Mathematical Sciences: Equivariant Index Theory on Non-Compact Manifolds

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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.