Mathematical Sciences: Equivariant Index Theory on Non-Compact Manifolds

Fox教授的课题是关于微分算子的指标理论和李群的表示理论的等测度。前一个主题可以被认为是流形（曲面及其高维类似物）上的微积分的深远发展。微积分的基本运算，微分和积分，在这种情况下以伪微分算子的形式打包在一起。这些运算符是根据流形的几何形状（局部尺度上的距离、曲率等）定义的，结果产生有关其拓扑结构的信息（总体形状和构象）。李群，另一个研究领域，作为对称群出现在许多数学情况中。一个例子（在目前的情况下不太适合，因为它恰好是紧致的）是一个球体的旋转群，其中群操作包括一个运动跟随另一个运动。当球对称存在时，这种特殊群的知识是非常有用的，李群的表示理论通常可以被视为对称的系统研究。福克斯项目的这两个主要方面紧密地交织在一起，因为要追求的指标理论的大部分是等价的，相对于一些（不一定是紧的）李群作用于流形。Kasparov的非紧群表示环是一个关注的焦点。本文将寻求狄拉克型几何算子在群作用下不变的指标公式，并将渐近伪微分算子的微积分推广到非紧流形。其中一些工作与物理学中量子霍尔效应的数学模型有关。

Professor Fox's project is concerned in about equal measure with index theory for differential operators and with the representation theory of Lie groups. The former subject may be thought of as a far-reaching outgrowth of calculus on manifolds (surfaces and their higher-dimensional analogues). The basic operations of calculus, differentiation and integration, are in this setting packaged together in the form of pseudodifferential operators. These operators, 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). Lie groups, the other area of investigation, arise in many mathematical situations as groups of symmetries. An example (not quite apt in the present context because it happens to be compact) is the group of rotations of a sphere, where the group operation consists of following one motion by another. Knowledge of this particular group is very helpful whenever spherical symmetry is present, and the representation theory of Lie groups generally may be regarded as the systematic study of symmetry. These two main aspects of Fox's project are closely interwoven in that much of the index theory to be pursued is equivariant with respect to some (not necessarily compact) Lie group acting on a manifold. Kasparov's representation ring for non-compact groups is a focus of interest. Index formulas for Dirac-type geometric operators invariant under group actions will be sought, as will an extension of the calculus of asymptotic pseudodifferential operators to non-compact manifolds. Some of this work bears on a mathematical model of the quantum Hall effect in physics.