Studies in Noncommutative Geometry

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ABSTRACT Moscovici The proposed research involves employing the unifying power of the concepts and methods of noncommutative geometry to address several problems which affect broad areas of mathematics, ranging from differential topology and geometry to operator algebras and mathematical physics. In collaboration with A. Connes, H. Moscovici will apply their noncommutative local index formalism to the study of diffeomorphism-equivariant characteristic classes, in particular connecting the Gelfand-Fuchs cohomology with the diffeomorphism-equivariant extension of the Atiyah-Singer index theorem. A second theme concerns the application of the straight Chern character construction to the theory of characteristic classes of pseudomanifolds, in order to develop a comprehensive approach encompassing both the classical Chern-Weil theory, in the smooth case, as well as Cheeger's formulae for the combinatorial Pontryagin classes. The third project focuses on the investigation of Connes' notion of noncommutative manifold and should ultimately lead to a complete classification of such objects in terms of a few basic noncommutative operations performed on ordinary manifolds. Noncommutative geometry is a newly emerging field of mathematics, rooted in the time-tested operator formalism of quantum mechanics, which -- in the vision of the renowned French mathematician Alain Connes -- provides a unified geometric framework able to treat the discrete on equal footing with the continuum as well as Einstein's general relativity on a par with the Planck-scale structure of space-time.