Mathematical Sciences: Variations on Hodge Structure

本研究将发展复代数变族的相对霍奇理论。更准确地说，与任何种类相关的一类物体被称为霍奇结构的分层变异。给定一个代数映射，沿着纤维的上同调的数据连同它们的混合Hodge结构应该产生这样一个对象。首席研究员将检查这种情况，以及考虑极化混合霍奇结构的类别。这是代数几何领域的研究，它是现代数学中最古老的部分之一。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。目前，该领域不仅使用代数方法，还使用分析和拓扑方法。它在这些数学领域以及理论计算机科学和机器人技术中得到了应用。

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.