Mathematical Sciences: Non-Commutative Differential Geometryof the Deformations of Commutative Rings

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This research is concerned with the noncommutative differential geometry of the deformations of commutative algebras; algebras of differential operators; symbols of pseudo-differential operators; coordinate algebras of quantum groups and spaces; and differential operators on quantum manifolds. The principal investigator will develop a Riemann-Roch theorem for quantum manifolds through his work on the cyclic homology theory of quantum groups and spaces. Applications of these results to arithmetic and conformal quantum field theory will also be explored. The research in this project is an outgrowth of calculus on manifolds. Roughly speaking, manifolds are surfaces and their higher-dimensional analogues. Differential operators, which come from calculus, see the manifold only in small pieces, but nevertheless contain information about its overall shape and conformation. One strategy for extracting this sort of information from differential operators involves assembling the operators into algebraic structures and then using algebraic machinery. This project is concerned with the development of this machinery.