, called rings, of systems of polynomial equations where one can add and multiply. The rings studied in this proposal usually come from a system of polynomial equations. 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 throughout this project. This project has a substantial educational component. The PI has served as PhD advisor and postdoc mentor to many researchers, and will continue this level of activity. The PI will investigate several aspects of commutative algebras which all fall under the rubric of uniformity. Three fundamental questions on symbolic powers of ideals and their relationship to regular powers are proposed. Significant partial results have been obtained. The PI will also investigate a fundamental question of Stillman concerning the complexity of equations. A variety of other problems will be studied, including classification of Golod rings, existence of rigid modules, the structure of iterated socles, and the development of Frobenius Betti numbers and their properties. Several of the projects are intended for collaboration with graduate students, postdocs, and young researchers.