The PI proposes to continue the study of problems in noncommutative geometry using tools from deformation theory, Lie groupoids, and operator algebras. They include the study of deformation quantizations, quantum groupoids and twisted cohomology over differentiable stacks. The project also involves studying the Chern-Connes character map for twisted K-theory, delocalized twisted equivariant cohomology, and its ring structure. It also aims to study C*-algebras associated to non-abelian gerbes and 2-groupoids.
The idea of noncommutative geometry is to study geometry via algebras of functions on ``noncommutative manifolds''. On such a ``noncommutative manifold'', the relevant objects are no longer points in a space, but rather an associative algebra, which may not be commutative. Many ``noncommutative spaces'' are obtained either as deformations of commutative algebras or as convolution algebras of groupoids. The theory of deformation quantization lies on the boundary between classical and quantum mechanics. The mathematical structures of the two theories are very different, so it is challenging to understand how the transition from classical to quantum takes place. Quantization, roughly speaking, is the study and prediction of quantum phenomena, which is normally described by noncommutative associative algebras, from the geometry of their underlying classical counterparts. The Kontsevich formality theorem in a certain sense confirms that such a prediction is indeed possible, which implies that there is a deep interplay between noncommutative algebras and geometry. Groupoid convolution algebras, substituting for algebras of functions (on badly behaved quotient spaces), play a central role in Connes' noncommutative geometry program. They also play an important role in the study of the stringy space-time, or gerbes. The purpose of the project, which is centered around the application of noncommutative geometry, is to investigate questions in these fields motivated by mathematical physics. More specifically, it is motivated by a combination of ideas from quantum groups, representation theory, differential geometry, noncommutative geometry, and operator algebras. Thus the interdisciplinary nature of the proposed project promotes further interaction between these fields.