Mathematical Sciences: Index Theorems on Noncompact Manifolds and for Actions of Noncompact Groups

哈斯克尔教授将研究指数理论 椭圆Fredholm算子在某些非紧流形上，和 非紧群作用于 流形前一个项目将特别关注 流形是奇点的光滑部分的情形 代数簇一个总的目标是将行为与 指数的整体拓扑的品种和几何 关于奇点的信息。第二个项目将是 从理论上用叉积构造法探讨 算子代数在这种情况下，椭圆算子给出 上升到交叉积代数的K-群上的映射， 哈斯克尔教授将试图更精确地解释其本质 阐明。交叉积研究的一个基本对象 是连通李群的卡斯帕罗夫表示环。这 环可以实现为几乎 酉表示之间的缠绕Fredholm算子 最后一个哈斯克尔教授将分析 这个环的某些半单群通过研究交织 具有附加结构的操作符。 涉及数学的三个基本学科领域 在这个项目中，即流形上的微分算子， 李群的表示理论和算子理论 代数第一个主题可能被认为是一个意义深远的 流形上的微积分的衍生（曲面及其高阶）， 尺寸类似物）。微积分的运算， 根据歧管的几何形状定义（距离， 曲率等）结果会产生 关于其拓扑结构的信息（整体形状和构象）。 当流形代数地出现时，比如说作为a的零集， 多项式，可以收获更多的信息， 学习微分算子。李群，以 挪威数学家Sophus Lie，被数学家用来 和物理学家来体现对称性， 允许的运动的一个给定的情况下，保持其 基本特征。李群经常作用在流形上， 有趣的方式，以及这种行动的重要特征 通常可以被某些微分算子检测到。的 算子代数理论，最后，提供了一个技术和 在这个概念框架内， 数学结构可以相互作用。

Professor Haskell will investigate the index theory of elliptic Fredholm operators on certain noncompact manifolds, and the index theory arising from actions of noncompact groups on manifolds. The former project will be particularly concerned with the case in which the manifold is the smooth part of a singular algebraic variety. A general aim is to relate the behavior of indices to the global topology of the variety and to geometric information about the singularities. The second project will be approached via the crossed product construction from the theory of operator algebras. Elliptic operators in this setting give rise to maps on the K-groups of the crossed product algebra, whose nature Professor Haskell will attempt more precisely to elucidate. A fundamental object in the study of crossed products is Kasparov's representation ring of a connected Lie group. This ring can be realized as equivalence classes of almost intertwining Fredholm operators between unitary representations of the group. Professor Haskell will analyze the structure of this ring for certain semisimple groups by studying intertwining operators with additional structure. Three fundamental subject areas in mathematics are involved in this project, namely differential operators on manifolds, the representation theory of Lie groups, and the theory of operator algebras. The first subject may be thought of as a far-reaching outgrowth of calculus on manifolds (surfaces and their higher- dimensional analogues). The operations of calculus, which are defined in terms of the geometry of the manifold (distance, curvature, and the like on a local scale) turn out to yield information about its topology (overall shape and conformation). When the manifold arises algebraically, say as the zero set of a polynomial, there is even more information that can be harvested from studying differential operators. Lie groups, named after the Norwegian mathematician Sophus Lie, are used by mathematicians and physicists to embody symmetry, roughly the totality of all allowable motions of a given situation that preserve its essential features. Lie groups often act on manifolds in interesting ways, and the important features of such an action can often be detected by certain differential operators. The theory of operator algebras, finally, provides a technical and conceptual framework within which apparently disparate mathematical structures can interact with one another.