Mathematical Sciences: Hodge Theory

卡普兰和卡塔尼教授将研究几个 关于平坦全纯丛性质的问题 携带额外的结构，例如霍奇的偏振变化， 结构或希格斯场他们计划研究当地的单值性 这种束与应用程序的模块化的精细结构， 品种，霍奇的最大变化的表征 结构，以及谐波度量的构造。 这项研究处于代数和 微分几何所研究的对象是代数的 品种，基本上是零套多项式，但 用于研究这些对象的方法是分析性的。这种混合的 代数和微分几何已经特别 成功地解决了那些不屈服于纯粹的 代数方法

Professors Kaplan and Cattani will investigate several questions concerning the properties of flat holomorphic bundles carrying additional structures such as polarized variation of Hodge structure or a Higgs field. They plan to study the local monodromy of such bundles with applications to the fine structure of modular varieties, the characterization of maximal variations of Hodge structure, and the construction of harmonic metrics. This research lies at the interface of algebraic and differential geometry. The objects under study are algebraic varieties, essentially the zero sets of polynomials, but the methods used to study these objects are analytic. This mix of algebraic and differential geometry has been particularly successful in attacking problems which did not yield to purely algebraic methods.