Mathematical Sciences: Non-commutative Differential Geometryof Deformations of Commutative Rings: Operations Index Theorems and Characteristic Classes

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This award is concerned with the study of the noncommutative differential geometry of deformations of commutative rings. The principal investigator has proved the algebraic index theorem for any deformation of the ring of functions on a symplectic manifold. The main technical tool was the theory of operations in cyclic homology. He plans to develop this construction further, which would yield the algebraic index theorem for families and foliations, the algebraic analogue of the Connes-Moscovici index theorem for an equivariant quantization of a ring of functions, and index and character formula representations of quantizations of commutative rings. These methods are also expected to yield various generalizations of the Riemann-Roch-Grothendieck theorem. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.