Mathematical Sciences: Non-Commutative Differential Geometry

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At a technical level, Professor Wodzicki's project is largely concerned with computing the cyclic homology of algebras that come, variously, from differential or algebraic geometry or from functional analysis. Algebras of differential operators, especially on smooth analytic varieties, play a major role in this research, but there is also an important aspect of it that deals with C*-algebras (certain algebras of operators on Hilbert space) and their K-theory and cyclic homology. The mathematical research here envisioned is a far-reaching outgrowth of calculus on manifolds, roughly 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, relevant to the work of Professor Wodzicki, for extracting this sort of information from differential operators involves assembling the operators into algebraic structures and then using algebraic machinery. The further development of this machinery, two important and intimately related parts of which are called K-theory and cyclic homology, and its use in elucidating geometric and topological structure constitute the agenda of the research supported by this award.