Mathematical Sciences: Hodge Theory

Kaplan和Cattani教授将研究带有附加结构的平坦全纯丛的性质的几个问题，例如Hodge结构的极化变化或Higgs场。他们计划研究这类丛的局部单调性，并将其应用于模簇的精细结构、Hodge结构的最大变差的刻画以及调和度量的构造。这项研究是在代数几何和微分几何的界面上进行的。所研究的对象是代数簇，本质上是多项式的零集，但用于研究这些对象的方法是解析的。这种代数和微分几何的混合在攻击那些没有屈服于纯代数方法的问题上特别成功。

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.