Hodge Theory, Commutative Algebra, and Geometry

格林教授将继续研究代数圈，特别是Abel-Jacobi映射象、高阶Abel-Jacobi映射象、算术Hodge结构的扩张类和算术Gauss-Manin联络。该项目还将考虑词典初始理想的几何用途，以及它们与某些规律性问题的联系。最后，格林教授将返回研究凸体的Steiner多项式的根。他希望最近的进展将允许将已建立的曲率不等式推广到更高的维度。这是代数几何中的一个项目。在现代数学的这个重要分支中，像曲线和曲面这样的几何对象是用代数结构来建模的。新的代数对象的抽象性质使数学家更容易研究它们。然而，新发现的代数模型的性质可以转化为几何曲线和曲面的新性质。这种几何代数方法的一个关键是研究对称性。数学对称比我们熟悉的几何对称更复杂、更抽象，但它们在简化我们对整个物体的看法方面也同样有用。霍奇结构是这个项目的一个焦点，它是对分层代数对称的一种系统计算。

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.