Homogeneous G-structures
Monday, 23.5.22, 16:15-17:15, Online / SR 125
G-structures unify several interesting geometries including: almost complex, Riemannian, almost symplectic geometry, etc., the integrable versions of which being complex, flat Riemannian, symplectic geometry, etc. Contact manifolds are odd dimensional analogues of symplectic manifolds but, despite this, there is no natural way to understand them as manifolds with an ordinary integrable G-structure. In this talk, we present a possible solution to this discrepancy. Our proposal is based on a new notion of homogeneous G-structures. Interestingly, besides contact, the latter include other nice (old and new) geometries including: cosymplectic, almost contact, and a curious “homogeneous version” of Riemannian geometry. This is joint work with A. G. Tortorella and O. Yudilevich.
The strong Homotopy Structure of Phase Space Reduction in Deformation Quantization
Monday, 30.5.22, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
A Hamiltonian action on a Poisson manifold induces a Poisson structure on a reduced manifold,\ngiven by the Poisson version of the Marsden-Weinstein reduction or equivalently the BRST-method.\nFor the latter there is a version in deformation quantization for equivariant star products, i.e. invariant\nunder the action and admitting a quantum momentum map which produces a star product on the\nreduced manifold.\nFixing a Lie group action on a manifold, one can define a curved Lie algebra whose Maurer-Cartan\nelements are invariant star products together with quantum momentum maps. Star products on the\nreduced manifold are Maurer-Cartan elements of the usual DGLA of polydifferential operators. Thus,\nreduction is just a map between these two sets of Maurer-Cartan elements. In my talk I want to show\nthat one can construct an \(L_\binfty\)-morphism, which on the level of Maurer Cartan elements provides a\nreduction map.\nThis a joint work with Chiara Esposito and Andreas Kraft (arXiv: 2202.08750).