Final Activity and Management Report Summary - NONCOMMGEOMETRY (Noncommutative geometry from spacetime curvature: Symplectic geometry of quantised geodesics on Riemann manifolds)
It was shown how quantisation by, first, the local linearisation of canonical quantisation by a Lie algebroid, then integrating the Lie algebroid to a Lie groupoid, and finally constructing the convolution algebra of the Lie groupoid can, in principle, produce a noncommutative geometry. In cases where the Lie algebroid is not integrable, it leads to the construction of groupoids internal to differentiable stacks. For gravitational fields, it sheds new light on the groupoid symmetry of the initial value formulation of general relativity.