Intersection Homogoly, Hodge Theory L2-Cohomology

9820958本提案有两个密切相关的中心主题。 首先是使用霍奇理论的代数品种，以完善研究问题的拓扑性质。 主要研究者将试图证明所谓的Zucker猜想的Hodge理论版本关于Shimura簇的平方可积上同调（1980年; 1987年肯定解决），并将混合Hodge理论应用于分析其边界的上同调。 第二个主题，几乎与第一个主题相反，是通过霍奇理论有效地扩展了以代数簇而闻名的性质的范围。 最好的例子是试图获得一类空间和映射的强大分解定理，而这些空间和映射比代数簇的态射更广泛。当数学家寻找几何空间的例子时，他们经常转向那些在任何数量的变量下多项式方程的解的集合，或者更一般地，许多这样的方程的联立解。 这些空间是数学分支代数几何的基石，代数几何的空间被称为代数簇。 虽然变量是以基本上直接的方式描述的，但通常很难看出这些信息如何转换为空间的重要几何属性。 当W.V.D.霍奇想出了理论，承担他的名字，它主要是为了应用代数几何。 Zucker教授提出的研究将是关于具有连续对称性的空间，以及关于由这些空间上的概念上简单的constructions所构成的代数簇。

9820958This proposal has two closely-related central themes. The first is the use of the Hodge theory for algebraic varieties to refine the study of questions of a topological nature. The Principal Investigator will attempt to prove the Hodge-theoretic version of the so-called Zucker Conjecture on the square-integrable cohomology of Shimura varieties (1980; settled affirmatively 1987), and to apply mixed Hodge theory to the analysis of the cohomology of their boundaries. The second theme, almost the reverse of the first, is to expand the scope of properties that are, effectively via Hodge theory, known for algebraic varieties. The best example of this is the attempt to obtain the powerful Decomposition Theorem for a class of spaces and mappings wider than morphisms of algebraic varieties.When mathematicians look for examples of geometric spaces, they often turn to those that are the set of solutions of a polynomial equation in any number of variables, or more generally, the simultaneous solutions of a number of such equations. These spaces are the building blocks of the branch of mathematics known as algebraic geometry, whose spaces are called algebraic varieties. Although varieties are described in a fundamentally direct manner, it is often hard to see how this information gets converted to the significant geometric properties of the space. When W.V.D. Hodge came up with the theory that bears his name, it was primarily intended for applications in algebraic geometry. The research proposed by Professor Zucker will be on spaces having continuous symmetries, and on the algebraic varieties that are made by conceptually simple contructions on those spaces.