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

现代科学最基本的发展之一是量子力学理论。对物理学和基本数学的理解的尝试直接导致了现代分析的两个重要领域--表示理论和算子代数--的当前前沿。表示理论是一种利用内在物理对称性的方法，而算子代数的概念是为了提供量子化的正确框架而发明的。最近，我们对量子化背后的数学原理的理解有了很大的扩展。表示理论和算子代数最近在一个强大的理论中结合在一起，该理论以著名的Atiyah-Singer指数定理为中心。这种不断发展的领域通常被称为非对易微分几何。表象理论涉及到局部对称空间上的群作用以及奇异代数簇在表象理论中的作用。目前的进展指向从有限离散群到无限离散群以及从紧致流形到非紧致奇异簇的运动。通过奇异簇的光滑点经常作为无限离散群在非紧流形上的作用商出现这一事实，这些联系在一起。空间的几何反映在非紧流形上的群表示理论和微分算子分析中。这些都是解决现代指数理论的问题。哈斯克尔教授是一名年轻的研究人员，他掌握了为这一领域做出贡献所需的广泛数学知识。调查员先前的工作，部分是合作的，为开展以下工作奠定了坚实的基础。Haskell教授将研究非紧流形上几何椭圆Fredholm型算子的指标理论。然后，他将把这项研究应用于非紧群作用的等变指数理论，代数簇的光滑部分的指数理论，以及局部对称空间。

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.