Mathematical Sciences: Variations on Hodge Structure

这项研究将发展一个相对霍奇理论， 复代数簇的族 更准确地说， 与任何种类的物体相关联的一类物体称为分层的 Hodge结构的变化。 给定一个代数地图， 沿着纤维的上同调以及它们的混合Hodge 结构应该产生这样一个物体。 校长 调查人员将审查这一情况，并考虑 一类可极化的混合Hodge结构。 这是代数几何领域的研究， 现代数学中最古老的部分之一 在其起源中，它 处理的图形，可以定义在平面上的最简单的 方程，即多项式。 如今，该领域利用 方法不仅从代数，而且从分析和拓扑。 它在这些数学领域中得到应用， 在理论计算机科学和机器人学中。

This research will develop a relative Hodge theory of families of complex algebraic varieties. More precisely, there is associated to any variety a category of objects called stratified variations of Hodge structure. Given an algebraic map, the data of the cohomology along the fibers together with their mixed Hodge structures should give rise to such an object. The principal investigator will examine this situation, as well as consider the category of polarizable mixed Hodge structures. This is research in the field of algebraic geometry, which is one of the oldest parts of modern mathematics. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but also from analysis and topology. It is finding application in those areas of mathematics as well as in theoretical computer science and in robotics.