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。对物理学和基础数学的理解的尝试直接导致了现代分析中两个重要领域的前沿：表示理论和算子代数。表征理论是利用固有的物理对称性的一种手段，而算子代数的概念是为了提供量化的正确框架而发明的。最近，我们对量化背后的数学的理解有了很大的扩展。最近，以著名的Atiyah-Singer指数定理为中心，表示理论和算子代数被结合在一起，形成了一个强大的理论。这个不断发展的领域通常被称为非交换微分几何。表征理论引入了群在局部对称空间上的作用，以及奇异代数变体在表征理论中的作用。当前的进展指出了从有限离散群到无限离散群以及从紧流形到非紧奇异变的运动。这些是通过奇异变量的光滑点经常作为非紧流形上无限离散群作用的商而出现的事实联系在一起的。非紧流形上的群表示理论和微分算子分析反映了空间的几何性质。这些都是现代指数理论要解决的问题。哈斯克尔教授是一位年轻的研究人员，他掌握了为这一领域做出贡献所必需的广泛的数学知识。研究者之前的部分合作工作为开展后续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型Fredholm算子的指数理论。然后，他将把这项研究应用于非紧群作用的等变指标理论、代数变簇光滑部分的指标理论和局部对称空间的指标理论。

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.