Intersection Homogoly, Hodge Theory L2-Cohomology

9820958本提案有两个密切相关的中心主题。第一种是使用代数簇的霍奇理论来改进对拓扑性问题的研究。主要研究人员将尝试证明所谓的Zucker猜想的Hodge理论版本(1980；1987年确定)，并将混合Hodge理论应用于它们边界的上同调分析。第二个主题，几乎与第一个主题相反，是扩大那些通过霍奇理论有效地以代数簇而闻名的性质的范围。最好的例子是试图得到一类比代数簇的态射更广的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，他们经常求助于那些多项式方程的任意变量的解的集合，或者更一般地，许多这样的方程的联立解。这些空间是被称为代数几何的数学分支的构建块，其空间被称为代数簇。尽管从根本上直接地描述了变化，但通常很难看出这种信息如何转化为空间的重要几何属性。当W·V·D·霍奇提出以他的名字命名的理论时，它主要是为了在代数几何中应用。Zucker教授提出的研究将是关于具有连续对称的空间，以及由这些空间上的概念上的简单构造构成的代数簇。

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.