Mathematical Sciences: Chern Characters and Correction Termsin Index Theory on Noncompact Manifolds

现代科学最基本的发展之一 就是量子力学理论 的尝试 理解物理学和基础数学 直接导致了两个重要领域的前沿 现代分析，表示论和算子 代数 表征理论是一种利用 固有的物理对称性，而运营商的概念 代数的发明是为了提供正确的框架， 量子化 最近，有一个很大的扩张， 我们对量子化背后数学的理解。 表示论和算子代数 最近在一个强有力的理论中， 著名的Atiyah-Singer指数定理。 这个不断发展的领域是 常被称为非交换微分几何。的 表示论在局部上进入群作用 对称空间以及奇异代数 多样性在表征理论中的作用 当前推进点 从有限离散群到无限离散群， 紧流形到非紧奇异簇这些是 通过光滑的奇异点 多样性常常作为无限行为的衍生物而出现， 非紧流形上的离散群 的几何形状 空间反映在群的表示理论中， 非紧流形上微分算子的分析 这些都是解决现代指数理论的问题。 哈斯克尔教授是一位年轻的研究员， 这需要大量的数学知识 领域 调查员先前的工作，部分是合作， 为下一步工作打下了坚实的基础。 哈斯克尔教授将研究几何的指数理论 非紧流形上的椭圆Fredholm算子 他将 然后将该研究应用于同变指数理论， 非紧群作用，指数理论的光滑部分， 代数簇和局部对称空间。

One of the most fundamental developments in modern science has been the theory of quantum mechanics. The attempt to understand both the physics and the underlying mathematics has led directly to current frontiers in two important areas of Modern Analysis, representation theory and operator algebras. Representation theory is a means of exploiting the inherent physical symmetries, while the concept of operator algebras was invented to provide the correct framework for quantization. Recently, there has been a great expansion in our understanding of the mathematics underlying quantization. Representation theory and operator algebras have been brought together recently in a powerful theory that centers around the celebrated Atiyah-Singer index theorem. This evolving field is often referred to as noncommutative differential geometry. The representation theory enters in the group actions on locally symmetric spaces as well as the role that singular algebraic varieties play in representation theory. Current advances point to the movement from finite to infinite discrete groups and from compact manifolds to noncompact singular varieties. These are tied together through the fact that smooth points of singular varieties often arise as quotients of actions of infinite discrete groups on noncompact manifolds. The geometry of the space is reflected in the representation theory of the group and analysis of differential operators on the noncompact manifold. These are the questions that address modern index theory. Professor Haskell is a young researcher who has mastered the broad range of mathematics necessary to contribute to this field. The investigator's prior work, in part collaborative, has laid a solid foundation for undertaking the following work. Professor Haskell will study the index theory of geometric elliptic Fredholm operators on noncompact manifolds. He will then apply this research to equivariant index theory of noncompact group actions, to index theory on the smooth parts of algebraic varieties, and on locally symmetric spaces.