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

Haskell教授将研究某些非紧流形上的椭圆Fredholm型算子的指标理论，以及非紧群在流形上的作用所产生的指标理论。前一个项目将特别关注流形是奇异代数簇的光滑部分的情况。一般目的是将指数的行为与变种的全局拓扑以及奇点的几何信息联系起来。第二个方案将通过从算子代数理论出发的交叉积构造来实现。在这种情况下，椭圆算子产生了交叉积代数的K-群上的映射，哈斯克尔教授将尝试更准确地阐明它的性质。研究交叉积的一个基本对象是连通李群的Kasparov表示环。这个环可以被实现为群的酉表示之间几乎相互缠绕的Fredholm型算子的等价类。哈斯克尔教授将通过研究具有附加结构的交织算符来分析某些半单群的环的结构。该项目涉及三个基本的数学学科领域，即流形上的微分算子、李群的表示理论和算子代数理论。第一个主题可能被认为是微积分在流形(曲面及其高维类似物)上的深远发展。根据流形的几何(局部尺度上的距离、曲率等)定义的微积分运算最终产生关于流形的拓扑(整体形状和构象)的信息。当流形以代数方式出现时，例如作为多项式的零集，从研究微分算子中可以获得更多的信息。李群是以挪威数学家索菲斯·李的名字命名的，数学家和物理学家使用它来体现对称性，对称性大致是给定情况下所有允许的运动的总和，这些运动保留了它的基本特征。李群经常以有趣的方式作用于流形，并且这种作用的重要特征通常可以通过某些微分算子来检测。最后，算子代数理论提供了一个技术和概念框架，在这个框架内，明显不同的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.