Hodge Theory, Commutative Algebra, and Geometry

9970307教授绿色将继续他的调查代数圈，特别是形象的阿贝尔-雅可比地图，更高的阿贝尔-雅可比地图，扩大类的算术霍奇结构，和算术高斯-马宁连接。 该项目还将考虑字典式初始理想的几何用途及其与某些正则性问题的联系。 最后教授绿色将返回研究根施泰纳多项式的凸机构。 他希望最近的进展将允许建立的曲率不等式推广到更高的维度。这是一个代数几何的项目。 在这个现代数学的重要分支中，曲线和曲面等几何对象使用代数构造来建模。 新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属性转化回几何曲线和曲面的新属性。 这种代数方法的一个关键是几何对称性的研究。 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同样有用。 霍奇结构是这个项目的一个重点，是一种分层代数对称性的系统会计。

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.