Mathematical Sciences: Non-Commutative Harmonic Analysis

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This project is mathematical research concerning smooth manifolds and certain differential operators associated with them. A manifold is a surface (or higher-dimensional analog) with the commonsensical property that in the near vicinity of any point, a coordinate system can be introduced that behaves like the ordinary system in two- (or n-) dimensional space. Differential operators can be defined in terms of these local coordinates. Certain numbers that can be extracted from the operators tell us very interesting things about the underlying manifold. Group representation theory comes into the picture when the manifold is a symmetric space. The noncommutative perspective afforded by recent developments in the theory of operator algebras is very helpful when the manifold carries some additional structure, such as a foliation with pathological leaf space. One specific objective of the project is to apply methods of harmonic analysis on semisimple Lie groups to the study of eta invariants and analytic torsion of locally symmetric manifolds with non-positive sectional curvature, aiming to express these invariants in terms of the geodesic flow. Another aspect of the work involves the Chern character and exponentially summable Fredholm modules in noncommutative differential geometry, with a view towards applications to operator algebras associated with certain groups.