Intersection Homogoly, Hodge Theory L2-Cohomology

9820958这项建议有两个密切相关的中心主题。第一个是利用霍奇代数变分理论来改进拓扑性质问题的研究。首席研究员将尝试证明关于志村变量的平方可积上同调的所谓Zucker猜想的Hodge理论版本（1980年；1987年确定），并将混合Hodge理论应用于其边界上同调的分析。第二个主题，几乎与第一个主题相反，是扩大性质的范围，这些性质实际上是通过霍奇理论，以代数变异而闻名的。这方面最好的例子是试图获得比代数变体的态射更宽的一类空间和映射的强大分解定理。当数学家寻找几何空间的例子时，他们通常会求助于任意数量变量的多项式方程的解的集合，或者更一般地说，许多这样的方程的联立解。这些空间是代数几何这一数学分支的组成部分，代数几何的空间被称为代数变体。尽管以一种基本直接的方式描述了变量，但通常很难看到如何将这些信息转换为空间的重要几何属性。当W.V.D. Hodge提出以他的名字命名的理论时，它主要是为了应用于代数几何。Zucker教授提出的研究将是关于具有连续对称性的空间，以及在这些空间上由概念上简单的构造产生的代数变异。

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.