Principal bundles in noncommutative differential geometry

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Noncommutative (NC) geometry is a generalisation of classical geometry that provides new mathematical tools for both mathematical problems and physical models by allowing for geometric spaces and spacetimes whose coordinates no longer necessarily commute. For example, NC geometry has been successfully applied both to solve major problems in foliation theory and to obtain a complete mathematical model of the integer quantum Hall effect in condensed matter physics. Typically, the basic strategy has been to compute quantities of interest as topological invariants of the relevant NC space. More recently, intriguing connections to number theory, theoretical physics, and the mathematics of signal processing have brought new significance to the differential geometry of NC spaces in its own right.Recent advances in unbounded KK-theory have provided powerful new tools for studying fibrations in Alain Connes's framework of spectral triples as NC manifolds. Bram Mesland and I have recently used them to investigate classical and -deformed smooth principal bundles with non-Abelian Lie structure group, providing crucial new evidence on how principal Lie actions and their good quotients manifest themselves in Connes's framework. Moreover, Steve Avsec and I have recently combined them with insights from NC harmonic analysis to produce a flexible framework for studying a large class of discrete group C*-algebras as compact quantum Lie groups. I propose to build on these advances to lay groundwork for an unbounded KK-theoretic theory of NC principal bundles with compatible NC ChernWeil theory that is capable of accommodating, for instance, NC quotient manifolds for suitable non-principal discrete group actions on manifolds.The first piece of groundwork, in collaboration with Bram Mesland, will be to develop a general unbounded KK-theoretic theory of NC differentiable principal bundles with Lie or finite quantum structure group. The second, in collaboration with Steve Avsec, will be to apply our earlier work to computing new invariants for certain classes of discrete groups and to generalise this work to certain quantum groups arising from NC probability. The third, in collaboration with Zhizhang Xie, will be to develop an NC generalisation of differential K-theory and compute it for key examples. All three projects will also provide a variety of research opportunities for graduate students. Besides substantially extending the current framework of NC differential geometry, these projects already promise potential applications to geometric group theory and NC harmonic analysis through new invariants for non-property (T) discrete group actions on manifolds and a new perspective on NC harmonic analysis on discrete classical and quantum groups. More generally, they will contribute to the advancement of NC geometry as a tool for mathematical physics, geometric group theory, and harmonic analysis.