Hodge Theory, Commutative Algebra, and Geometry

点击翻译按钮获取中文摘要

9970307Professor Green will continue his investigations of algebraic cycles, and in particular the image of the Abel-Jacobi map, higher Abel-Jacobi maps, extension classes of arithmetic Hodge structures, and the arithmetic Gauss-Manin connection. The project will also consider the geometric uses of lexicographic initial ideals and their connection with certain regularity questions. Finally Professor Green will return to the study the roots of Steiner polynomials of convex bodies. He hopes that recent advances will allow the generalization of the established curvature inequalities to higher dimensions.This is a project in algebraic geometry. In this important branch of modern mathematics, geometric objects like curves and surfaces are modeled using algebraic constructions. The abstract nature of the new algebraic objects makes them easier for mathematicians to study. Nevertheless, newly discovered properties of the algebraic models translate back to new properties of the geometric curves and surfaces. One key to this algebraic approach to geometry is the study of symmetries. Mathematical symmetries are more complex and abstract than the familiar geometric symmetries, but they are just as useful in simplifying our view of the an entire object. The Hodge structures that are a focus of this project are a kind of systematic accounting of tiered algebraic symmetries.