Algebraic Geometry, Commutative Algebra, and Algorithmic Methods

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

David Eisenbud will work on three groups of problems from commutative algebra and algebraic geometry: The use of exterior algebra methods in projective geometry and their extension to toric geometry; the study of infinite free resolutions; and the study of codimensions of determinantal (and related) ideals. This last includes the circle of problems around the existence of vector bundles of low rank on projective spaces. His activity will also include the development of fundamental algorithms in computational algebraic geometry and their implementation in the Macaulay2 package written by Grayson and Stilllman. By far the largest portion of the project budget is devoted to the support of graduate students. This and the computational tools produced by the project will broaden its impact. Algebraic geometry deals with geometric forms defined by simple equations (polynomials). The subject is central in mathematics because these forms include the fundamental examples in most disciplines of mathhematics; and it is important in the applications of mathematics because these forms provide the models most often used in representing nature in equations or in the computer. Over the last 150 years the group of techniques called commutative algebra has been developed to study and unify this and problems arising in number theory. Much more recently it was realized that an extension of commutative algebra into the noncommutative domain of exterior algebras gives a fruitful new approach to some old problems in this field. Independently, computers and algorithms became powerful enough to help research in algebraic geometry. Eisenbud's proposed work has to do with some of the most classical problems in algebraic geometry as well as with new uses of exterior algebra and computation in this domain, and he will also continue his work developing computational tools for others to use. The largest part of his grant will be devoted to the support of graduate students, whom he will train in these techniques and methods.